Sequential subspace optimization for nonlinear inverse problems

TL;DR

This paper extends the sequential subspace optimization (SESOP) method from linear to nonlinear inverse problems, introducing multiple search directions and a fast algorithm with convergence and regularization guarantees.

Contribution

It adapts SESOP to nonlinear inverse problems by designing search directions that account for nonlinearity and noise, and proposes a fast two-direction algorithm with proven convergence.

Findings

Convergence of the proposed method is established.

Regularization properties are demonstrated.

A fast algorithm using two search directions is introduced.

Abstract

In this work we discuss a method to adapt sequential subspace optimization (SESOP), which has so far been developed for linear inverse problems in Hilbert and Banach spaces, to the case of nonlinear inverse problems. We start by revising the well-known technique for Hilbert spaces. In a next step, we introduce a method using multiple search directions that are especially designed to fit the nonlinearity of the forward operator. To this end, we iteratively project the initial value onto stripes whose shape is determined by the search direction, the nonlinearity of the operator and the noise level. We additionally propose a fast algorithm that uses two search directions. Finally we will show convergence and regularization properties for the presented method.