Rate-optimal sparse approximation of compact break-of-scale embeddings

TL;DR

This paper introduces hyperbolic wavelet-based methods for sparse approximation of functions with mixed regularity, achieving optimal convergence rates independent of dimension, relevant for electronic Schrödinger equations.

Contribution

It develops new Besov- and Triebel-Lizorkin-type spaces using hyperbolic wavelets for energy norm approximation of functions with mixed smoothness, providing explicit algorithms with sharp convergence rates.

Findings

Achieves dimension-independent convergence rates.

Develops new function spaces for hybrid regularity.

Provides explicit algorithms for sparse approximation.

Abstract

The paper is concerned with the sparse approximation of functions having hybrid regularity borrowed from the theory of solutions to electronic Schr\"odinger equations due to Yserentant [43]. We use hyperbolic wavelets to introduce corresponding new spaces of Besov- and Triebel-Lizorkin-type to particularly cover the energy norm approximation of functions with dominating mixed smoothness. Explicit (non-)adaptive algorithms are derived that yield sharp dimension-independent rates of convergence.