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

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

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Submission date: 02. Oct. 2003 (revised version: June 2004)
Pages: 21
published in: Computing, 74 (2005) 2, p. 131-157 
DOI number (of the published article): 10.1007/s00607-004-0086-y
Bibtex
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: hierarchical matrices, kronecker tensor products, high space dimensions
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Abstract:
The class of -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the -matrix technique combined with the Kronecker tensor-product approximation to represent the inverse of a discrete elliptic operator in a hypercube 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 , where is the discrete problem size.