English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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