We present IDR(s), a new family of efficient, short-recurrence methods for large nonsymmetric systems of linear equations. The new methods are based on the induced dimension reduction (IDR) method proposed by Sonneveld in 1980. IDR(s) generates residuals that are forced to be in a sequence of nested subspaces. Although IDR(s) behaves like an iterative method, in exact arithmetic it computes the true solution using at most N + N/s matrix-vector products, with N the problem size and s the codimension of a fixed subspace. We describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications. Our experiments show that IDR(s) is competitive with or superior to most Bi-CG-based methods and outperforms Bi-CGSTAB when s > 1.

Abstract. Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm, we are able to provide an optimal value for the shift and to explain the mesh-dependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem. 1. Introduction. In this paper we investigate the spectral behavior of iterative methods applied to the time-harmonic wave equation in heterogeneous media. The underlying equation governs wave propagation and scattering phenomena arising in acoustic problems in many areas, e.g., aeronautics, marine technology, geophysics, and optical problems. In particular, we look for solutions of the Helmholtz equation discretized by using finite difference, finite volume, or finite element discretizations. Since the number of grid points per wavelength should be sufficiently large to result in acceptable solutions, for very high frequencies the discrete problem becomes extremely large, prohibiting the use of direct solution methods. Krylov subspace iterative methods are an interesting alternative. However, Krylov subspace methods are not competitive without a good preconditioner.Finding a suitable preconditioner for the Helmholtz equation is still an area of active research; see, for example, [7]. A class of preconditioners that has recently attracted considerable attention is the class of shifted Laplace preconditioners. Preconditioning of the Helmholtz equation using the Laplace operator without shift was first suggested in [1]. This approach has been enhanced in [8,9] by adding a positive shift to the Laplace operator, resulting in a positive definite preconditioner. In [2,3,4,13] the class of shifted Laplace preconditioners is further generalized by also considering general complex shifts.It is well known that the spectral properties of the preconditioned matrix give important insight in the convergence behavior of the preconditioned Krylov subspace methods. Spectral analyses for the Helmholtz equation preconditioned by a shifted

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.

Abstract. IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008)

We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear systems arising at each iteration of the iterative rational Krylov algorithm (IRKA) [Gugercin, Antoulas, and Beattie, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 609-638] for H 2optimal model reduction of linear systems. The idea is to construct in advance bases for the two preconditioned Krylov subspaces (one for the matrix and one for its adjoint). By exploiting the shift-invariant property of Krylov subspaces, these bases are then reused inside the model reduction methods for the other shifts. The polynomial preconditioner is chosen to maintain this shift-invariant property. This means that the shifted systems can be solved without additional matrix-vector products. The performance of our proposed implementation is illustrated through numerical experiments.