Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

TL;DR

This paper develops a generalized Newton-type iterative method for solving inverse problems with nonlinear equations, incorporating various data misfit functionals, and applies it to Poisson data scenarios like phase retrieval and obstacle scattering.

Contribution

It extends the iteratively regularized Gauss-Newton method to general data misfit functionals, including the Kullback-Leibler divergence, and provides convergence analysis for these generalized methods.

Findings

Proven convergence and rates for the generalized method.

Application to Poisson data problems like phase retrieval.

Numerical examples demonstrating effectiveness.

Abstract

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples.