Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

TL;DR

This paper develops and analyzes rational Krylov and ADI iterative methods for solving semi-infinite quasi-Toeplitz matrix equations, with applications in PDEs and Markov chains, supported by convergence theory and numerical validation.

Contribution

It introduces a theoretical framework for infinite-dimensional quasi-Toeplitz equations and adapts rational Krylov and ADI methods for their efficient solution.

Findings

Convergence of the proposed methods is established.

Numerical experiments demonstrate effectiveness and accuracy.

The approach is applicable to PDEs and Markov chain problems.

Abstract

We consider a class of linear matrix equations involving semi-infinite matrices which have a quasi-Toeplitz structure. These equations arise in different settings, mostly connected with PDEs or the study of Markov chains such as random walks on bidimensional lattices. We present the theory justifying the existence in an appropriate Banach algebra which is computationally treatable, and we propose several methods for their solutions. We show how to adapt the ADI iteration to this particular infinite dimensional setting, and how to construct rational Krylov methods. Convergence theory is discussed, and numerical experiments validate the proposed approaches.