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Solving a class of infinite-dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

Journal Article · · Numerical Linear Algebra with Applications
DOI:https://doi.org/10.1002/nla.2573· OSTI ID:2551769
 [1];  [2];  [3];  [1]
  1. Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
  2. Visage Technologies (Croatia)
  3. Ecole Polytechnique Federale Lausanne (EPFL) (Switzerland)
+ Show Author Affiliations

Here, we examine a method for solving an infinite-dimensional tensor eigenvalue problem Hx = λx, 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 to ensure accurate and rapid convergence of the power method.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)
Grant/Contract Number:
AC02-05CH11231
OSTI ID:
2551769
Journal Information:
Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 6 Vol. 31; ISSN 1070-5325
Publisher:
WileyCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
infinite eigenvalue problem
matrix product states
tensor eigenvalue problem
tensor ring
tensor train
translational invariance