On the iteratively regularized Gauss--Newton method in Banach spaces with applications to parameter identification problems

TL;DR

This paper extends the iteratively regularized Gauss-Newton method to Banach spaces using convex optimization, allowing non-smooth penalties like L1 and TV to recover features such as sparsity and discontinuities, with applications to PDE parameter identification.

Contribution

It introduces a Banach space version of the method with non-smooth penalties and provides convergence analysis and practical numerical experiments.

Findings

Method effectively reconstructs sparse and discontinuous features.

Convergence is established under new Banach space framework.

Numerical tests demonstrate practical applicability in PDE problems.

Abstract

In this paper we propose an extension of the iteratively regularized Gauss--Newton method to the Banach space setting by defining the iterates via convex optimization problems. We consider some a posteriori stopping rules to terminate the iteration and present the detailed convergence analysis. The remarkable point is that in each convex optimization problem we allow non-smooth penalty terms including $L^1$ and total variation (TV) like penalty functionals. This enables us to reconstruct special features of solutions such as sparsity and discontinuities in practical applications. Some numerical experiments on parameter identification in partial differential equations are reported to test the performance of our method.