Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
48/2003

Existence and Computation of a Low Kronecker-Rank Approximant to the Solution of a Tensor System with Tensor Right-Hand Side

Lars Grasedyck

Abstract

In this paper we construct an approximation to the solution $x$ of a linear system of equations $Ax=b$ of tensor product structure as it typically arises for finite element and finite difference discretisations of partial differential operators on tensor grids. For a right-hand side $b$ of tensor product structure we can prove that the solution $x$ can be approximated by a sum of ${\cal O}(\log(\varepsilon)^{2})$ tensor product vectors where $\varepsilon$ is the relative approximation error. Numerical examples for systems of size $1024^{256}$ indicate that this method is suitable for high-dimensional problems.

Received:
May 28, 2003
Published:
May 28, 2003
MSC Codes:
15A69, 65F05, 65N22
Keywords:
high-dimensional problems, kronecker product, low rank approximation

Related publications

inJournal
2004 Repository Open Access
Lars Grasedyck

Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure

In: Computing, 72 (2004) 3/4, pp. 247-265