Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems

TL;DR

This paper introduces an efficient iterative method for solving nonlinear inverse problems using Ivanov regularization, involving trust region subproblems and convergence analysis, with demonstrated numerical effectiveness.

Contribution

The paper develops a novel approach combining trust region methods with quadratic approximations for nonlinear inverse problems, including convergence analysis and numerical validation.

Findings

Effective solution of non-convex trust region subproblems

Convergence of the proposed iterative method

Numerical experiments demonstrating practical performance

Abstract

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a non-convex cost function under a norm constraint, where non-convexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (non-convex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.