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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