An efficient two-level preconditioner for multi-frequency wave propagation problems

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

“…The seed frequency is chosen at τ = (1 − 0.5i)ω max for which we recorded optimal convergence behavior. When the number of frequencies is increased, we observe an improved performance compared to an extrapolation of the N ω = 2 following [7] improves convergence, and allows for larger frequency ranges case. We also observed that the size of interval in which the different frequencies range is crucial for the convergence behavior.…”

“…Spectral anlysis shows that the angles ϕ k can be chosen such that the spectrum of the preconditioned operator is rotated and convergence is improved, cf. [7]. In the present case of global IDR(s) (Algorithm 1) combined with an inexact MSSS preconditioner (19), we record a reduction to 60% of the CPU time when spectral rotation is applied to the N ω = 10 case, cf.…”

“…We also observed that the size of interval in which the different frequencies range is crucial for the convergence behavior. In [7], we describe how the convergence of global GMRES [17] can be improved by scaling the k-th column of the block unknown X by e −iϕ k . Spectral anlysis shows that the angles ϕ k can be chosen such that the spectrum of the preconditioned operator is rotated and convergence is improved, cf.…”

Flexible and deflated variants of the block shifted GMRES method

“…Inspired by [21], in our experiments we choose τ k near the shifts of interest as this strategy has been shown to yield often fast convergence rates. However, we acknowledge that the choice of τ k for multi-shifted linear systems is still an open problem that deserves closer attention; e.g., see [44] for a strategy to select the optimal parameter τ for a single shift-and-invert preconditioner applied to the solution of shifted linear systems. For the PDBGMRES-Sh method, we choose a right preconditioner in shifted form A − τ I with shifts τ = −0.002, −0.006, −2.…”

Convergence and complexity study of GMRES variants for solving multi-frequency elastic wave propagation problems

“…• Global GMRES [21] suitable for the matrix equation approach (2) (cf. Algorithm 1 and [6]), • Polynomial preconditioners [1,8] for multi-shift GMRES (cf. Algorithm 2), • Nested multi-shift FOM-FGMRES as presented in [7] (cf.…”