All-at-once versus reduced iterative methods for time dependent inverse problems

TL;DR

This paper compares all-at-once and reduced regularization methods for dynamic inverse problems, focusing on iterative algorithms like Landweber and Gauss-Newton, and analyzes their convergence properties with an example involving nonlinear diffusion.

Contribution

It provides a detailed comparison of all-at-once versus reduced iterative regularization methods for nonlinear time-dependent inverse problems, including convergence analysis and function space setting.

Findings

All-at-once and reduced methods exhibit different convergence behaviors.

Convergence conditions are established within Hilbert space regularization theory.

An example with a nonlinear diffusion equation illustrates the theoretical results.

Abstract

In this paper we investigate all-at-once versus reduced regularization of dynamic inverse problems on finite time intervals $(0,T)$. In doing so, we concentrate on iterative methods and nonlinear problems, since they have already been shown to exhibit considerable differences in their reduced and all-at-once versions, whereas Tikhonov regularization is basically the same in both settings. More precisely, we consider Landweber iteration, the iteratively regularized Gauss-Newton method, and the Landweber-Kaczmarz method, the latter relying on cyclic iteration over a subdivision of the problem into subsequent subintervals of $(0,T)$. Part of the paper is devoted to providing an appropriate function space setting as well as establishing the required differentiability results needed for well-definedness and convergence of the methods under consideration. Based on this, we formulate and compare the above mentioned iterative methods in their all-at-once and their reduced version. Finally, we provide some convergence results in the framework of Hilbert space regularization theory and illustrate the convergence conditions by an example of an inverse source problem for a nonlinear diffusion equation.