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Data-sparse approximation to the operator-valued functions of elliptic operator


Authors: Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij
Journal: Math. Comp. 73 (2004), 1297-1324
MSC (2000): Primary 47A56, 65F30; Secondary 15A24, 15A99
DOI: https://doi.org/10.1090/S0025-5718-03-01590-4
Published electronically: July 29, 2003
MathSciNet review: 2047088
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Abstract: In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator $\mathcal{L}.$ The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent $\left( zI-\mathcal{L}\right) ^{-1},$ $z\in\mathbb{C}.$

In the present paper, we consider various operator functions, the operator exponential $e^{-t\mathcal{L}},$ negative fractional powers ${\mathcal{L} }^{-\alpha}$, the cosine operator function $\cos(t\sqrt{\mathcal{L} })\mathcal{L}^{-k}$ and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents $\left( z_{k}I-\mathcal{L}\right) ^{-1}$ mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.


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Additional Information

Ivan P. Gavrilyuk
Affiliation: Berufsakademie Thüringen, Am Wartenberg 2, D-99817 Eisenach, Germany
Email: ipg@ba-eisenach.de

Wolfgang Hackbusch
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
Email: wh@mis.mpg.de

Boris N. Khoromskij
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
Email: bokh@mis.mpg.de

DOI: https://doi.org/10.1090/S0025-5718-03-01590-4
Keywords: Operator-valued function, elliptic operator, $\mathcal{H}$-matrices
Received by editor(s): July 9, 2002
Received by editor(s) in revised form: January 10, 2003
Published electronically: July 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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