Convergence analysis of multi-step one-shot methods for linear inverse problems

TL;DR

This paper analyzes the convergence of multi-step one-shot methods for linear inverse problems, showing that few inner iterations suffice for effective inversion, with theoretical conditions and numerical validation.

Contribution

It provides a convergence analysis for two variants of multi-step one-shot methods, establishing conditions on the descent step and demonstrating their efficiency.

Findings

Few inner iterations on the forward problem ensure good convergence.

Theoretical eigenvalue analysis supports convergence conditions.

Numerical experiments compare favorably with classical gradient descent.

Abstract

In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.