Conjugate gradient based acceleration for inverse problems

TL;DR

This paper enhances the conjugate gradient method for inverse problems by addressing non-differentiability issues through reweighted least squares and convolution smoothing, demonstrating improved results in geotomography and synthetic reconstructions.

Contribution

It introduces novel algorithms combining conjugate gradient with reweighting and smoothing techniques for non-differentiable functionals in inverse problems.

Findings

Improved convergence in geotomographical reconstructions

Effective handling of non-differentiable functionals

Enhanced regularization parameter estimation

Abstract

The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite precision arithmetic) the exact solution after a finite number of iterations. It is thus well suited for many types of inverse problems. On the other hand, the method requires the computation of the gradient. Here difficulty can arise, since the functional of interest to the given inverse problem may not be differentiable. In this paper, we review two approaches to deal with this situation: iteratively reweighted least squares and convolution smoothing. We apply the methods to a more generalized, two parameter penalty functional. We show advantages of the proposed algorithms using examples from a geotomographical application and for synthetically constructed multi-scale reconstruction and regularization parameter estimation.