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Comments on the GMRES convergence for preconditioned systems. (English) Zbl 1229.65063

Lirkov, Ivan (ed.) et al., Large-scale scientific computing. 6th international conference, LSSC 2007, Sozopol, Bulgaria, June 5–9, 2007. Revised papers. Berlin: Springer (ISBN 978-3-540-78825-6/pbk). Lecture Notes in Computer Science 4818, 40-51 (2008).
Summary: The purpose of this paper is to comment a frequent observation by the engineers studying acoustic scattering. It is related to the convergence of the GMRES method when solving systems \(Ax = b\) with \(A = I - B\). The paper includes a theorem which expresses the convergence rate when some eigenvalues of \(B\) have modulus larger than one; that rate depends on the rate measured when solving the system obtained by spectral projection onto the invariant subspace corresponding to the other eigenvalues. The conclusion of the theorem is illustrated on the Helmholtz equation.
For the entire collection see [Zbl 1157.65003].

MSC:

65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F25 Orthogonalization in numerical linear algebra
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