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

Abstract

We consider the computation of u(t) = exp(−tA)ϕ using rational Krylov subspace reduction for 0 ≤ t < ∞, where u(t), ϕ ∈ R N and 0 < A = A * ∈ R N×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.

The structure of matrices in rational Gauss quadrature

“…Functions f that arise in applications include the exponential, the logarithm, and the square root; see, e.g., [1,3,5,7,10,15,19]. We formulate our results for the case when A is positive definite.…”

“…For this reason an m-point rational Gauss rule, which is designed to exactly integrate rational functions with a specified bounded pole, may yield a more accurate approximation of (1.3) than a standard Gauss rule (1.5) with the same number of nodes for this type of integrand; see also [4,10] for error bounds for polynomial and rational approximation. The application of rational Gauss rules is particularly attractive when the matrix A in the functional (1.1) has a structure that allows efficient evaluation of A −1 w for arbitrary vectors w, e.g., by factorization or by application of an iterative method.…”

Automated parameter selection for rational Arnoldi approximation of Markov functions

“…[4,6]). Other examples are the computation of Neumann-to-Dirichlet and Dirichlet-to-Neumann maps [1,5].…”

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

“…[15,17]). Functions of this type also arise in the context of computation of Neumann-to-Dirichlet and Dirichlet-to-Neumann maps [16,3], the solution of systems of stochastic differential equations [2], and in quantum chromodynamics [22].…”