Regularization based on all-at-once formulations for inverse problems

TL;DR

This paper explores all-at-once regularization methods for inverse problems, comparing their computational aspects and convergence properties to traditional reduced approaches, with a focus on nonlinear models.

Contribution

It provides convergence results and implementation comparisons for all-at-once regularization methods in inverse problems, highlighting differences from reduced formulations.

Findings

All-at-once methods have distinct convergence properties for nonlinear models.

Implementation requirements differ significantly between all-at-once and reduced approaches.

Numerical comparisons illustrate practical advantages and challenges of all-at-once formulations.

Abstract

Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularization method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this paper we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-one and reduced versions and provide some numerical comparison.