Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings II

TL;DR

This work develops a comprehensive width-based framework to compare linear and nonlinear approximation of elliptic problems, identifying when nonlinear (best $n$-term) approximations outperform linear ones. Using Besov spaces and wavelet-based discretizations, it establishes sharp asymptotics for linear, nonlinear, and manifold widths, showing that for right-hand sides in $B^{-s+t}_{q}(L_p(\Omega))$ with $p<2$ nonlinear methods can achieve faster convergence than linear ones, while for $p\ge 2$ the convergence rates coincide. The Poisson equation on Lipschitz and polygonal domains is treated to illustrate practical nonlinear approximation via wavelets, with adaptive residual-based schemes achieving near best-$n$-term performance and favorable complexity. The results are complemented by extensive analysis of widths for embeddings of weighted sequence spaces and Besov spaces, providing a detailed map of how approximation quality scales with dimension, smoothness, and integrability parameters. Together, these insights yield sharp benchmarks for algorithm design, adaptive schemes, and the theoretical limits of elliptic problem approximation.

Abstract

We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings. We identify those cases where optimal nonlinear approximation is better than optimal linear approximation.

Theorems & Definitions (85)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Remark 3
• Lemma 1
• Remark 4
• Lemma 2
• Theorem 1
• Lemma 3