Conceitos Básicos
Deriving an effective preconditioning procedure for the correction equation in Jacobi-Davidson type methods to substantially accelerate the inner iterations and thus improve the overall efficiency of computing partial singular value decompositions of large matrices.
Resumo
The content discusses preconditioning techniques for accelerating Jacobi-Davidson type methods in computing partial singular value decompositions (SVDs) of large matrices.
Key highlights:
In Jacobi-Davidson type methods for SVD problems (JDSVD), a large symmetric and generally indefinite correction equation needs to be approximately solved iteratively at each outer iteration, which dominates the overall efficiency.
The authors analyze the convergence of the MINRES method for solving the correction equation, and show that it may converge very slowly when the desired singular values are clustered around the target.
To address this issue, the authors derive a preconditioned correction equation that extracts useful information from the current searching subspaces to construct effective preconditioners. This preconditioned correction equation is proved to retain the same convergence of the outer iterations of JDSVD.
The resulting method, called inner-preconditioned JDSVD (IPJDSVD), is shown to have much faster convergence of the inner iterations compared to the standard JDSVD.
The authors also propose a new thick-restart IPJDSVD algorithm with deflation and purgation that simultaneously accelerates the outer and inner convergence and computes several singular triplets of a large matrix.
Numerical experiments justify the theory and demonstrate the considerable superiority of IPJDSVD over JDSVD.