A rational Arnoldi approach for ill-conditioned linear systems
This work addresses the challenge of solving ill-posed linear systems for researchers in numerical linear algebra and inverse problems, offering a novel iterative refinement approach that integrates with standard regularization techniques.
The paper presents a new Arnoldi-based method for solving ill-conditioned linear systems that reconstructs the true solution via a matrix function, functioning as iterative refinement. Numerical experiments on integral equations and interpolation problems demonstrate its effectiveness, and the method is extended to handle noisy right-hand sides with Tikhonov regularization.
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.