Data-sparse approximation to the operator-valued functions of elliptic operator
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- by Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij PDF
- Math. Comp. 73 (2004), 1297-1324 Request permission
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
- Received by editor(s): July 9, 2002
- Received by editor(s) in revised form: January 10, 2003
- Published electronically: July 29, 2003
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2047088