New Jacobi--Davidson type methods for the large SVD computations

Abstract

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, by fully exploiting useful information from current subspaces, a new effective correction equation is derived at each outer iteration, leading to a new variant of JDSVD, called JDSVD-V. It is proved that JDSVD-V retains the same convergence of the outer iterations as JDSVD. A substantial advantage of JDSVD-V over JDSVD is that the new correction equations in JDSVD-V are much easier to iteratively solve than the standard ones in JDSVD: the MINRES method for the new correction equations converges much faster when there is a cluster of singular values closest to a given target, a typical case in applications. A new thick-restart JDSVD-V algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of JDSVD-V to JDSVD, and demonstrate that a similar two-stage JDSVD-V algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.