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MiS Preprint
16/2004
Hierarchical Kronecker tensor-product approximation to a class of nonlocal operators in high dimensions
Wolfgang Hackbusch and Boris N. Khoromskij
Abstract
The class of $\mathcal{H}$-matrices allows an approximate matrix arithmetic with almost linear complexity. The combination of the hierarchical and tensor-product format offers the opportunity for efficient data-sparse representations of integral operators and the inverse of elliptic operators in higher dimensions. In the present paper, we apply the $\mathcal{H}$-matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions $\mathcal{F}(A)$ 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 $d$. In particular, we approximate the functions $A^{-1}$ and $sign(A)$ of a finite difference discretisations $A\in\mathbb{R}^{N\times N}$ with rather general location of the spectrum. 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 one space direction.
