Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings I

TL;DR

This work analyzes the optimal approximation of the solution map $S={\mathcal A}^{-1}$ for elliptic-type operator equations in Hilbert spaces, comparing linear and nonlinear approximation schemes via widths. It establishes that in Hilbert spaces the best linear and continuous widths match the nonlinear widths up to constants when Bernstein widths satisfy a regularity condition, and shows that for $H^{s+t}$-regular elliptic problems the optimal rate is $n^{-t/d}$. It also demonstrates that allowing only function evaluations can degrade the rate to $n^{(s-t)/d}$, and it investigates the Poisson equation with wavelet-based best $n$-term methods, showing potential suboptimality in general but near-optimal performance on polygonal domains in $\mathbb{R}^2$ due to Besov regularity and singularity decompositions. The results highlight when nonlinear approximation offers advantages and emphasize Besov- and wavelet-based techniques for elliptic problems.

Abstract

We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings.

Theorems & Definitions (32)

• Remark 1
• Definition 1
• Remark 2
• Remark 3
• Remark 4
• Definition 2
• Remark 5
• Lemma 1
• proof
• Lemma 2