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 flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

“…As an outlook for the future, the advanced development of preconditioning strategies (such as the polynomial preconditioning [6,26], the nested iterative technique [46] and other preconditioning strategies [22,47]) for solving shifted linear systems remains an meaningful topic of further research.…”

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

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“…the considerations in [6,38]. The efficient application of the preconditioner (10) for problems of dimension d = 2 and d = 3 on a structured domain is presented in Section 4, and the choice of τ is discussed in Section 5.2.…”

“…Note that an alternative approach to efficiently solve (9) at multiple frequencies (N ω > 1) leads to the solution of shifted linear systems as presented in [6,Section 4.2] and the references therein. The memory-efficient approach followed by [6] relies on the shift-invariance property of the Krylov spaces belonging to different frequencies. Some …”

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

“…Shifted extensions of the restarted generalized minimum residual (GMRES) method [39][40][41][42][43][44], the restarted full orthogonalization (FOM) method [18][19][20] and the restarted Hessenberg method [21] are some relevant examples built upon the wellknown Arnoldi procedure. On the other hand, shifted versions of the quasi-minimal residual (QMR) method and its transpose-free variant (TFQMR) [14], the induced dimension reduction (IDR(s)) [15,16] and its QMR form [17], the biconjugate gradient (BiCG) method and its stabilized and generalized product-type extensions (BiCGStab, BiCGStab(ℓ) and GPBiCG) [12,13,25], the biconjugate residual (BiCR) method and its stabilized form (BiCRSTAB) [24] are built upon short-term vector recurrences such as the Bi-Lanczos [22] and the A-biorthogonalization [23] procedures. In [26,27], recycling variants of BiCG and BiCGSTAB have been applied to the solution of multi-shifted non-Hermitian linear systems arising in model reduction applications.…”