Functional error estimators for the adaptive discretization of inverse problems

TL;DR

This paper develops functional error estimators for adaptive discretization in inverse problems, applicable to various regularization terms including nonsmooth penalties, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces a novel approach to estimate discretization errors in inverse problems with diverse regularization terms using residual-based functional error estimators.

Findings

Effective error estimation for sparsity regularization demonstrated

Residual-based estimators provide explicit error bounds

Numerical results confirm estimator reliability

Abstract

So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.