Scientific Computation and Applied & Industrial Mathematics: Andy Wathen
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Such comments apply quite generally, however there is one class of nonsymmetric matrices for which we have recently been able to rigorously prove descriptive convergence bounds, namely real Toeplitz (constant diagonal) matrices. Our results apply regardless of non-normality or any `degree' of nonsymmetry.
Gil Strang proposed the use of circulant matrices (and the FFT) for preconditioning symmetric Toeplitz matrix systems in 1986 and there is now a well-developed theory which guarantees rapid convergence of the conjugate gradient method for such preconditioned positive definite symmetric systems.
In this talk we describe our recent approach which provides a preconditioned MINRES method with the same guarantees for real nonsymmetric Toeplitz systems regardless of the non-normality, and demonstrate the application of the approach for time-dependent PDE problems.
This is joint work with Jennifer Pestana (Strathclyde University) and Elle McDonald (Oxford University).
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Andy Wathen, Oxford University