A rational Arnoldi approach for ill-conditioned linear systems

TL;DR

This paper introduces a new Arnoldi-based iterative method for solving ill-conditioned linear systems, capable of reconstructing true solutions and extending to Tikhonov regularization with noisy data.

Contribution

It presents a novel Arnoldi approach that reconstructs solutions of ill-posed systems and integrates with Tikhonov regularization for noisy problems.

Findings

Successfully reconstructs true solutions in numerical experiments

Extends to regularized systems with noisy data

Demonstrates effectiveness on integral equations and interpolation problems

Abstract

For the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix. In this sense the method can be referred to as an iterative refinement process. Numerical experiments arising from integral equations and interpolation theory are presented. Finally, the method is extended to work in connection with the standard Tikhonov regularization with a right hand side contaminated by noise.