An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

TL;DR

This paper introduces a non-stationary Tikhonov method with inexact step computation for large-scale nonlinear ill-posed problems, improving computational efficiency while maintaining regularization properties.

Contribution

It develops a novel inexact, non-stationary Tikhonov approach using Lanczos-based elliptical trust-region subproblems for large-scale nonlinear ill-posed problems.

Findings

Provides computational savings over exact methods.

Ensures regularization properties through trust-region radius control.

Successfully applied to parameter identification and image registration.

Abstract

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.