Randomized Core Reduction for Discrete Ill-Posed Problem

TL;DR

This paper introduces a randomized algorithm for efficiently approximating solutions to large-scale discrete ill-posed problems, specifically targeting total least squares, with theoretical error bounds and numerical validation.

Contribution

It proposes a novel regularization method using randomization and subspace iteration to approximate the core problem in large-scale TLS problems.

Findings

Provides upper bounds for solution and residual errors based on singular values.

Demonstrates effectiveness through numerical examples and comparisons.

Offers a scalable approach for large-scale discrete ill-posed problems.

Abstract

In this paper, we apply randomized algorithms to approximate the total least squares (TLS) solution of the problem $Ax\approx b$ in the large-scale discrete ill-posed problems. A regularization technique, based on the multiplicative randomization and the subspace iteration, is proposed to obtain the approximate core problem.In the error analysis, we provide upper bounds %in terms of the $(k\!\!+\!\!1)$-th singular value of $A$ for the errors of the solution and the residual of the randomized core reduction. Illustrative numerical examples and comparisons are presented.