Multigrid methods combined with low-rank approximation for tensor structured Markov chains

TL;DR

This paper introduces a novel tensor-based algorithm combining multigrid and low-rank approximation techniques to efficiently solve large, structured Markov chains, overcoming previous computational limitations.

Contribution

The work presents a new tensorized multigrid method integrated with AMEn for solving large tensor-structured Markov chains, enabling handling of unprecedentedly large models.

Findings

Combination improves convergence over individual methods

Enables solving larger Markov chain models

Demonstrates effectiveness across various applications

Abstract

Markov chains that describe interacting subsystems suffer, on the one hand, from state space explosion but lead, on the other hand, to highly structured matrices. In this work, we propose a novel tensor-based algorithm to address such tensor structured Markov chains. Our algorithm combines a tensorized multigrid method with AMEn, an optimization-based low-rank tensor solver, for addressing coarse grid problems. Numerical experiments demonstrate that this combination overcomes the limitations incurred when using each of the two methods individually. As a consequence, Markov chain models of unprecedented size from a variety of applications can be addressed.