Multigrid methods for tensor structured Markov chains with low rank approximation

TL;DR

This paper introduces a multigrid approach tailored for tensor-structured Markov chains, leveraging low-rank tensor approximations to efficiently compute stationary distributions in high-dimensional models.

Contribution

It adapts algebraic multigrid methods to tensor structures, incorporating tensor coarse-grid operators and low-rank representations for improved efficiency.

Findings

Efficient computation of stationary distributions in high-dimensional tensor Markov chains.

Demonstrated effectiveness of tensor-based multigrid methods in reducing computational complexity.

Applicable to complex systems like production and telephone networks.

Abstract

Tensor structured Markov chains are part of stochastic models of many practical applications, e.g., in the description of complex production or telephone networks. The most interesting question in Markov chain models is the determination of the stationary distribution as a description of the long term behavior of the system. This involves the computation of the eigenvector corresponding to the dominant eigenvalue or equivalently the solution of a singular linear system of equations. Due to the tensor structure of the models the dimension of the operators grows rapidly and a direct solution without exploiting the tensor structure becomes infeasible. Algebraic multigrid methods have proven to be efficient when dealing with Markov chains without using tensor structure. In this work we present an approach to adapt the algebraic multigrid framework to the tensor frame, not only using the tensor structure in matrix-vector multiplications, but also tensor structured coarse-grid operators and tensor representations of the solution vector.