- Lecture on Numerical Linear Algebra
- 12th floor, Lecture room 1 (Room 1212), National Institute of Informatics
- July 20th (Friday), 2-3pm
- Balanced Incomplete Factorization Preconditioners
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.
- July 20th (Friday), 3-4pm
- On Preconditioned Iterative Methods for Sinc Systems of Linear Third-Order ODEs
Dr. Zhi-Ru Ren (Assistant Professor, Institute of Computational Mathematics, 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.
Contact: Ken Hayami(NII, firstname.lastname@example.org)