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

Authors:
Ivan P. Gavrilyuk;Wolfgang Hackbusch;Boris N. Khoromskij
Affiliations:
Berufsakademic Thüringen, Staatliche Studienakademie, Am Wartenberg, Eisenach, Germany;Max-Planck-Institut für Mathematik in den, Naturwissenschaften, Inselstr., Leipzing, Germany;Max-Planck-Institut für Mathematik in den, Naturwissenschaften, Inselstr., Leipzing, Germany
Venue:
Computing
Year:
2005

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Abstract

The class of H -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the H-matrix technique combined with the Kronecker tensor-product approximation (ef. [2. 20]) to represent the inverse of a discrete elliptic operator in a hypercube 0.1 d = Kd on the ease 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 C(dn log2 n) where N - n2 is the discrete problem size.