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
83/2003
Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High Dimensional Elliptic Problems
Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij
Abstract
The class of $\mathcal{H}$-matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the $\mathcal{H}$-matrix technique combined with the Kronecker tensor-product approximation to represent the inverse of a discrete elliptic operator in a hypercube $\left( 0,1\right) ^{d} \in\mathbb{R}^{d}$ in the case of a high spatial dimension $d$. In this data-sparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by $\mathcal{O}(dn\log^{q}n)$, where $N=n^{d}$ is the discrete problem size.
