A fast subspace optimization method for nonlinear inverse problems in Banach spaces with an application in parameter identification

TL;DR

This paper presents a fast iterative method using multiple search directions and Bregman projections to efficiently solve nonlinear inverse problems in Banach spaces, applicable to both exact and noisy data, with demonstrated application in parameter identification.

Contribution

The paper introduces a novel, fast subspace optimization method based on sequential Bregman projections that reduces iteration count for nonlinear inverse problems in Banach spaces.

Findings

Reduces total number of iterations compared to standard methods

Applicable to problems with noisy data as a regularization technique

Effective in parameter identification in elliptic boundary value problems

Abstract

We introduce and analyze a fast iterative method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. The key idea, in contrast to the standard Landweber method, is to use multiple search directions per iteration in combination with a regulation of the step width in order to reduce the total number of iterations. This method is suitable for both exact and noisy data. In the latter case, we obtain a regularization method. An algorithm with two search directions is used for the numerical identification of a parameter in an elliptic boundary value problem.