An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties

An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties

Van Gijzen, M.B.
Sonneveld, P.

Electrical Engineering, Mathematics and Computer Science

2010-07-31

The IDR(s) method that is proposed in [18] is a very efficient limited memory method for solving large nonsymmetric systems of linear equations. IDR(s) is based on the induced dimension reduction theorem, that provides a way to construct subsequent residuals that lie in a sequence of shrinking subspaces. The IDR(s) algorithm that is given in [18] is a direct translation of the theorem into an algorithm. This translation is not unique. This paper derives a new IDR(s) variant, that imposes (one-sided) bi-orthogonalization conditions on the iteration vectors. The resulting method has lower overhead in vector operations than the original IDR(s) algorithms. In exact arithmetic, both algorithms give the same residual at every s + 1-st step, but the intermediate residuals, and also the numerical properties differ. We show through numerical experiments that the new variant is more stable and more accurate than the original IDR(s) algorithm, and that it outperforms other state-of-the-art techniques for realistic test problems.

Iterative methods
IDR
IDR(s)
Krylov-subspace methods
large sparse nonsymmetric linear systems

http://resolver.tudelft.nl/uuid:30374db4-61bb-4c96-8f97-cf4464219dd9

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

1389-6520

Reports of the Department of Applied Mathematical Analysis, 10-16

Institutional Repository

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(c)2010 Van Gijzen, M.B., Sonneveld, P.