Iterative solution of dense linear systems
Jussi Rahola(CERFACS, Toulouse)
Friday October 9, 11.00 a.m. Parallel Algorithms Seminar CERFACS Conference Room
Abstract :
Discretizations of integral equations produce large dense systems of linear equations that can be very expensive to solve in terms of CPU time and memory consumption. For dense linear systems preconditioned iterative solvers can outperform direct solvers (e.g., LAPACK). In some cases, the iterative solvers can converge in a relatively low number of iterations. Furthermore, with iterative solvers, the matrix-vector products can be computed approximately without forming the coefficient matrix explicitly. These approximate techniques can give huge CPU and memory savings because the full coefficient need not be stored.
In this talk I will present two case studies of the application of iterative solvers to dense linear systems. The first problem arises from the discretization of the surface integral equations of electromagnetic scattering, which produce a complex symmetric (non-Hermitian) coefficient matrix. These systems have been solved with a complex symmetric version of QMR with a sparse approximate inverse preconditioner. The matrix-vector products are computed with the fast multipole method. These techniques make it possible to solve dense linear systems with hundreds of thousands of unknowns.
The second case study arises from the biomagnetic inverse problem where the electric activity in the brain is localized from the weak magnetic fields that are measured from outside the head. The localization process involves the repeated solution of a potential integral equation on the surface of the brain. The dense linear systems arising from the discretization of the integral equations have been solved with the iterative method Bi-CGSTAB together with an incomplete LU preconditioner. For this solver, the number of iterations is very low and is practically independent of the number of unknowns.
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