On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems

Abstract

We compute 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. 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.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…It was further developed in reduced basis models for elliptic PDEs in [218], the steady incompressible Navier-Stokes equations in [217], and parabolic PDEs in [111,112]. It has since been applied in conjunction with POD methods [61,122,230] and rational interpolation methods [79]. The key advantage of the greedy approach is that the search over the parameter space embodies the structure of the problem, so that the underlying system dynamics guide the selection of appropriate parameter samples.…”

Automated parameter selection for rational Arnoldi approximation of Markov functions

“…Note that, in contrast to the algorithms presented in [7,8], we do not require any estimation for the spectral interval of A; in fact, we will demonstrate in the following section that our algorithm performs well also for highly nonsymmetric and nonnormal matrices. A detailed analysis of this algorithm will be subject of future work.…”

“…This choice is a straightforward adaption of a pole selection heuristic proposed in [7,8], where the nodal function has to be large on a negative real interval Γ and small on −Γ. In our case we do not have such a symmetry, but still we hope that our nodal rational function s n is large on Γ and small on some "relevant subset" of the numerical range of A (recall from above that s n (A)v is minimal!).…”

“…This necessity for choosing optimal parameters is a serious problem that prevents rational Arnoldi from being used in practice more widely. Recently, interesting strategies for the adaptive selection of the poles ξ j have been proposed for the exponential function [7] and the transfer function [8]. We shall build on these ideas for proposing an adaptive pole selection strategies for functions of type (1).…”

Efficient iterative algorithms for linear stability analysis of incompressible flows

“…In Druskin et al (2010) (for symmetric S) and (for general S), adaptive and parameter-free approaches for generating the shifts s were proposed. Both approaches require some knowledge of the spectrum of S. In , the first shift s 1 is chosen to be a rough estimate of either −Re min (θ) or −Re max (θ), where Re min (θ) and Re max (θ) denote the minimum and maximum real parts of the eigenvalues of S, respectively.…”