On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems
We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, a...
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| Format: | Article |
| Language: | en_US |
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Society of Industrial and Applied Mathematics (SIAM)
2011
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| Online Access: | http://hdl.handle.net/1721.1/60578 |
| _version_ | 1811070579357253632 |
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| author | Druskin, Vladimir Lieberman, Chad E. Zaslavsky, Mikhail |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Druskin, Vladimir Lieberman, Chad E. Zaslavsky, Mikhail |
| author_sort | Druskin, Vladimir |
| collection | MIT |
| description | We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760–3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator $A$. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra. |
| first_indexed | 2024-09-23T08:38:17Z |
| format | Article |
| id | mit-1721.1/60578 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:38:17Z |
| publishDate | 2011 |
| publisher | Society of Industrial and Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/605782022-09-30T10:08:14Z On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems Druskin, Vladimir Lieberman, Chad E. Zaslavsky, Mikhail Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Lieberman, Chad E. Lieberman, Chad E. We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760–3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator $A$. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra. 2011-01-14T18:40:10Z 2011-01-14T18:40:10Z 2010-08 2009-10 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/60578 Druskin, Vladimir, Chad Lieberman, and Mikhail Zaslavsky. “On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems.” SIAM Journal on Scientific Computing 32.5 (2010): 2485-2496. Print. en_US http://dx.doi.org/10.1137/090774082 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society of Industrial and Applied Mathematics (SIAM) SIAM |
| spellingShingle | Druskin, Vladimir Lieberman, Chad E. Zaslavsky, Mikhail On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title | On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title_full | On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title_fullStr | On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title_full_unstemmed | On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title_short | On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems |
| title_sort | on adaptive choice of shifts in rational krylov subspace reduction of evolutionary problems |
| url | http://hdl.handle.net/1721.1/60578 |
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