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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
30/2005

Low-Rank Kronecker Product Approximation to Multi-Dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

Wolfgang Hackbusch and Boris N. Khoromskij

Abstract

This article is the second part continuing Part I. We apply the $\mathcal{H}$-matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions $F(A)$ of a discrete elliptic operator $A$ in a hypercube $\left( 0,1\right) ^{d}\in\mathbb{R}^{d}$ in the case of a high spatial dimension $d$. We focus on the approximation of the operator-valued functions $A^{-\mu}$, $\mu>0$, and $\operatorname{sign}(A)$ for a class of finite difference discretisations $A\in\mathbb{R}^{N\times N}$. The asymptotic complexity of our data-sparse representations can be estimated by $\mathcal{O}(n^{p}\log^{q}n)$, $p=1,2$, with $q$ independent of $d$, where $n=N^{1/d}$ is the dimension of the discrete problem in \emph{one} space direction.

Received:
Apr 14, 2005
Published:
Apr 14, 2005
MSC Codes:
65F50, 65F30, 46B28, 47A80
Keywords:
hierarchical matrices, kronecker tensor-product, high spatial dimension, sinc interpolation, sinc quadrature

Related publications

inJournal
2006 Repository Open Access
Wolfgang Hackbusch and Boris N. Khoromskij

Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. Pt. 2 : HKT representation of certain operators

In: Computing, 76 (2006) 3/4, pp. 203-225