Balanced Incomplete Factorization Preconditioners

2-3pm, July 20 (Fri)
Lecture Room 1 (1212), 12th floor, National Institute of Informatics

José Mas Marí (Professor, Departament de Matemàtica Aplicada, Universitat Politècnica de València, Spain)

 BIF is an incomplete factorization of a square matrix into triangular factors in which standard LU or LDL^T factors (direct factors) and their inverses (inverse factors) can be obtained at the same time. This method is derived from the approach based on the Sherman-Morrison formula. Direct and inverse factors directly influence each other throughout the computation, and consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete LDL^T factorization, we also briefly analyze the nonsymmetric case and how to apply the preconditioner to least square problems. Experimental results will be presented.

On Preconditioned Iterative Methods for Sinc Systems of Linear Third-Order ODEs


3-4pm, July 20 (Fri)
Lecture Room 1 (1212), 12th floor, National Institute of Informatics

Zhi-Ru Ren (Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

 In this talk, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method.

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H-matrices and their Schur complements


2-4pm, July 27 (Fri)
Lecture Room 1 (1212), 12th floor, National Institute of Informatics

José Mas Marí (Professor, Departament de Matemàtica Aplicada, Universitat Politècnica de València, Spain)

 What happens if the comparison matrix of a given matrix is a singular M-matrix? This question will be answered and characterizations of H-matrices with singular or nonsingular comparison matrix will be analyzed. In particular, the case of irreducible matrices will be analyzed and some insights into the reducible case will be given. The spectral radius of the Jacobi matrix of the comparison matrix and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

  In the second part we will focus on the Schur complement of H-matrices. It is well-known that the Schur complement of some H-matrices is an H-matrix. We will study the Schur complement of any general H-matrixt. In particular it is proved that the Schur complement, if it exists, is an H-matrix and it is studied to which class of H-matrix the Schur complement belongs to. In addition, results are given for singular irreducible H-matrices and for the Schur complement of nonsingular irreducible H-matrices.

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