Bi-level iterative regularization for inverse problems in nonlinear PDEs

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.

References in corpus (4)

• All-at-once versus reduced iterative methods for time dependent inverse problems
• Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: All-at-once versus Reduced version
• On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging
• Newton's methods for solving linear inverse problems with neural network coders

Cited by in corpus (3)

• Sequential bi-level regularized inversion with application to hidden reaction law discovery
• On the convergence analysis of one-shot inversion methods
• Data assimilation via model reference adaptation for linear and nonlinear dynamical systems