The least error method for sparse solution reconstruction

TL;DR

This paper introduces a least error regularization method in an  setting for reconstructing sparse solutions of ill-posed linear problems, with convergence analysis and practical discretization strategies.

Contribution

It formulates and analyzes a novel -based least error method with convergence guarantees and discretization rules for sparse solution reconstruction.

Findings

Achieves linear or sublinear convergence rates under source conditions.

Provides discretization strategies with a priori and a posteriori rules.

Analyzes the structure of approximate solutions and source elements.

Abstract

This work deals with a regularization method enforcing solution sparsity of linear ill-posed problems by appropriate discretization in the image space. Namely, we formulate the so called least error method in an $\ell^1$ setting and perform the convergence analysis by choosing the discretization level according to an a priori rule, as well as two a posteriori rules, via the discrepancy principle and the monotone error rule, respectively. Depending on the setting, linear or sublinear convergence rates in the $\ell^1$-norm are obtained under a source condition yielding sparsity of the solution. A part of the study is devoted to analyzing the structure of the approximate solutions and of the involved source elements.