Bi-level iterative regularization for inverse problems in nonlinear PDEs

TL;DR

This paper introduces a bi-level iterative regularization method for solving ill-posed inverse problems involving nonlinear PDEs, combining reduced and all-at-once approaches for improved parameter recovery.

Contribution

It proposes a novel bi-level Landweber scheme that integrates parameter and state approximation, enabling convergence analysis and application to magnetic particle imaging.

Findings

Derived stopping rules for iterative convergence.

Proved convergence of the bi-level method.

Demonstrated applicability to magnetic particle imaging.

Abstract

We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.