Preconditioning correction equations in Jacobi--Davidson type methods for computing partial singular value decompositions of large matrices

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, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, at each outer iteration a new correction equation is derived that extracts useful information from current subspaces to construct effective preconditioners for the correction equation and is proven to retain the same convergence of outer iterations of JDSVD.The resulting method is called inner preconditioned JDSVD (IPJDSVD) method; it is also a new JDSVD method, and any viable preconditioner for the correction equations in JDSVD is straightforwardly applicable to those in IPJDSVD. Convergence results show that MINRES for the new correction equation can converge much faster when there is a cluster of singular values closest to a given target. A new thick-restart IPJDSVD 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 IPJDSVD to JDSVD, and demonstrate that a similar two-stage IPJDSVD algorithm substantially outperforms the most advanced PRIMME_SVDS software nowadays for computing the smallest singular triplets.