Mstab: Stabilized Induced Dimension Reduction for Krylov Subspace Recycling

Abstract

We introduce Mstab, a Krylov subspace recycling method for the iterative solution of sequences of linear systems, where the system matrix is fixed and is large, sparse, and nonsymmetric, and the right-hand-side vectors are available in sequence. Mstab utilizes the short-recurrence principle of induced dimension reduction-type methods, adapted to solve sequences of linear systems. Using IDRstab for solving the linear system with the first right-hand side, the proposed method then recycles the Petrov space constructed throughout the solution of that system, generating a larger initial space for subsequent linear systems. The richer space potentially produces a rapidly convergent scheme. Numerical experiments demonstrate that Mstab often enters the superlinear convergence regime faster than other Krylov-type recycling methods.1. Introduction. We consider iterative methods for the solution of sequences of large sparse nonsymmetric linear systemswith fixed nonsingular A ∈ C N ×N , where the right-hand sides b (ι) ∈ C N are provided in sequence. Such situations occur, for example, when applying an implicit time stepping scheme to numerically solve a transient partial differential equation (PDE). Relevant applications are, e.g., topology optimization [9], model reduction [6], structural dynamics [18], quantum chromodynamics [7], electrical circuit analysis [29], fluid dynamics [15], and optical tomography [13]. In all the aforementioned references a technique called Krylov subspace recycling (KSSR) is used.It is useful to start our discussion by establishing the notation and a few basic principles for solving a single linear system, Ax = b. Suppose x 0 is an initial guess of the solution, and let r 0 = b − Ax 0 be the initial residual. We define, as usual, the Krylov subspace of degree k ∈ N associated with A and r 0 as K k (A; r 0 ) := span{r 0 , Ar 0 , . . . , A k−1 r 0 }.Standard Krylov subspace methods solve a single linear system by searching in the kth iteration an approximate solution x k ∈ x 0 + K k (A; r 0 ) such that the residual r k *

A survey of subspace recycling iterative methods

“…More efficient versions of recycling MINRES, especially including efficient ways of computing and updating recycle spaces, were proposed in [100,153]. Extensions for recycling in BiCG [60], BiCGStab [150], and IDR(s) [142] have also been proposed [5,6,108]. The idea to use projection on a search space of old solutions for initial vectors was also proposed in [59], but this paper does not involve recycling or augmentation.…”

“…Obviously, recycling versions of QMR [64] and TFQMR [63] can developed as well. Recently, a recycling IDR(s) variant was developed [108].…”