A Flexible Power Method for Solving Infinite Dimensional Tensor Eigenvalue Problems

01/30/2021
by   Roel Van Beeumen, et al.
0

We propose a flexible power method for computing the leftmost, i.e., algebraically smallest, eigenvalue of an infinite dimensional tensor eigenvalue problem, H x = λ x, where the infinite dimensional symmetric matrix H exhibits a translational invariant structure. We assume the smallest eigenvalue of H is simple and apply a power iteration of e^-H with the eigenvector represented in a compact way as a translational invariant infinite Tensor Ring (iTR). Hence, the infinite dimensional eigenvector can be represented by a finite number of iTR cores of finite rank. In order to implement this power iteration, we use a small parameter t so that the infinite matrix-vector operation e^-Htx can efficiently be approximated by the Lie product formula, also known as Suzuki–Trotter splitting, and we employ a low rank approximation through a truncated singular value decomposition on the iTR cores in order to keep the cost of subsequent power iterations bounded. We also use an efficient way for computing the iTR Rayleigh quotient and introduce a finite size iTR residual which is used to monitor the convergence of the Rayleigh quotient and to modify the timestep t. In this paper, we discuss 2 different implementations of the flexible power algorithm and illustrate the automatic timestep adaption approach for several numerical examples.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/07/2021

Bivariate collocation for computing R_0 in epidemic models with two structures

Structured epidemic models can be formulated as first-order hyperbolic P...
research
03/02/2023

Noda Iteration for Computing Generalized Tensor Eigenpairs

In this paper, we propose the tensor Noda iteration (NI) and its inexact...
research
12/04/2019

Finding entries of maximum absolute value in low-rank tensors

We present an iterative method for the search of extreme entries in low-...
research
05/02/2023

Avoiding discretization issues for nonlinear eigenvalue problems

The first step when solving an infinite-dimensional eigenvalue problem i...
research
08/13/2021

A Parallel Distributed Algorithm for the Power SVD Method

In this work, we study how to implement a distributed algorithm for the ...
research
10/29/2020

A passivation algorithm for linear time-invariant systems

We propose and study an algorithm for computing a nearest passive system...

Please sign up or login with your details

Forgot password? Click here to reset