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Lecture on Numerical Linear Algebra

Lectures on Numerical Linear Algebra



February 5th (Friday)
National Institute of Informatics
12F Room 1212 (Lecture room 1)
Professor Jose Mas
Departament de Matemàtica Aplicada Universitat Politècnica de València
Preconditioners for electromagnetism applications.
Computational electromagnetism applications based on the solution of the integral form of Maxwell's equations with boundary element methods require the solution of large and dense linear systems.
For large-scale problems the solution is obtained using iterative Krylov type methods provided that a fast method for performing matrix-vector products is available. For ill conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance, but sometimes additional techniques must be applied to get convergence.

First we will analyze the application of spectral low-rank updates (SLRU) to a previously computed sparse approximate inverse preconditioner.
The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used with the IRA’s method implemented in ARPACK. In this work we investigate the use of a Jacobi– Davidson method, in particular its JDQR variant. The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA’s method.

Second: For many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix. In this context, approximate inverse preconditioners based on Frobenius-norm minimization have emerged as a robust and highly parallel alternative. In this work we propose a two-level ILU preconditioner for the preconditioned GMRES method. The computation and application of the preconditioner is based on graph partitioning techniques. Numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ILU preconditioners that perform competitively compared with Frobenius-norm minimization preconditioners.


Prof. Ken Hayami
hayami[at] *Please replace [at] with @.


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