Interpreting IDR(s) as a deflation method

Interpreting IDR(s) as a deflation method

Collignon, T.P.
Sleijpen, G.L.G.
Van Gijzen, M.B.

Electrical Engineering, Mathematics and Computer Science

2010-10-30

In this paper the IDR(s) method is interpreted in the context of deflation methods. It is shown that IDR(s) can be seen as a Richardson iteration preconditioned by a variable deflation–type preconditioner. The main result of this paper is the IDR projection theorem, which relates the spectrum of the deflated system in each IDR(s) cycle to all previous cycles. The theorem shows that this so–called active spectrum becomes increasingly more clustered. This clustering property may serve as an intuitive explanation for the excellent convergence properties of IDR(s). These remarkable spectral properties exist whilst using a deflation subspace matrix of fixed rank. Variants of explicitly deflated IDR(s) are compared to IDR(s) in which the IDR deflation subspace matrix is augmented with “traditional” deflation vectors. The theoretical results are illustrated by numerical experiments.

iterative methods
numerical linear algebra
nonsymmetric linear systems
IDR(s)
deflation

http://resolver.tudelft.nl/uuid:1f831163-38b5-4d18-9497-5bc41e3b25f3

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, 10-21

Institutional Repository

report

(c)2010 Collignon, T.P., Sleijpen, G.L.G., Van Gijzen, M.B.