MiS Preprint Repository

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MiS Preprint

Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High Dimensional Elliptic Problems

Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij


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.

MSC Codes:
65F50, 65F30, 46B28, 47A80
hierarchical matrices, kronecker tensor products, high space dimensions

Related publications

2005 Repository Open Access
Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij

Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

In: Computing, 74 (2005) 2, pp. 131-157