-matrix approximation for the operator exponential with applications

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

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Submission date: 06. Jul. 2000
Pages: 16
published in: Numerische Mathematik, 92 (2002) 1, p. 83-111 
DOI number (of the published article): 10.1007/s002110100360
Bibtex
MSC-Numbers: 65F50, 65F30, 15A09, 15A24, 15A99
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Abstract:
We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator.

In preceding papers a class of matrices (-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrix-matrix product with such matrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the -matrix techniques to approximate the exponent of an elliptic operator.

Starting with the Dunford-Cauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the -matrices. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linear-logarithmic complexity of the method.