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

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is approximately 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, a preconditioned correction equation is derived that extracts useful information from current searching subspaces to construct effective preconditioners for the correction equation and is proved to retain the same convergence of outer iterations of JDSVD. The resulting method is called inner preconditioned JDSVD (IPJDSVD) method. Convergence results show that MINRES for the preconditioned correction equation can converge much faster when there is a cluster of singular values closest to a given target, so that IPJDSVD is more efficient than JDSVD. 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 of a large matrix. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD.