Nonlinear approximation spaces for inverse problems

TL;DR

This paper develops certified recovery bounds for inverse problems using nonlinear approximation spaces, demonstrating improved performance over linear methods, especially for piecewise smooth functions and shape recovery from cell-average data.

Contribution

It introduces certified recovery bounds for nonlinear approximation spaces in inverse problems and applies these to bidimensional shape recovery from cell-average data.

Findings

Nonlinear spaces outperform linear spaces in approximating piecewise smooth functions.

Certified recovery bounds are established for nonlinear approximation-based inversion.

Application to shape recovery demonstrates practical effectiveness.

Abstract

This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were introduced in [MPPY15, BCDDPW17] with certified recovery bounds. It is however known that linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions that typically occur in hyperbolic PDEs (shocks) or images (edges). For such families, nonlinear spaces [Devore98] are known to significantly improve the approximation performance. The first contribution of this paper is to provide with certified recovery bounds for inversion procedures based on nonlinear approximation spaces. The second contribution is the application of this framework to the recovery of general bidimensional shapes from cell-average data. We also discuss how the application of our results to n-term approximation relates to classical results in compressed sensing.