diff --git "a/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt" "b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt" @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf,len=1156 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='05540v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='NA] 13 Jan 2023 SOLVING PDES WITH INCOMPLETE INFORMATION PETER BINEV, ANDREA BONITO, ALBERT COHEN, WOLFGANG DAHMEN RONALD DEVORE, AND GUERGANA PETROVA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Introduction The questions we investigate sit in the broad research area of using measurements to enhance the numer- ical recovery of the solution u to a PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The particular setting addressed in this paper is to numerically approximate the solution to an elliptic boundary value problem when there is insufficient information on the boundary value to determine a unique solution to the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In place of complete boundary information, we have a finite number of data observations of the solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This data serves to narrow the set of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We ask what is the optimal accuracy to which we can recover u and what is a near optimal numerical algorithm to approximate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Problems of this particular type arise in several fields of science and engineering (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' [28, 3, 7] for examples in fluid dynamics), where a lack of full information on boundary conditions arises for various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, the correct physics might not be fully understood [22, 24], or the boundary values are not accessible [11], or they must be appropriately modified in numerical schemes [8, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Other examples of application domains for the results of the present paper can be found in the introduction of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A model for PDEs with incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this paper, we consider the model elliptic problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) − ∆u = f in Ω, u = g on Γ := ∂Ω, where Ω ⊂ Rd is a bounded Lipschitz domain with d = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Lax-Milgram theorem [29] implies the existence and uniqueness of a solution u from the Sobolev space H1(Ω) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1), once f and g are prescribed in H−1(Ω) (the dual of H1 0(Ω)) and in H1/2(Γ) (the image of H1(Ω) by the trace operator), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that the trace operator T is defined on a function w ∈ C(¯Ω) as the restriction of w to Γ and this definition is then generalized to functions in Sobolev spaces by a denseness argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, the trace operator is well defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any function v in H1(Ω) we denote by vΓ its trace, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) vΓ := T (v) = v|Γ, v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Lax-Milgram analysis also yields the inequalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) c0∥v∥H1(Ω) ≤ ∥∆v∥H−1(Ω) + ∥vΓ∥H1/2(Γ) ≤ c1∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here the constants c0, c1 depend on Ω and on the particular choice of norms employed on H1(Ω) and H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our interest centers on the question of how well we can numerically recover u in the H1 norm when we do not have sufficient knowledge to guarantee a unique solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There are many possible settings to which our techniques apply, but we shall focus on the following scenario: Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This research was supported by the NSF Grants DMS 2110811 (AB), DMS 2038080 (PB and WD), DMS-2012469 (WD), DMS 21340077 (RD and GP), the MURI ONR Grant N00014-20-1-278 (RD and GP), the ARO Grant W911NF2010318 (PB), and the SFB 1481, funded by the German Research Foundation (WD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1 (i) We have a complete knowledge of f but we do not know g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (ii) The function g belongs to a known compact subset KB of H 1 2 (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, membership in KB describes our knowledge of the boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function u we wish to recover comes from the set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) K := {u : u solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) for some g ∈ KB}, which is easily seen from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) to be a compact subset of H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iii) We have access to finitely many data observations of the unknown solution u, in terms of a vector (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) λ(u) := (λ1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm(u)) ∈ Rm, where the λj are fixed and known linear functionals defined on the functions from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Natural candidates for the compact set KB are balls of Sobolev spaces that are compactly embedded in H 1 2 (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We thus restrict our attention for the remainder of this paper to the case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) KB := U(Hs(Γ)), for some s > 1 2,where the precise definition of Hs(Γ) and its norm ∥ · ∥Hs(Γ) is described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that U(Y ) denotes the unit ball of a Banach space Y with respect to the norm ∥ · ∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The optimal recovery benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let wj := λj(u), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) w := (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , wm) = λ(u) ∈ Rm, be the vector of data observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, the totality of information we have about u is that it lies in the compact set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) Kw := {u ∈ K : λ(u) = w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our problem is to numerically find a function ˆu ∈ H1(Ω) which approximates simultaneously all the u ∈ Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is a special case of the problem of optimal recovery from data (see [15, 27, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The optimal recovery, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' the best choice for ˆu, has the following well known theoretical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let B(Kw) be a smallest ball in H1(Ω) which contains Kw and let R(Kw) := R(Kw)H1(Ω) be its radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, R(Kw) is the optimal recovery error, that is, the smallest error we can have for recovering u in the norm of H1(Ω), and the center of B(Kw) is an optimal recovery of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in understanding how small R(Kw) is and what are the numerical algorithms which are near optimal in recovering u from the given data w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We say that an algorithm w �→ ˆu = ˆu(w) delivers near optimal recovery with constant C if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥u − ˆu(w)∥H1(Ω) ≤ CR(Kw), w ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, we want C to be a reasonable constant independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our results actually deliver a recovery estimate of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ∥u − ˆu(w)∥H1(Ω) ≤ R(Kw) + ε, w ∈ Rm, where ε > 0 can made arbitrarily small at the price of higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this sense, the recovery is near optimal with constant C > 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) that can be made arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A connection with the recovery of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a natural restatement of our recovery problem in terms of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let f be the right side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1), where f is a known fixed element of H−1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let u0 be the function in H1(Ω) which is the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) with g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, we can write any function u ∈ K as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) u = u0 + uH, where uH is a harmonic function in H1(Ω) which has boundary value g = T (uH) with g ∈ KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall our assumption that KB is the unit ball of Hs(Γ) with s > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2 Let Hs(Ω) denote the set of harmonic functions v defined on Ω for which vΓ ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer the reader to [2], where a detailed study of spaces like Hs(Ω) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define the norm on Hs(Ω) to be the one induced by the norm on Hs(Γ), namely, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) ∥v∥Hs(Ω) := ∥vΓ∥Hs(Γ), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There exist several equivalent definitions of norms on Hs(Γ), as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the moment, observe that from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) it follows the existence of a constant Cs such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥v∥H1(Ω) ≤ Cs∥v∥Hs(Ω), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, the space Hs(Ω) is a Hilbert space that is compactly embedded in H1(Ω), as a consequence of the compact embedding of Hs(Γ) in H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We denote by KH the unit ball of Hs(Ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) KH := U(Hs(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since the function u0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) is fixed, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) R(Kw) = R(KH w′)H1(Ω), w′ := λ(uH) = w − λ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There are two conclusions that can be garnered from this reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first is that the optimal error in recovering u ∈ Kw is the same as that in recovering the harmonic function uH ∈ KH w′ in the H1(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The harmonic recovery problem does not involve f except in determining w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second point is that one possible numerical algorithm for our original problem is to first construct a sufficiently accurate approximation ˆu0 to u0 and then to numerically implement an optimal recovery of a harmonic function in KH from data observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical approach requires the computation of w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In theory, u0 is known to us since we have a complete knowledge of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, u0 must be computed and any approximation ˆu0 will induce an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Although this error can be made arbitrarily small, it means that we only know w′ up to a certain numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One can thus view the harmonic reformulation as an optimal recovery problem with perturbed observations of w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical algorithm presented here follows this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Its central constituent, namely the recovery of harmonic functions from a finite number of noisy observations, can be readily employed as well in a number of different application scenarios described e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Objectives and outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our main goal is to create numerical algorithms which are guaranteed to produce a function ˆu which is near optimal and to discuss their practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We begin in §2 with some remarks on the definition of the space Hs(Γ) and its norm, which are of importance both in the accuracy analysis and the practical implementation of recovery algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The general approach for optimal recovery that was introduced in [15, 14] is recalled in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We describe a solution algorithm which takes into consideration the effect of numerical perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first consider the case when the linear functionals λj are defined on all of H1(Ω) and then adapt this algorithm to the case when the linear functionals are point evaluations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) λj(u) := u(xj), xj ∈ Ω, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluations are not defined on all of H1(Ω) when d > 1, however, they are defined on K when the smoothness order s is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The critical ingredient in our proposed algorithm is the numerical computation of the Riesz representers φj of the restrictions of λj to the Hilbert space Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Each of these Riesz representers is characterized as a solution to an elliptic problem and can be computed offline since it does not involve the measurement vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our suggested numerical method for approximating φj is based on finite element discretizations and is discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We establish quantitative error bounds for the numerical approximation in terms of the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Numerical illustrations of the optimal recovery algorithm are given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the optimal recovery error over the class K strongly depends on the choice of the linear functionals λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, in the case of point evaluation, this error can be very large if the data sites {xj}m j=1 are poorly positioned, or small if they are optimally positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This points to the importance of the Gelfand widths and sampling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' They describe the optimal recovery error over K with optimal choice of functionals in the general case and the point evaluation case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical behaviour of these quantities in our specific setting is discussed in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The spaces Hs(Γ) and Hs(Ω) In this section, we discuss the definition and basic properties of the spaces Hs(Γ) and Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer to [1] for a general treatment of Sobolev spaces on domains D ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that for fractional orders r > 0, the norm of Hr(D) is defined as ∥v∥2 Hr(D) := ∥v∥2 Hk(D) + � |α|=k � D×D |∂αv(x) − ∂αv(y)|2 |x − y|d+2(r−k) dxdy, where k is the integer such that k < r < k+1, and ∥v∥2 Hk(D) := � |α|≤k ∥∂αv∥2 L2(D) is the standard Hk-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Equivalent definitions of Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let Ω be any bounded Lipschitz domain in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We recall the trace operator T introduced in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One first possible definition of the space Hs(Γ), for any s ≥ 1 2, is as the restriction of Hs+ 1 2 (Ω) to Γ, that is, Hs(Γ) = T (Hs+ 1 2 (Ω)), with norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) ∥g∥Hs(Γ) := min � ∥v∥Hs+ 1 2 (Ω) : vΓ = g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The resulting norm is referred to as the trace norm definition for Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a second, more intrinsic way to define Hs(Γ), by properly adapting the notion of Sobolev smoothness to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This can be done by locally mapping the boundary onto domains of Rd−1 and requiring that the pullback of g by such transformation have Hs smoothness on such domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer the reader to [10] and [17] for the complete intrinsic definition, where it is proved to be equivalent to the trace definition for a range of s that depends on the smoothness of the boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For small values of s, Sobolev norms for Hs(Γ) may also be equivalently defined without the help of local parameterizations, as contour integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, if 0 < s < 1 and Ω is a Lipschitz domain, we define ∥g∥2 Hs(Γ) := ∥g∥2 L2(Γ) + � Γ×Γ |g(x) − g(y)|2 |x − y|d−1+2s dxdy, and if s = 1 and Ω is a polygonal domain, we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) ∥g∥2 H1(Γ) := ∥g∥2 L2(Γ) + ∥∇Γg∥2 L2(Γ), where ∇Γ is the tangential gradient, and likewise ∥g∥2 Hs(Γ) := ∥g∥2 H1(Γ) + � Γ×Γ |∇Γg(x) − ∇Γg(y)|2 |x − y|d−1+2(s−1) dxdy, for 1 < s < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the numerical illustration given in §5, we will specifically take the value s = 1 and a square domain, using the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' When Ω has smooth boundary, it is known that the trace definition and intrinsic definition of the Hs(Γ) norms are equivalent for all s ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, when Ω does not have a smooth boundary, it is easily seen that the two definition are not equivalent unless restrictions are made on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Consider for example the case of polygonal domains of R2: it is easily seen that the trace vΓ of a smooth function v ∈ C∞(Ω) has a tangential gradient ∇ΓvΓ that generally has jump discontinuities at the corner points and thus does not belong to H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, the equivalence between the trace and intrinsic norms only holds for s < 3 2 and in such case we limit the value of s to this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The same restriction s < 3/2 applies to a polyhedral domain in the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The regularity of functions in Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next give some remarks on the Sobolev smoothness of functions from the space Hs(Ω) when s > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly such harmonic functions are infinitely smooth inside Ω and also belong to H1(Ω), but one would like to know for which value of r they belong to Hr(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To answer this question, we consider v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By the definition of Hs(Ω), v is harmonic in Ω and vΓ ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4 Having assumed that s in the admissible range where all above definitions of the Hs(Γ) norms are equivalent, and using the first one, we know that there exists a function ˜v ∈ Hs+ 1 2 (Ω) such that ˜vΓ = vΓ ∥˜v∥Hs+ 1 2 (Ω) = ∥vΓ∥Hs(Γ) = ∥v∥Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define v := v − ˜v so that v = ˜v +v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in the regularity of v since it will give the regularity of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Notice that vΓ = 0 and −∆v = f := ∆˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function f belongs to the Sobolev space Hs− 3 2 (Ω) and we are left with the classical question of the regularizing effect in Sobolev scales when solving the Laplace equation with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Obviously, when Ω is smooth, we find that v ∈ Hs+ 1 2 (Ω) and so we have obtained the continuous embedding Hs(Ω) ⊂ Hr(Ω), r = s + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For less smooth domains, the smoothing effect is limited (in particular by the presence of singularities on the boundary of Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', v is only guaranteed to be in Hr(Ω) where r may be less than s + 1/2, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely Hs(Ω) ⊂ Hr(Ω), where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) r := min � s + 1 2, r∗� , Here, r∗ = r∗(Ω) is the limiting bound for the smoothing effect: (i) For smooth domains r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (ii) For convex domains r∗ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iii) For non-convex polygonal domains in R2, or a polyhedron in R3, one has 3/2 < r∗ < 2 where the value of r∗ depends on the reentrant angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iv) In particular for polygons, we can take r∗ = 1 + π ω − ε, for any ε > 0 where ω is the largest inner angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that r∗ could be strictly smaller than s + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In summary, for an admissible range of r > 1 that depends on s and Ω one has the continuous embedding Hs(Ω) ⊂ Hr(Ω), and so there exists a constant C1 that depends on (r, s) and Ω, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥v∥Hr(Ω) ≤ C1∥v∥Hs(Ω) = C1∥vΓ∥Hs(Γ), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A near optimal recovery algorithm In this section, we present a numerical algorithm for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) when the information about the bound- ary value g is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first work under the assumption that the λj’s are continuous over H1(Ω), and assumed to be linearly independent (linear independence can be guaranteed by throwing away dependent functionals when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We prove that the proposed numerical recovery algorithm is near optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then adapt our approach to the case where the λj’s are point evaluations, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16), and therefore not continuous over H1(Ω) when d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Minimum norm data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As noted in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3, the problem of recovering u ∈ Kw is directly related to the problem of recovering the harmonic component uH ∈ KH from the given data observations w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that KH is the unit ball of the Hilbert space Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a general approach for optimal recovery from data observations in this Hilbert space setting, as discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first describe the general principles of this technique and then apply them to our specific setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let H be any Hilbert space and suppose that λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm ∈ H∗ are linearly independent functionals from H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let X be a Banach space such that H is continuously embedded in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in optimal recovery of a function v in the norm ∥ · ∥X, knowing that v ∈ K := U(H), the unit ball of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If w ∈ Rm is the vector of observations, we define the minimal norm interpolant as v∗(w) = argmin{∥v∥H : v ∈ H and λ(v) = w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5 It is easily checked that when Kw is non-empty, the function v∗(w) coincides with the Chebyshev center of Kw in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To see this, first note that any v ∈ Kw may be written as v = v∗(w) + η where η belongs to the null space N of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Because v∗(w) has minimal norm, v − v∗(w) = η is orthogonal to v∗(w) and hence from the Pythagorean theorem ∥v − v∗∥2 H = ∥v∥2 H − ∥v∗(w)∥2 H ≤ 1 − ∥v∗(w)∥2 H =: r2, because ∥v∥H ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Notice that v∗(w) − η is also in Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that Kw is precisely the ball in the affine space v∗(w) + N centered at v∗(w) and of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, Kw is centrally symmetric around v∗(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, v∗(w) is the Chebyshev center for Kw for any norm, in particular for the ∥ · ∥X norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, ∥v − v∗(w)∥X ≤ R(Kw)X, v ∈ Kw, that is, the minimal norm interpolant gives optimal recovery with constant C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Standard Hilbert space analysis shows that the mapping w �→ v∗(w) is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More importantly, it has a natural expression that is useful for numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Namely, from the Riesz representation theorem each λj can be described as λj(v) = ⟨v, φj⟩H, v ∈ H, where φj ∈ H is called the Riesz representer of λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The minimal norm interpolant has the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) v∗ = m � j=1 a∗ jφj, where a∗ = (a∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , a∗ m) solves the system of equations Ga∗ = w, G := (⟨φi, φj⟩H)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, with G being the Gramian matrix associated to φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , φm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case where H is a more general Banach space, we are still ensured that the minimal norm interpolation is a near-optimal recovery with constant C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, its dependence on the data w is no longer linear and the above observation regarding its computation does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us now apply this general principle to our particular setting in which the Hilbert space H is Hs(Ω) and X = H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let φj ∈ Hs(Ω) be the Riesz representer of the functional λj when viewed as a functional on Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In other words λj(v) = ⟨v, φj⟩Hs(Ω), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We assume that the λj are linearly independent on Hs(Ω) and thus the Gramian matrix G = � gi,j � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, gi,j := ⟨φi, φj⟩Hs = λj(φi), is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Now, let u = u0 + uH, with uH ∈ KH = U(Hs(Ω)) be the function in K that gave rise to our data observation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' So, we have w′ = w − λ(u0) = λ(uH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If a∗ is the vector in Rm which satisfies Ga∗ = w′, then u∗ H := �m j=1 a∗ jφj is the function of minimum Hs(Ω) norm which satisfies the data w′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', λ(u∗ H) = w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen that ∥uH − u∗ H∥H1(Ω) ≤ R(KH w′)H1(Ω), namely, u∗ H is the optimal recovery of the functions in KH w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the recovery error is measured in H1 not in Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15), the function u∗ := u∗ H + u0 is the optimal recovery for functions in Kw: ∥u − u∗∥H1(Ω) ≤ R(Kw)H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The idea behind our proposed numerical method is to numerically construct a function ˆu ∈ H1 that approximates u∗ well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If, for example, we have for ε > 0 the bound ∥u∗ − ˆu∥H1(Ω) ≤ ε, then for any u ∈ K, we have by the triangle inequality ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6 Given any C > 1, by taking ε small enough, we have that ˆu is a near best recovery of the functions in Kw with constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical recovery algorithm for H1-continuous functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Motivated by the above analysis, we propose the following numerical algorithm for solving our recovery problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The algorithm involves approximations of the function u0 and the Riesz representers φj, typically computed by finite element discretizations, and the application of the linear functionals λj to these approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to avoid extra technicalities, we make here the assumption that the applications of the functionals to a known finite element function can be exactly computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first work under the additional assumption that the linear functionals λj are not only defined on K but that they are continuous over H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define Λ as the maximum of the norms of the λj on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) |λj(v)| ≤ Λ∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In what follows, throughout this paper, we use the following weighted ℓ2 norm on Rm, ∥z∥ := \uf8eb \uf8ed 1 m m � j=1 |zj|2 \uf8f6 \uf8f8 1/2 = m−1/2∥z∥ℓ2, z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , zm) ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, we have ∥λ(v)∥ ≤ Λ∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given a user prescribed accuracy ε > 0, our algorithm does the following four steps involving intermediate tolerances (ε1, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 1: We numerically find an approximation ˆu0 to u0 which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) ∥u0 − ˆu0∥H1(Ω) ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To find such a ˆu0, we use standard or adaptive FEM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given that ˆu0 has been constructed, we define ˆw := w − λ(ˆu0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, for w′ := w − λ(u0) we have, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥w′ − ˆw∥ ≤ Λε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, since |λj(v)| ≤ Λ∥v∥H1(Ω) ≤ Λs∥v∥Hs(Ω) ≤ Λs, where Λs := CsΛ, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14), we derive that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) ∥w′∥ ≤ Λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus by triangle inequality, we also find that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ∥ ˆw∥ ≤ Λs + Λε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 2: For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, we numerically compute an approximation ˆφj ∈ H1(Ω) to φj which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) ∥φj − ˆφj∥H1(Ω) ≤ ε2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical computation is crucial and is performed during the offline phase of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We detail it in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the ˆφj’s are not assumed to be in Hs(Ω), and in particular not assumed to be harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 3: We define and compute the matrix ˆG = (ˆgi,j)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, ˆgi,j := λj(ˆφi), and thus |ˆgi,j − gi,j| ≤ Λε2 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 7 It follows that for the matrix R := G − ˆG we have ∥R∥1 ≤ mΛε2, where we use the shorthand notation ∥ · ∥1 := ∥ · ∥ℓ1→ℓ1 for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since G is invertible, we are ensured that ˆG is also invertible for ε2 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define M := ∥G−1∥1, ˆ M := ∥ ˆG−1∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While these two norms are finite, their size will depend on the nature and the positioning of the linear functionals λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, as it will be seen in the section on numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' These two numbers are close to one another when ε2 is small since ˆ M converges towards the unknown quantity M as ε2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, we have |M − ˆ M| = |∥G−1∥1 − ∥ ˆG−1∥1| ≤ ∥G−1 − ˆG−1∥1 = ∥ ˆG−1RG−1∥1 ≤ M ˆ MmΛε2, from which we obtain that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) M ≤ ˆ M 1 − m ˆ MΛε2 and ˆ M ≤ M 1 − mMΛε2 , provided that mMΛε2 < 1 and m ˆ MΛε2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We also have the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥ ˆG−1 − G−1∥1 ≤ M 2 1 − mMΛε2 mΛε2 =: δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is important to observe that δ can be made arbitrarily small by diminishing ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 4: We numerically solve the m×m algebraic system ˆGˆa = ˆw, thereby finding a vector ˆa = (ˆa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , ˆam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then define ˆuH := �m j=1 ˆaj ˆφj and our recovery of u is ˆu := ˆu0 + ˆuH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This step can be implemented by standard linear algebra solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One major advantage of the above algorithm is that Steps 1-2-3 can be performed offline since they do not involve the data w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' That is, we can compute ˆu0, the approximate Riesz representers ˆφj and the approximate Gramian ˆG and its inverse without knowing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this way, the computation of ˆu from given data w can be done fast online by Step 4 which only involves solving an m × m linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This may be a significant advantage, for example, when having to process a large number of measurements for the same set of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A near optimal recovery bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The following theorem shows that a near optimal recovery of u can be reached provided that the tolerances in the above described algorithm are chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any prescribed ε > 0, if the tolerances (ε1, ε2), are small enough such that mMΛε2 < 1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, where C0 := maxj=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m ∥φj∥H1(Ω) and δ := M2 1−mMΛε2 mΛε2, then the function ˆu generated by the above algorithm satisfies ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε, for every u ∈ Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, for any C > 1 it is a near optimal recovery of u with constant C provided ε is taken sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let u = u0 + v be our target function in Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define w′ = w − λ(u0) and v∗ := v∗(w′) which is the Chebyshev center of KH w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We recall the algebraic system Ga∗ = w′ associated to the characterization of v∗ (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) ∥u∗ H− ˆuH∥H1(Ω) ≤ ��� m � j=1 a∗ j(φj − ˆφj) ��� H1(Ω) + ��� m � j=1 (a∗ j −ˆaj)ˆφj ��� H1(Ω) ≤ ∥a∗∥ℓ1ε2+∥a∗−ˆa∥ℓ1(C0 +ε2), where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) and the fact that ∥ˆφj∥H1(Ω) ≤ ∥φj∥H1(Ω) + ∥φj − ˆφj∥H1(Ω) ≤ C0 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 8 Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) ∥a∗∥ℓ1 = ∥G−1w′∥ℓ1 ≤ M∥w′∥ℓ1 ≤ Mm∥w′∥ ≤ mMΛs, where we have used that ∥w′∥ℓ1 ≤ m∥w′∥ and inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥u∗ H − ˆuH∥H1 ≤ mMΛsε2 + ∥a∗ − ˆa∥ℓ1(C0 + ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the estimation of ∥a∗ − ˆa∥ℓ1, we introduce the intermediate vector ˜a ∈ Rm, which is the solution to the system G˜a = ˆw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly, ∥˜a − a∗∥ℓ1 = ∥G−1( ˆw − w′)∥ℓ1 ≤ M∥ ˆw − w′∥ℓ1 ≤ Mm∥ ˆw − w′∥ ≤ mMΛε1, where we invoked (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6), we have ∥˜a − ˆa∥ℓ1 = ∥(G−1 − ˆG−1) ˆw∥ℓ1 ≤ δ∥ ˆw∥ℓ1 ≤ mδ∥ ˆw∥ ≤ m(Λs + Λε1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these two estimates, we find that ∥a∗ − ˆa∥ℓ1 ≤ mMΛε1 + m(Λs + Λε1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now insert this bound into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) to obtain ∥u∗ H − ˆuH∥H1(Ω) ≤ mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, for u∗ := u0 + u∗ H and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), we have ∥u∗ − ˆu∥H1(Ω) ≤ ∥u0 − ˆu0∥H1(Ω) + ∥u∗ H − ˆuH∥H1(Ω) ≤ ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) Since u = u0 + uH, we have ∥u − u∗∥H1(Ω) = ∥uH − u∗ H∥H1(Ω) ≤ R(KH w′)H1(Ω) = R(Kw)H1(Ω), and the statement of the theorem follows from this inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that in numerical computations the quantity ˆ M is available while M is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus in practice, in order to achieve the prescribed accuracy ε, we can first impose that ε2 < (2m ˆ MΛ)−1 and derive the inequalities, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8), M ≤ ˆ M 1 − m ˆ MΛε2 ≤ 2 ˆ M, ∥G−1 − ˆG−1∥1 ≤ ˆ M 2 1 − m ˆ MΛε2 mΛε2 ≤ 2 ˆ M 2mΛε2 =: ˆδ, where the last inequality is proven in a similar fashion to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If we then follow the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2, the requirement in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) can be substituted by ε1 + 2m ˆ MΛsε2 + (C0 + ε2)(2m ˆ MΛε1 + m(Λs + Λε1)ˆδ) ≤ ε, and thus all participating quantities are computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 can easily be extended to the case of noisy data, that is, to the case when the observations ˜w = w + η, where η is a noise vector of norm ∥η∥ ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, the application of the algorithm to this noisy data leads to finding in Step 1 the vector ˆw := w + η − λ(ˆu0) that satisfies ∥w′ − ˆw∥ ≤ Λε1 + κ, and ∥ ˆw∥ ≤ Λs + ε1Λ + κ, where w′ = w − λ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Inspection of the above proof shows that under the same assumption as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2, one has the recovery bound ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε + Cκ, for every u ∈ Kw, where C := (M + δ)m(C0 + ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 9 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, we did not introduce in the above analysis the possible errors in the application of the λi to the approximations ˆu0 and ˆφj, and in the numerical solution to the system ˆGˆa = ˆw, which would simply result in similar conditions involving the extra tolerance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now want to extend the numerical algorithm and its analysis to the case when the data functionals λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, are point evaluations λj(h) := h(xj), xj ∈ Ω, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course these functionals are not defined for general functions h from H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, we can formulate the recovery problem whenever the functionals λj are well defined on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now discuss settings when this is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that any u ∈ K can be written as u = u0 + uH, where u0 is the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) with right side f and g = 0 and uH ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluation is well defined for the harmonic functions uH ∈ Hs(Ω), provided the points are in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition, they are well defined for points on the boundary Γ if the space Hs(Ω) continuously embeds into C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For d = 2, this is the case when s > 1/2 and when d = 3, this is the case when s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Concerning u0, we will need some additional assumption to guarantee that point evaluation of u0 makes sense at the data sites xj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, it is enough to assume that u0 is globally continuous or at least in a neighborhood of each of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This can be guaranteed by assuming an appropriate regularity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we assume that one of these settings holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then write w′ j := uH(xj) = wj − u0(xj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and follow the algorithm of the previous section with the following simple modifications: Modified Step 1: We numerically find an approximation ˆu0 to u0, which in addition to ∥u0 − ˆu0∥H1(Ω) ≤ ε1, satisfies the requirement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m |u0(xi) − ˆu0(xi)| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To find such a ˆu0 we use standard or adaptive FEM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given that ˆu0 has been constructed, we define ˆwj := wj − ˆu0(xj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and thus, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15), we have ∥w′ − ˆw∥ ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Modified Step 2: For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, we numerically compute an approximation ˆφj to φj, which in addition to ∥φj − ˆφj∥H1(Ω) ≤ ε2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m |φj(xi) − ˆφj(xi)| ≤ ε2, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) ensures that in Step 3 we can choose the entries ˆgi,j of the matrix ˆG as ˆgi,j = ˆφj(xi), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Steps 3 and 4 of our algorithm remain the same as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' With the above modifications, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 holds with the exact same statement in this point evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The proof is the same as that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finite element approximations of the Riesz representers The computation of an approximation ˆu0 to u0, required in Step 1 of the algorithm, can be carried out by standard finite element Galerkin schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Depending on our knowledge on f one can resort to known a priori estimates for ε1, or may employ standard a posteriori estimates to ensure that the underlying discretization provides a desired target accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, in the remainder of this section, we focus on a numerical implementation of Step 2 of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our proposed numerical algorithm for Step 2 is to use finite element methods to generate the approx- imations ˆφj of the Riesz representers φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that each of the functions φj is harmonic on Ω but we do not require that the sought after numerical approximation ˆφj is itself harmonic but only that it provides an accurate H1(Ω) approximation to φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This allows us to use finite element approximations which are themselves not harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, the ˆφj will necessarily have to be close to being harmonic since they approximate a harmonic function in the H1(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our numerical approach to constructing a ˆφj, discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1, is to use discretely harmonic finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, ˆφj is a discrete harmonic extension of a finite element approximation to the trace ψj = T (φj) computed by solving a Galerkin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to reduce computational cost (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), we incorporate discrete harmonicity as constraints and introduce in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 an equivalent saddle point formulation that has the same solution ˆφj, and which is the one that we practically employ in the numerical experiments given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We give in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 an a priori analysis with error bounds for ∥φj − ˆφj∥H1 in terms of the finite element mesh size, in the case where the measurement functionals are continuous on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' These error bounds can in turn be used to ensure the prescribed accuracy ε2 in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We finally discuss in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5 the extensions to the point value case where pointwise error bounds on |ˆφj(xi) − ˆφj(xi)| are also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to simplify notation, we describe these procedures for finding an approximation ˆφ to the Riesz representer φ ∈ Hs = Hs(Ω) of a given linear functional ν on Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical procedure is then applied with ν = λj, to find the numerical approximations ˆφj to the Riesz representer φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, throughout this section, we work under the assumption that Ω is a polygonal domain of R2 or polyhedral domain of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This allows us to define finite element spaces based on triangular or simplicial partitions of Ω that in turn induce similar partitions on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We assume that 1 2 < s < 3 2, which is the relevant range for such domains, as explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our analysis can be extended to more general domains with smooth or piecewise smooth boundaries, for example by using isoparametric elements near the boundary, however at the price of considerably higher technicalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A Galerkin formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let s > 1/2 be fixed and assume that ν is any linear form continuous on Hs(Ω) with norm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) Cs := max{ν(v) : ∥v∥Hs(Ω) = 1} In view of the the definition of the Hs norm, the representer φ ∈ Hs(Ω) of ν for the corresponding inner product can be defined as φ = Eψ, where E is the harmonic extension operator of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) below and where ψ ∈ Hs(Γ) is the solution to the following variational problem: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) ⟨ψ, η⟩Hs(Γ) = µ(η) := ν(Eη), η ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that this problem admits a unique solution and we have ∥ψ∥Hs(Γ) = ∥φ∥Hs(Ω) = Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) Eg := argmin{∥∇v∥L2(Ω) : vΓ = g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function Eg is characterized by T (Eg) = g and � Ω ∇Eg · ∇v = 0, v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 11 From the left inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥Eg∥H1(Ω) ≤ CE∥g∥H1/2(Γ), g ∈ H1/2(Γ), where CE can be taken to be the inverse of the constant c0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, one approach to discretizing this problem is the following: consider finite element spaces Vh associated to a family of meshes {Th}h>0 of Ω, where as usual h denotes the maximum meshsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define Th to be the space obtained by restriction of Vh on the boundary Γ, that is, Th = T (Vh) Since we have assumed that Ω is a polygonal or polyhedral domain, the space Th is a standard finite element space for the boundary mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Having also assumed that s < 3/2, when using standard H1 conforming finite elements such as Pk-Lagrange finite elements, we are ensured that Th ⊂ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We denote by Wh := {vh ∈ Vh : T (vh) = 0}, the finite element space with homogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define the discrete harmonic extension operator Eh associated to Vh as follows : for gh ∈ Th, Ehgh := argmin{∥∇vh∥L2(Ω) : vh ∈ Vh, T (vh) = gh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that Ehgh is not harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to E, the function Ehgh is characterized by T (Ehgh) = gh and � Ω ∇Ehgh · ∇vh = 0, vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, we define the approximation φh ∈ Vh to φ as φh = Ehψh, where ψh ∈ Th is the solution to the following variational problem: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) ⟨ψh, gh⟩Hs(Γ) = µh(gh) := ν(Ehgh), gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here we are assuming that, in addition to be defined on Hs(Ω), the functional ν is also well defined on the space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We shall further consider separately two instances where this is the case : (i) ν is a continuous functional on H1(Ω) and (ii) ν is a point evaluation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) is not the straightforward Galerkin approximation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), since µh differs from µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This complicates somewhat the further conducted convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical method we employ for computing φh is to numerically solve an equivalent saddle point problem described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We apply the strategy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) to ν := λj for each j and thereby obtain the corresponding approximations ˆφj := φh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since Step 2 requires that we guarantee the error ∥φj − ˆφj∥H1 ≤ ε2, our main goal in this section is to establish a quantitative convergence bound for ∥φ − φh∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We also need to establish a pointwise convergence bound for |φ(x) − φh(x)| when considering the modified version of Step 2 in the case that the measurements are point values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to E, it will be important in our analysis to control the stability of Eh in the sense of a bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ∥Ehgh∥H1(Ω) ≤ DE∥gh∥H1/2(Γ), gh ∈ Th, with a constant DE that is independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, such a uniform bound is not readily inherited from the stability of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As observed in [6], its validity is known to depend on the existence of uniformly H1-stable linear projections onto Vh preserving the homogeneous boundary condition, that is, projectors Ph onto Vh that satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) Ph(H1 0(Ω)) = Wh and ∥Phv∥H1(Ω) ≤ B∥v∥H1(Ω), v ∈ H1(Ω), for some B independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One straightforward consequence of this is that if v ∈ H1(Ω) with v|Γ ∈ Th then Ph(v)|Γ = v|Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next show that the existence of such projectors is sufficient to guarantee the stability of Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this, suppose (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) holds and gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then PhEgh ∈ Vh and the trace of PhEgh is equal to gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥Ehgh − PhEgh∥H1(Ω) ≤ CP ∥∇Ehgh − ∇PhEgh∥L2(Ω) ≤ CP ∥∇Ehgh∥L2(Ω) + CP ∥∇PhEgh∥L2(Ω), ≤ 2CP ∥PhEgh∥H1(Ω), 12 where CP is the Poincar´e constant for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, the last inequality follows from the minimizing property of Ehgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, by triangle inequality, one has ∥Ehgh∥H1(Ω) ≤ (1 + 2CP )∥PhEgh∥H1(Ω) ≤ (1 + 2CP )B∥Egh∥H1(Ω) ≤ (1 + 2CP )BCE∥gh∥H1/2(Γ), which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) with DE = (1 + 2CP )BCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The requirement of uniformly stable projectors Ph with the property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) is satisfied by projectors of Scott-Zhang type [26] when the family of meshes {Th}h>0 is shape regular, that is, when all elements T have a uniformly bounded ratio between their diameters h(T ) and the diameter ρ(T ) of their inner circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In other words, the shape parameter (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) σ = σ({Th}h>0) := sup h>0 max T ∈Th h(T ) ρ(T ), is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In all that follows in the present paper, we work under such an assumption on the meshes Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) holds when Vh is subordinate to such partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A saddle point formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Before attacking the convergence analysis, we need to stress an impor- tant computational variant of the above described Galerkin method, that leads to the same solution φh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is based on imposing harmonicity via a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this purpose, we introduce the Hilbert space Xs(Ω) that consists of all v ∈ H1(Ω) such that vΓ ∈ Hs(Γ), and equip it with the norm ∥v∥Xs(Ω) := � ∥vΓ∥2 Hs(Γ) + ∥∇v∥2 L2(Ω) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, the Riesz representer φ is equivalently determined as the solution of the saddle point problem: find (φ, π) ∈ Xs(Ω) × H1 0(Ω) such that a(φ, v) + b(v, π) = ν(v), v ∈ Xs(Ω) b(φ, z) = 0, z ∈ H1 0(Ω), where the bilinear forms are given by a(φ, v) := ⟨φΓ, vΓ⟩Hs(Γ) and b(v, π) := ⟨∇v, ∇π⟩L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly the second equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) means that φ is harmonic and testing the first equation with a v ∈ Hs(Ω) shows that φ is the Riesz representer of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This saddle point formulation is well-posed: the bilinear forms a and b obviously satisfies the continuity properties a(φ, v) ≤ ∥φΓ∥Hs(Γ)∥vΓ∥Hs(Γ) ≤ ∥φ∥Xs(Ω)∥v∥Xs(Ω), φ, v ∈ Xs(Ω), and for the standard norm ∥v∥H1 0 (Ω) = ∥∇v∥L2(Ω), b(v, π) ≤ ∥∇v∥L2(Ω)∥∇π∥L2(Ω) ≤ ∥v∥Xs(Ω)∥π∥H1 0(Ω), v ∈ Xs(Ω), π ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition, for all v ∈ Hs(Ω), one has ∥v∥2 Xs(Ω) ≤ ∥vΓ∥2 Hs(Γ) + ∥v∥2 H1(Ω) ≤ ∥vΓ∥2 Hs(Γ) + C2 E∥v∥2 H1/2(Γ) ≤ (1 + C2 E)a(v, v), which shows that a is coercive on the null space of b in Xs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finally, the bilinear form b satisfies the inf-sup condition inf π∈H1 0 (Ω) sup v∈Xs(Ω) b(v, π) ∥v∥Xs(Ω)∥π∥H1 0(Ω) ≥ inf π∈H1 0 (Ω) b(π, π) ∥π∥Xs(Ω)∥π∥H1 0(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore the standard LBB theory ensures existence and uniqueness of the solution pair (φ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now discretize the saddle point problem by searching for (φh, πh) ∈ Vh × Wh such that a(φh, vh) + b(vh, πh) = ν(vh), vh ∈ Vh b(φh, zh) = 0, zh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The equivalence with the previous derivation of φh by the Galerkin approach is easily checked: the second equation tells us that the solution φh is discretely harmonic, and therefore equal to Ehψh for some ψh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then taking vh of the form Ehgh for gh ∈ Th gives us exactly the Galerkin formulation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 13 This discrete saddle point problem is uniformly well-posed when we equip the space Wh with the H1 0 norm, and the space Vh with the Xs norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The continuity of a and b, and the inf-sup condition for b follow by the exact same arguments applied to the finite element spaces, with the same constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, we need to check the uniform ellipticity of a in the space VH h ⊂ Vh of discretely harmonic functions, which can be defined as VH h := {vh ∈ Vh : b(vh, zh) = 0, zh ∈ Wh}, or equivalently as the image of Th by the operator Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For all vh ∈ Vh,H and gh = T (uh), we write ∥vh∥2 Xs(Ω) ≤ ∥gh∥2 Hs(Γ) + ∥vh∥2 H1(Ω) ≤ ∥gh∥2 Hs(Γ) + D2 E∥gh∥2 H1/2(Γ) ≤ (1 + D2 E)a(vh, vh), where we have used the discrete stability of Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In practice, we use this discrete saddle point formulation for the computation of φh rather than the equivalent Galerkin formulation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let Nh := dim Vh, Mh := dim Wh, and Ph := dim Th = Nh − Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Computing the right hand side load vector in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) requires computing discretely harmonic extensions of Ph basis functions, which means solving Ph linear systems of dimension Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition one has to solve the sparse linear system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) of size Ph followed by another system of size Mh to compute φh = Ehψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Using optimal iterative solvers of linear complexity the minimum amount of work needed to compute one representer scales then like PhMh ∼ N 1+ d−1 d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' while solving the saddle point problem requires the order of Nh + Mh ∼ Nh operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand the characterization of φh through (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) appears to be more convenient when deriving error bounds for ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is the objective of the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Preparatory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the derivation of error bounds for ∥φ − φh∥H1(Ω), we will need several ingre- dients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first is the following lemma that quantifies the perturbation induced by using Eh in place of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any gh ∈ Th, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥(E − Eh)gh∥H1(Ω) ≤ C2hr−1∥gh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' where C2 depends on r and s, the shape-parameter σ, and on the geometry of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From the properties of E and Eh, one has ⟨∇(E − Eh)gh, ∇vh⟩ = 0, vh ∈ Wh This orthogonality property shows that ∥∇(Egh − Ehgh)∥L2(Ω) ≤ ∥∇(Egh − Ehgh − vh)∥L2(Ω), vh ∈ Wh, and therefore ∥∇(Egh − Ehgh)∥L2(Ω) ≤ min vh∈Vh,T (vh)=gh ∥∇(Egh − vh)∥L2(Ω) ≤ ∥∇(Egh − PhEgh)∥L2(Ω), where Ph is the stable projector that preserves homogeneous boundary condition, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥∇(Egh − Ehgh)∥L2(Ω) ≤ (1 + B) min vh∈Vh ∥Egh − vh∥H1(Ω), where B is the uniform H1-stability bound on Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By standard finite element approximation estimates and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), we have min vh∈Vh ∥Egh − vh∥H1(Ω) ≤ Chr−1∥Egh∥Hr(Ω) ≤ CC1hr−1∥gh∥Hs(Γ), where the constant C depends on r and on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) follows by Poincar´e inequality since Egh − Ehgh ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second ingredient concerns the regularity of the solution to the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ⟨κ, v⟩Hs(Γ) = γ(v), v ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 14 For a general linear functional γ ∈ H−s(Γ), that is, continuous on Hs(Γ), we are only ensured that the solution κ is bounded in Hs(Γ), with ∥κ∥Hs(Γ) = ∥γ∥H−s(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, if γ has some extra regularity, this then translates into additional regularity of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As a simple example, consider the case where γ is in addition continuous on L2(Γ), that is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) γ(v) = ⟨g, v⟩L2(Γ), for some g ∈ L2(Γ), and assume that we work with s = 1 and a polygonal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then the variational problem has a solution κ ∈ H1(Γ) and in addition κ ∈ H2(E) for each edge E with weak second derivative given by −κ′′ = g − κ ∈ L2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, standard finite element approximation estimates yield min κh∈Th ∥κ − κh∥H1(Γ) ≤ Ch∥g∥L2(Γ), with a constant C that depends on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, gain of regularity theorems for elliptic problems are known in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, we have not found a general treatment of gain of regularity that addresses the setting of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In going forward, we do not wish to systematically explore this gain in regularity and approximability for more general values of s and smoothness of γ since this would significantly enlarge the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Instead, we state it as the following general assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Assumption R: for s > 1 2 and δ > 0, there exists r(s, δ) > 0 such that if γ ∈ H−s+δ(Γ) for some δ > 0, then the solution κ to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) min κh∈Th ∥κ − κh∥Hs(Γ) ≤ Chr(s,δ)∥γ∥H−s+δ(Γ), with a constant C that depends on s, δ, and on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The above example shows that r(1, 1) = 1 for a polygonal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We expect that this assumption always holds for the range 1 2 < s < 3 2 that is considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' An a priori error estimate for ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we work under the assumption that the linear functional ν is continuous on H1(Ω) with norm Cν := max{ν(v) : ∥v∥H1(Ω) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us first check that this assumption implies a uniform a priori bound on ∥ψh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, we may write ∥ψh∥2 Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ CνDE∥ψh∥H1/2(Γ) ≤ CνDE∥ψh∥Hs(Γ), where the first inequality used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥ψh∥Hs(Γ) ≤ CνDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen in §2 that the function φ belongs to the standard Sobolev space Hr(Ω) for r defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We use this r throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), there exists a constant C1 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) ∥Ew∥Hr(Ω) ≤ C1∥w∥Hs(Γ), w ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As noted in §2, the amount of smoothness r depends both on s and on the geometry of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' What is important for us is that since s > 1/2, we have shown in (2) that r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, for smooth domains it is r = s+ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The fact that φ ∈ Hr(Ω) hints that the finite element approximation φh to φ should converge with a certain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is indeed the case as given in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Under Assumption R, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) ∥φ − φh∥H1(Ω) ≤ CCνht, where t = min{r − 1, r(s, s + 1 2) + r(s, s − 1 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The constant C depends on s and on the geometry of Ω, and on the family of meshes through the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We use the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) φ − φh = Eψ − Ehψh = E(ψ − ψh) + (E − Eh)ψh, The second term can be estimated with the help of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3 applied to gh = ψh which gives ∥(E − Eh)ψh∥H1(Ω) ≤ C2hr−1∥ψh∥Hs(Γ) ≤ C2DECνhr−1, from the a priori estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) for ψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We thus have obtained a bound in O(hr−1) for the H1 norm of the second term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term, we know that ∥E(ψ − ψh)∥H1(Ω) ≤ CE∥ψ − ψh∥H1/2(Γ), and so we are led to estimate ψ − ψh in the H1/2(Γ) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this purpose, we introduce the intermediate solution ψh ∈ Th to the problem ⟨ψh, gh⟩Hs(Γ) = µ(gh) = ν(Egh), gh ∈ Th, and we use the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17) ψ − ψh = (ψ − ψh) + (ψh − ψh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We estimate the second term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17) by noting that for any gh ∈ Th, ⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ Cν∥(E − Eh)gh∥H1(Ω) ≤ CνC2hr−1∥gh∥Hs(Γ), where we have again used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Taking gh = ψh − ψh we obtain a bound O(hr−1) for its Hs(Γ) norm, and in turn for its H1/2(Γ) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It remains to estimate ∥ψ − ψh∥H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that ψh is exactly the Galerkin approximation of ψ since we use the same linear form µ in both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In fact, we have ⟨ψ − ψh, gh⟩Hs(Γ) = 0, gh ∈ Th, that is ψh is the Hs-orthogonal projection of ψ onto Th and therefore ∥ψ − ψh∥Hs(Γ) = min κh∈Th ∥ψ − κh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since the linear form µ satisfies |µ(g)| = |ν(Eg)| ≤ Cν∥Eg∥H1(Ω) ≤ CνCE∥g∥H1/2(Γ), and thus belongs to H−1/2(Γ), we may apply the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) to γ = ν, κ = ψ, δ = s − 1 2 > 0, to reach (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18) ∥ψ − ψh∥H1/2(Γ) ≤ ∥ψ − ψh∥Hs(Γ) ≤ CCνCEhr(s,s− 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This proves the theorem for the value t = min{r − 1, r(s, s − 1 2)} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We finally improve the value of t by using a standard Aubin-Nitsche duality argument as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now take κ to be the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) with γ(v) = ⟨ψ − ψh, v⟩H1/2(Γ), v ∈ H1/2(Γ), where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='⟩H1/2(Γ) stands for the H1/2 scalar product associated with the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='∥H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then write ∥ψ − ψh∥2 H1/2(Γ) = ⟨ψ − ψh, ψ − ψh⟩H1/2(Γ) = ⟨κ, ψ − ψh⟩Hs(Γ) = ⟨κ − κh, ψ − ψh⟩Hs(Γ), where the last equality comes from Galerkin orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥ψ − ψh∥2 H1/2(Γ) ≤ ∥κ − κh∥Hs(Γ)∥ψ − ψh∥Hs(Γ) ≤ Chr(s,s+ 1 2 )∥ψ − ψh∥H1/2(Γ)∥ψ − ψh∥Hs(Γ), where we have again used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) now with δ = s+ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Using the already established estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18), it follows that ∥ψ − ψh∥H1/2(Γ) ≤ CCECνh˜t, with ˜t := r(s, s + 1 2) + r(s, s − 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' With all such estimates, the desired convergence bound follows with t := min{r − 1, ˜t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 16 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case of a polygonal domain and s = 1 which is further considered in our numerical experiments, we know that r = 3 2 and r(1, 1) = 1 so that ˜t ≥ r(1, 3 2) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn the convergence bound is established with t = r − 1 = 1 2, a rate that we observe in practice, see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of point value evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We discuss now the case where ν(v) = δz(v) = v(z), for some z ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to guarantee that point evaluation is a continuous functional on Hs, we assume that s > d − 1 2 , that is s > 1 2 for d = 2, and s > 1 for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We want to find the Riesz representer of such a point evaluation functional on Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that our assumption on s ensures the continuous embeddings Hs(Γ) ⊂ C(Γ), as well as Hs(Ω) ⊂ Hr(Ω) ⊂ C(Ω), since in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) r = min � s + 1 2, r∗� > d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' where in the inequality we recall that r∗ > 3 2 for polygonal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The point evaluation functional ν is thus continuous on Hs(Ω) with norm Cs bounded independently of the position of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, the Galerkin scheme analyzed above for ν ∈ H1(Ω)∗ continues to make sense since ν is well defined on the space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As explained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4, the prescriptions in Step 2 of the recovery algorithm need to be strengthened in the point evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, we are interested in bounding the pointwise error |φ(x) − φh(x)| at the measurement points, in addition to the H1-error ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In what follows, we establish a modified version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 in the point value setting that gives a convergence rate for ∥φ − φh∥H1(Ω), and in addition for ∥φ − φh∥L∞(Ω) ensuring the pointwise error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We stress that the numerical method remains unchanged, that is, φh is defined in the exact same way as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The new ingredients that are needed in our investigation are two classical results on the behavior of the finite element method with respect to the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first one is the so-called weak discrete maximum principle which states that there exists a constant Cmax such that, for all h > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) ∥Ehgh∥L∞(Ω) ≤ Cmax∥gh∥L∞(Γ), gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This result was first established in [4] with constant Cmax = 1 for piecewise linear Lagrange finite elements under acuteness assumptions on the angles of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The above version with Cmax ≥ 1 is established in [25] for Lagrange finite elements of any degree on 2d polygonal domains, under the more general assumption that the meshes {Th}h>0 are quasi-uniform (in addition to shape regularity, all elements of Th have diameters of order h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A similar result is established in [13] on 3d convex polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second ingredient we need is a stability property in the L∞ norm of the Galerkin projection Rh : H1 0(Ω) → Wh where Rhv, v ∈ H1 0(Ω), is defined by � Ω ∇Rhv · ∇vh = � Ω ∇v · ∇vh, vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Specifically, this result states that there exists a constant Cgal and exponent a ≥ 0 such that, for all h > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20) ∥Rhv∥L∞(Ω) ≤ Cgal(1 + | ln(h)|)a∥v∥L∞(Ω), v ∈ L∞(Ω) ∩ H1 0(Ω), that is, the Ritz projection is stable and quasi-optimal, uniformly in h, up to a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This result is established in [25] for Lagrange finite elements on 2d polygonal domains and quasi-uniform partitions, with a = 1 in the case of piecewise linear elements and a = 0 for higher order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A similar result is established in [13] with a = 0 for convex polygons and polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In going further, we assume that the choice of finite element meshes ensures the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 17 We begin our analysis with the observation that under the additional mesh assumptions, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3 can be adapted to obtain an estimate on ∥(E − Eh)gh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any gh ∈ Th, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='21) ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d 2 ∥gh∥Hs(Γ), where C3 depends on (r, s), the geometry of Ω, and the family of meshes through Cgal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any vh ∈ Vh such that T (vh) = gh, we write ∥(E − Eh)gh∥L∞(Ω) ≤ ∥Egh − vh∥L∞(Ω) + ∥Ehgh − vh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is readily seen that Ehgh − vh = Rh(Ehgh − vh) = Rh(Egh − vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed RhEhgh − RhEgh ∈ Wh and � Ω ∇(Rh(Ehgh − Egh)) · ∇vh = � Ω ∇(Ehgh − Egh) · ∇vh = 0 for all vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20), we obtain ∥(E − Eh)gh∥L∞(Ω) ≤ (1 + Cgal(1 + | ln(h)|)a) min vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, we are ensured that Egh belongs to Hr(Ω) where r > d 2, and therefore has H¨older smoothness of order r − d 2 > 0 with ∥Egh∥Cr− d 2 (Ω) ≤ Ce∥Egh∥Hr(Ω) ≤ CeC1∥gh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By standard finite element approximation theory, min vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω) ≤ Chr− d 2 ∥Egh∥Cr− d 2 (Ω), where C depends on r and the shape-parameter σ and therefore we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are now in position to give an adaptation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 to the point value setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Under Assumption R, for any t1 < min{r − d 2, r(s, s + 1 2) + r(s, s − 1 2)}, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='22) ∥φ − φh∥H1(Ω) ≤ Cht1, and for any t2 < min{r − d 2, 2r(s, s − d−1 2 )}, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) ∥φ − φh∥L∞(Ω) ≤ Cht2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The constant C depends in both cases on s, t1 and t2, on the geometry of Ω, as well as on the family of meshes through the constants Cmax and Cgal, and the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We estimate ∥φ − φh∥H1(Ω) by adapting certain steps in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first change lies in the a priori estimate of the Hs(Γ) norm of ψh that was previously given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) which is not valid anymore since Cν = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Instead, we write ∥ψh∥2 Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ ∥Ehψh∥L∞(Ω) ≤ Cmax∥ψh∥L∞(Γ) ≤ CmaxBs∥ψh∥Hs(Γ), where we have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) and where Bs is the continuous embedding constant between Hs(Γ) and L∞(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, we find that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='24) ∥ψh∥Hs(Γ) ≤ CmaxBs, which results in the slightly modified estimate ∥(E − Eh)ψh∥H1(Ω) ≤ C2CmaxBshr−1, for the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term E(ψ − ψh), we proceed in a similar manner to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Namely, we estimate the H1/2(Γ) norms of two summands in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate of ∥ψh − ψh∥H1/2(Γ) is modified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We note that for any gh ∈ Th, ⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d 2 ∥gh∥Hs(Γ), 18 where we have now used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Taking gh = ψh − ψh we obtain a bound of order O(hr− d 2 ) up to logarithmic factors for its Hs norm, and in turn for its H1/2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate of ∥ψ − ψh∥H1/2(Γ) is left unchanged and of order O(h˜t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these various estimates, we have established (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='22) for any t1 < min{r − d 2, ˜t}, with ˜t := r(s, s + 1 2) + r(s, s − 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next estimate ∥φ − φh∥L∞(Ω) by the following adaptation of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term (E − Eh)ψh of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6 combined with the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='24) of ψh which give us ∥(E − Eh)ψh∥L∞(Ω) ≤ CmaxBsC3(1 + | ln(h)|)a)hr− d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the second term E(ψ − ψh), we use the continuous maximum principle to obtain ∥E(ψ − ψh)∥L∞(Ω) ≤ ∥ψ − ψh∥L∞(Γ) ≤ ∥ψh − ψh∥L∞(Γ) + ∥ψ − ψh∥L∞(Γ) For the first summand, we write ∥ψh − ψh∥L∞(Γ) ≤ Ce∥ψh − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant, and we have already observed that ∥ψh − ψh∥Hs(Γ) satisfies a bound in O(hr− d 2 ) up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the second summand, we may write ∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since ν belongs to H−s+δ(Γ) for all δ < s − d−1 2 , we can apply the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) to reach a convergence bound ∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ), where C depends on the closeness of δ to s − d−1 2 , and on the family of meshes through the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these estimates then gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) for any t2 < min{r − d 2, ˜t} where ˜t = r(s, s − d−1 2 ), since δ can be picked arbitrarily close to s − d−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We can improve the range of t2 as follows: pick any s such that d−1 2 < s < s and write ∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then apply a similar Aubin-Nitsche argument to derive an estimate ∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ)+r(s,s−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these estimates gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) for any t2 < min{r − d 2, t}, where t := 2r(s, s − d−1 2 ) since s can be picked arbitrarily close to d−1 2 and δ arbitrarily close to s − d−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Numerical Illustrations In this section, we implement some examples of our numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this, we have to specify the domain Ω, the functionals λj, and a function u ∈ H1(Ω) which gives rise to the data vector w = λ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While our numerical method can be applied to general choices for these quantities, in our illustrations we make these choices so that the computations are not too involved but yet allow us the flexibility to illustrate certain features of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The specific choices we make for our numerical example are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The domain: In order to simplify the presentation, we restrict ourselves when Ω = (0, 1)2 but point out again that the algorithm can be extended to more general domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function u: For the function u we choose the harmonic function u = uH where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) uH(x, y) = ex cos(y), (x, y) ∈ Ω := (0, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This choice means that u0 = 0 and therefore allows us not to deal with the computation of ˆu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This choice corresponds to the right side f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the trace of uH on the boundary Γ is piecewise smooth and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, we have T (uH) ∈ H1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We take s = 1 as our assumption on the value of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that we shall seek Riesz representor for the functionals given below when viewed as acting on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of linear functionals defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we consider numerical experiments for linear functionals defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In our illustrative example, we relabel these functionals by double indices associated with a regular square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) λi,j(v) := 1 √ 2πr2 � Ω v(z)e− 1 2 |z−zi,j|2 r2 dz, v ∈ H1(Ω), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, we assume that m is a square integer and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The centers zi,j ∈ Ω are uniformly distributed (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) zi,j := 1 √m + 1(i, j), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that our numerical algorithm as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 is based on finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Specifically, we us the finite element spaces Vh := � vh ∈ C0(Ω) : vh|T ∈ Q1, T ∈ Th � , where Th are subdivisions of Ω made of squares of equal side length h and Q1 denotes the space of polynomials of degree at most 1 in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to study the effect of the mesh-size we specifically consider h = hn := 2−n, n = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , 9, that is, bilinear elements on uniformly refined meshes with mesh-size 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We display in Table 1 the results of our numerical recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The entries in the table are the recovery errors e(m, n) := ∥uH − ˆuH∥H1(Ω), where ˆuH ∈ Vhn is the recovery for the particular values of m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' n m 4 9 16 25 36 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='73 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='43 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='31 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='79 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='06 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='06 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recovery error e(m, n) for different amounts of Gaussian measurements m and finite element refinements n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have proven in this paper that our numerical recovery algorithm is near optimal with constant C that can be made arbitrarily close to one by choosing n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that the error e(m, n) satisfies e(m, n) ≤ CR(KH w )H1(Ω) for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Increasing the number m of measurements is expected to decrease this Chebyshev radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While one is tempted to think that the entries in each column of the table provides an upper bound for the Chebyshev radius of KH w for these measurements, this is not guaranteed since we are only measuring the error for one function from Kw, namely uH, and not all possible functions from Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, the entries in any given column provide a lower bound for the Chebyshev radius of KH w provided n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Increasing the number m of measurements requires a finer resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', increasing n, of the finite element discretization until the perturbation ε in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is indeed confirmed by the results in Table 1 where stagnating error bounds (in each fixed column) indicate the corresponding tip-over point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We notice in particular that for small values of n, the error becomes very large as m grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is explained by the fact that the Gramian matrix G becomes severely ill-conditioned, and in turn the prescriptions on ∥G − ˆG∥1 cannot be fulfilled when using finite element approximation of the Riesz representers on too coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' An overall convergence of the recovery error to zero can, of course, only take place when both m and n increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of point value measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we describe our numerical experiments in the case where the linear functionals λi,j are point evaluations at points from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that while the λi,j are not defined for general functions in H1(Ω) they are defined for functions in the model class KH := U(Hs(Ω)) provided s is sufficiently large (s > 1/2 for d = 2 and s > 1 for d = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that the optimal recovery problem is well posed in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 sufficient conditions on a numerical algorithm to give near optimal recovery and then we have gone on to show in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5 that our proposed numerical algorithm based on discrete harmonics converges to a near optimal recovery with any constant C > 1 provided that the finite element spaces are discretized fine enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the numerical experiments of this section, we again take Ω = (0, 1)2, s = 1, and the data to be the point values of the harmonic function uH defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We choose the evaluation points to be the zi,j of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now provide in Table 2 the recovery error e(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The observed behavior is similar to the case of Gaussian averages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' n m 4 9 16 25 36 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='43 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='49 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='02 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='27 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='89 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recovery error e(m, n) for different amounts of point evaluation measurements m and refinements n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Additional comments on the approximation of Riesz representers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finally, we provide a little more information on the computations that may be of interest to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We work in the same setting as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us begin with the rate of convergence of our numerical approximations to the Riesz representers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first consider the computation of the Riesz representer for the Gaussian measurement functional centered at z = zi,j := (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let φn ∈ Vhn be the approximation to the Riesz representer φ produced by the finite element computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 shows the error ∥φn − φ9∥H1(Ω), n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This graph indicates an error decay Ch1/2 n which matches the rate guaranteed by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4, see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Next consider the computation of the Riesz representer for point evaluation at the same z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 reports the numerical computations of error in both the H1(Ω) and L∞(Ω) norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Again, the graph indicates an error decay Ch1/2 n for the H1(Ω) norm which matches the rate guaranteed by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 and a decay rate closer to Chn for the L∞(Ω) norm (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 only guarantees Ch1/2 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal data sites: Gelfand widths and sampling numbers In this section, we make some comments on the number of measurements m that are needed to guarantee a prescribed error in the recovery of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Bounds on m are known to be governed by the Gelfand width for the case of general linear functionals and by sampling numbers when the functionals are required to be point evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We explain what is known about these quantities for our specific model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As we shall see these issues are not completely settled for the model classes studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The problem of finding the best choice of functionals, respectively point evaluations, is in need of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen that the accuracy of the optimal recovery of u ∈ Kw is given by the Chebyshev radius R(Kw) := R(Kw)H1(Ω) or equivalently R(KH w ) := R(KH w )H1(Ω) for the harmonic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The worst case recovery error R(K) over the class K is defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) R(K)H1(Ω) := sup w∈Rm R(Kw)H1(Ω), Notice that this worst case error fixes the measurement functionals but allows the measurements w to come from any function in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Both the individual error R(Kw) and the worst case error R(K) are very dependent 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='001000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='010000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='100000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000000 100 1000 dim(Vhn) Gaussian: H1 error Point evaluation: L∞ error Point evaluation: H1 error order 1 2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Approximation errors for the Riesz representers of the Gaussian and point evaluation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' on the choice of the data functionals λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, in the case that these functionals are point evaluations at points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , zm ∈ ¯Ω, then R(Kw) and R(K) will depend very much on the positioning of these points in ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case of general linear functionals, one may fix m and then search for the λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm that minimize the worst case recovery error over the class K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This minimal worst case error is called the Gelfand width of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If we restrict the linear functionals to be given by point evaluation, we could correspondingly search for the sampling points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm minimizing the worst case recovery error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This minimal error is called the deterministic sampling number of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The goal of this section is not to provide new results on Gelfand widths and sampling numbers, since we regard this as a separate issue in need of a systematic study, but to discuss what is known about them in our setting and refer the reader to the relevant papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us recall that R(Kw) is equivalent to R(KH w )H1 and so we restrict our discussion in what follows to sampling of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal choice of functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Suppose we fix the number m of observation to be allowed and ask what is the optimal choice for the λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and what is the optimal error of recovery for this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The answer to the second question is given by the Gelfand width of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given a compact set K of a Banach space X, we define the Gelfand width of K in X by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) dm(K)X := inf λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',λm R(K)X where the infimum is taken over the linear functionals defined on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us mention that this definition differs from that employed in the classical literature [21] where dm(K)X is defined as the infimum over all spaces F of codimension n of max{∥v∥X : v ∈ K ∩F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The two definitions are equivalent in the case where K is a centrally symmetric set such that K + K ⊂ CK for some constant C ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Any set of functionals which attains the infimum in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) would be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Gelfand width is often used as a benchmark for performance since it says that no matter how the m functionals λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm are chosen, the error of recovery of u ∈ K cannot be better than dm(K)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' When X is a Hilbert space and K is the ball of a Hilbert space Y with compact embedding in X, it is known that the Gelfand width coincides with the Kolmogorov width, that is dm(K)X = dm(K)X := inf dim(E)=m dist(K, E)X = inf dim(E)=m max{∥v − PEv∥X : v ∈ K}, where the infimum is taken over all linear spaces E of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is precisely our setting as discussed in §3: taking X = H1 := H1(Ω) and K as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) dm(K)H1(Ω) = dm(KH)H1(Ω) = dm(KH)H1(Ω) ∼ dm(KB)H1/2(Γ) = dm(KB)H1/2(Γ), 22 where the equivalence follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Upper and lower bounds for the Gelfand width of KB in L2(Γ) are explicitely given in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We can estimate the rate of decay of the Kolmogorov and Gelfand width of KB in H1/2(Γ) by the following general argument: as explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1, for the admissible range of smoothness, the Sobolev spaces Hs(Γ) have an intrinsic description by locally mapping the boundary onto domains of Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely, in [17] and [10], the Hs(Γ) norm of g is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥g∥Hs(Γ) := � J � j=1 ∥gj∥2 Hs(Rj) �1/2 , where the Rj are open bounded rectangles of Rd−1 that are mapped by transforms γj into portions Γj that constitute a covering of Γ, and gj = g ◦ γj are the local pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From this it readily follows that the Gelfand and Kolmogorov m-width of the unit ball of Hs(Γ) in the norm Ht(Γ), with 0 ≤ t < s behaves similar to that of the unit ball of Hs(R) in the norm Ht(R) where R is a bounded rectangle of Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The latter is known to behave like m− s−t d−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, for KH = U(Hs) with s > 1 2 in the admissible range allowed by the boundary smoothness, one has (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) cm− s−1/2 d−1 ≤ dm(KH)H1(Ω) ≤ Cm− s−1/2 d−1 , m ≥ 1, where c and C are positive constants depending only on Ω and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have already observed in §2 that the space Hs(Ω) is continuously embedded in the Sobolev space Hr(Ω) with r := max{s+ 1 2, r∗} and in particular r = s+ 1 2 for smooth domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However the Gelfand and Kolmogorov widths of the unit ball of Hr(Ω) in H1(Ω) have the slower decay rate m− r−1 d = m− s−1/2 d compared to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) for Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This improvement reflects the fact that the functions from Hs(Ω) have d variables but are in fact determined by functions of d − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The reduction in dimension from d to d − 1 is related to the fact that in our formulation of our problem we have complete knowledge of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal choice of sampling points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We turn to the particular setting where the λj are point evaluations functionals, λj(v) = v(xj), at m points xj ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to the Gelfand width, the deterministic sampling numbers are defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ρm(K)X := inf x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',xm R(K)X, A variant of this is to measure the worst case expected recovery error when the m points are chosen at random according to a probabilty distribution and search for the distribution that minimizes this error, leading to the randomized sampling number of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Obviously, one has (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) ρm(K)X ≥ dm(K)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the majority of the literature, deterministic and randomized sampling numbers are studied with error measured in the L2(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this setting, concrete strategies for optimal deterministic and randomized point design have been given when K is the unit ball of a reproducing kernel Hilbert space H defined on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, the recent results in [16, 12, 18, 5] show that under the assumption � m>0 |dm(K)L2(Ω)|2 < ∞, then, for all t > 1 2, sup m≥1 mtdm(K)L2(Ω) < ∞ =⇒ sup m≥1 mtρm(K)L2(Ω) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In words, under the above assumptions, optimal recovery in L2(Ω) has the same algebraic convergence rate when using optimally chosen point values compared to an optimal choice of general linear functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While similar general results have not been established for Gelfand width and sampling numbers in the H1 norm, we argue that they hold in our particular setting where H = Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, as in §4, we consider a domain that is either a polygon when d = 2 or polyhedron when d = 3, and thus consider the range d−1 2 < s < 3 2 where the restriction from below ensures that Hs(Ω) ⊂ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recalling the 23 finite element spaces Vh and their traces Th on the boundary, based on quasi-uniform meshes {Th}h>0, we consider for a given h > 0 the measurement points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm that are the mesh vertices located on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By the quasi-uniformity property the number m = m(h) of these points satisfies ch1−d ≤ m ≤ Ch1−d, for some c, C > 0 independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If v ∈ Hs(Ω), its trace vΓ belongs to Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, denoting by Ih the piecewise linear interpolant on the boundary, standard finite element approximation theory ensures the estimate ∥vΓ − IhvΓ∥H1/2(Γ) ≤ Chs− 1 2 ∥vΓ∥Hs(Γ) = Chs− 1 2 ∥v∥Hs(Ω), for some C that only depends on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, introducing ˜v := EIhv, one has ∥v − ˜v∥H1(Ω) ≤ CE∥vΓ − IhvΓ∥H1/2(Γ) ≤ CDEm− s−1/2 d−1 ∥v∥Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since ˜v only depends on the value of v at the points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm, we have thus proved an upper bound of order m− s−1/2 d−1 for ρm(KH)H1(Ω), and in turn for ρm(K)H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5), a lower bound of the same order must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In summary, similar to the Gelfand widths, the sampling numbers satisfy (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) ˜cm− s−1/2 d−1 ≤ ρm(K)H1(Ω) ≤ ˜Cm− s−1/2 d−1 , m ≥ 1, where ˜c and ˜C are positive constants depending only on Ω and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' References [1] R.' metadata={'source': 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+page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Andrea Bonito, Department of Mathematics, Texas A&M University, College Station, TX 77843, bonito@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Albert Cohen, Labortoire Jacques-Louis Lions, Sorbonne Universi´e, 4, Place Jussieu, 75005 Paris, France, albert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='cohen@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='fr Wolfgang Dahmen, Department of Mathematics, University of South Carolina, Columbia, SC 29208, dahmen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Ronald DeVore, Department of Mathematics, Texas A&M University, College Station, TX 77843, rdevore@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Guergana Petrova, Department of Mathematics, Texas A&M University, College Station, TX 77843, gpetrova@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='tamu.' metadata={'source': 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