Nested Krylov Methods for Shifted Linear Systems

Abstract

SummaryShifted linear systems are of the formwhere A ∈ C N ×N , b ∈ C N and {σ k } Nσ k=1 ∈ C is a sequence of numbers, called shifts. In order to solve (1) for multiple shifts efficiently, shifted Krylov methods make use of the shift-invariance property of their respective Krylov subspaces, i.e.and, therefore, compute one basis of the Krylov subspace (2) for all shifted systems. This leads to a significant speed-up of the numerical solution of the shifted problems because obtaining a basis of (2) is computational expensive. However, in practical applications, preconditioning of (1) is required, which in general destroys the shift-invariance. One of the few known preconditioners that lead to a new, preconditioned shifted problem is the so-called shift-and-invert preconditioner which is of the form (A − τ I) where τ is the seed shift. Most recently, multiple shift-and-invert preconditioners have been applied within a flexible GMRES iteration, cf. [10,20]. Since even the one-time application of a shift-and-invert preconditioner can be computationally costly, polynomial preconditioners have been developed for shifted problems in [1]. They have the advantage of preserving the shift-invariance (2) and being computational feasible at the same time.The presented work is a new approach to the iterative solution of (1). We use nested Krylov methods that use an inner Krylov method as a preconditioner for an outer Krylov iteration, cf. [17,24] for the unshifted case. In order to preserve the shift-invariance property, our algorithm only requires the inner Krylov method to produce collinear residuals of the shifted systems. The collinearity factor is then used in the (generalized) Hessenberg relation of the outer Krylov method. In this article, we present two possible combinations of nested Krylov algorithms, namely a combination of inner multi-shift FOM and outer multi-shift GMRES as well as inner multi-shift IDR(s) and outer multi-shift QMRIDR(s). However, we will point out that in principle every combination is possible as long as the inner Krylov method leads to collinear residuals. Since multi-shift IDR [31] does not lead to collinear residuals by default, the development of a collinear IDR variant that can be applied as an inner method within the nested framework is a second main contribution of this work.The new nested Krylov algorithm has been tested on several shifted problems. In particular, the inhomogeneous and time-harmonic linear elastic wave equation leads to shifts that directly correspond to different frequencies of the waves.

A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right‐hand sides

“…Special preconditioners that preserve property are shift‐and‐invert, multishift, and polynomial preconditioners . We refer the reader to other works for some discussions on preconditioning techniques for solving shifted linear systems.…”

“…The first obvious approach is to solve each of the p multishifted linear systems independently. In this case, shifted methods that take advantage of the shift‐invariance property of the Krylov subspace may be of good choice . Recently, Darnell et al have proposed an efficient method based on deflated restarting that corrects the computed solution for an extra right‐hand side at moderate cost.…”

“…Shift‐and‐invert and polynomial preconditioners are two viable options. Recently, other preconditioning strategies have been proposed for solving shifted linear systems, see, for example, other works . In this paper, we combine our BGMRES‐DR‐Sh method with the shift‐and‐invert preconditioner.…”

A Fast Iterative Solution of the Time-harmonic Wave Equation with MSSS-preconditioned IDR(s)

“…with seed frequency τ. This preconditioner has already been used by Baumann and Van Gijzen (2015) where (1) is treated as a sequence of shifted linear systems. The spy plot of P(τ) in Figure 1 shows the hierarchical structure of the preconditioner when a Cartesian grid is used.…”

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

“…The focus of the present work, however, lies on situations where the discretization matrices K and M stem from a discretization of the time-harmonic elastic wave equation [13]. Depending on the specific choice of boundary conditions, the structure of the matrices varies, and the shifts s k are either equal to the (angular) wave frequencies [6,41] or to the squared (angular) wave frequencies [2,30]. For both situations, we will consider viscous damping by substituting s k → (1 − i)s k , where > 0 is the damping parameter and i ≡ √ −1, cf.…”