Convergence analysis of multilevel sequentially semiseparable preconditioners

Convergence analysis of multilevel sequentially semiseparable preconditioners

Qiu, Y.
van Gijzen, M.B.
van Wingerden, J.W.
Verhaegen, M.
Vuik, C.

Electrical Engineering, Mathematics and Computer Science

Delft Institute of Applied Mathematics

2015-12-31

Multilevel sequentially semiseparable (MSSS) matrices form a class of structured matrices that have low-rank off-diagonal structure, which allows the matrix-matrix operations to be performed in linear computational complexity. MSSS preconditioners are computed by replacing the Schur complements in the block LU factorization of the global linear system by MSSS matrix approximations with low off-diagonal rank. In this manuscript, we analyze the convergence properties of such preconditioners. We show that the spectrum of the preconditioned system is contained in a circle centered at (1, 0) and give an analytic bound of the radius of this circle. This radius can be made arbitrarily small by properly setting a parameter in the MSSS preconditioner. Our results apply to a wide class of linear systems. The system matrix can be either symmetric or unsymmetric, definite or indefinite. We demonstrate our analysis by numerical experiments.

multilevel sequentially semiseparable preconditioners
convergence analysis
saddlepoint systems
Helmholtz equation

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Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Delft Institute of Applied Mathematics, 15-01

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