Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts

Abstract

We consider the computation of $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\le t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. 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.