Randomized Algorithms for Large-scale Inverse Problems with General Regularizations

TL;DR

This paper develops randomized algorithms to efficiently solve large-scale inverse problems with general regularizations, improving computational speed and robustness while maintaining solution accuracy.

Contribution

It introduces new randomized algorithms and a generalized SVD method for large-scale inverse problems, enhancing efficiency and stability over traditional approaches.

Findings

Algorithms significantly reduce computational time.

Solutions maintain accuracy comparable to classical methods.

Enhanced robustness and stability in large-scale settings.

Abstract

We shall investigate randomized algorithms for solving large-scale linear inverse problems with general regularizations. We first present some techniques to transform inverse problems of general form into the ones of standard form, then apply randomized algorithms to reduce large-scale systems of standard form to much smaller-scale systems and seek their regularized solutions in combination with some popular choice rules for regularization parameters. Then we will propose a second approach to solve large-scale ill-posed systems with general regularizations. This involves a new randomized generalized SVD algorithm that can essentially reduce the size of the original large-scale ill-posed systems. The reduced systems can provide approximate regularized solutions with about the same accuracy as the ones by the classical generalized SVD, and more importantly, the new approach gains obvious robustness, stability and computational time as it needs only to work on problems of much smaller size. Numerical results are given to demonstrated the efficiency of the algorithms.