Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts
- 1 January 2009
- journal article
- research article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Scientific Computing
- Vol. 31 (5), 3760-3780
- https://doi.org/10.1137/080742403
Abstract
We consider the computation of u(t) - exp(-tA)phi using rational Krylov subspace reduction for 0 <= t < infinity, where u(t),phi is an element of R-N and 0 < A = A* is an element of R-NxN. The objective of this work is the optimization of the shifts for the rational Krylov subspace (RKS). We consider this problem in the frequency domain and reduce it to a classical Zolotaryov problem. The latter yields an asymtotically optimal solution with real shifts. We also construct an infinite sequence of shifts yielding a nested sequence of the RKSs with the same (optimal) Cauchy-Hadamard convergence rate. The effectiveness of the developed approach is demonstrated on an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration.Keywords
This publication has 30 references indexed in Scilit:
- On monotonicity of the Lanczos approximation to the matrix exponentialLinear Algebra and its Applications, 2008
- Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projectionGeophysical Journal International, 2008
- Numerical range and functional calculus in Hilbert spaceJournal of Functional Analysis, 2007
- Krylov subspace techniques for reduced-order modeling of large-scale dynamical systemsApplied Numerical Mathematics, 2002
- Sharp constants for rational approximations of analytic functionsSbornik: Mathematics, 2002
- Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex PlaneSIAM Journal on Numerical Analysis, 2000
- Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related FunctionsSIAM Journal on Matrix Analysis and Applications, 1998
- Two polynomial methods of calculating functions of symmetric matricesUSSR Computational Mathematics and Mathematical Physics, 1989
- Approximation of e−x by rational functions with concentrated negative polesJournal of Approximation Theory, 1981
- Chebyshev rational approximations to e−x in [0, +∞) and applications to heat-conduction problemsJournal of Approximation Theory, 1969