Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex Plane
SIAM Journal on Numerical Analysis
Preconditioning Lanczos Approximations to the Matrix Exponential
SIAM Journal on Scientific Computing
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space
SIAM Journal on Scientific Computing
Acceleration Techniques for Approximating the Matrix Exponential Operator
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Numerical Analysis
Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts
SIAM Journal on Scientific Computing
Error Estimates and Evaluation of Matrix Functions via the Faber Transform
SIAM Journal on Numerical Analysis
Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation
SIAM Journal on Numerical Analysis
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We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq tSIAM 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.