Solving a Class of Infinite-Dimensional Tensor Eigenvalue Problems by Translational Invariant Tensor Ring Approximations

TL;DR

This paper introduces a novel method for solving infinite-dimensional tensor eigenvalue problems with translational invariance using tensor ring approximations, enabling efficient computation and convergence monitoring.

Contribution

It presents a new approach combining tensor ring structures, low-rank approximations, and a power method to efficiently solve infinite-dimensional eigenvalue problems with translational symmetry.

Findings

Efficient eigenvector approximation using translational invariant tensor rings.

Automatic time step adjustment improves convergence speed.

Numerical results demonstrate accuracy and efficiency of the method.

Abstract

We examine a method for solving an infinite-dimensional tensor eigenvalue problem $H x = \lambda x$, where the infinite-dimensional symmetric matrix $H$ exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to $e^{-Ht}$ is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step $t$ to ensure accurate and rapid convergence of the power method.