A rational Krylov methods for large scale linear multidimensional dynamical systems

TL;DR

This paper introduces tensor rational Krylov subspace methods for efficient model reduction of large-scale multidimensional linear systems, including algorithms and adaptive strategies, validated through numerical experiments.

Contribution

It develops novel tensor rational Krylov algorithms and an adaptive interpolation point selection method for large-scale system reduction.

Findings

Effective reduction of large-scale MLTI systems

Successful application to Lyapunov tensor equations

Numerical experiments confirm efficiency and accuracy

Abstract

In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time-invariant (MLTI) systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the systems solution within a low-dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large-scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well-known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.