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Band-Toeplitz preconditioners for ill-conditioned Toeplitz systems

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Abstract

Preconditioning for Toeplitz systems has been an active research area over the past few decades. Along this line of research, circulant preconditioners have been recently proposed for the Toeplitz-like system arising from discretizing fractional diffusion equations. A common approach is to combine a circulant preconditioner with the preconditioned conjugate gradient normal residual (PCGRN) method for the coefficient system. In this work, instead of using PCGRN for the normal equation system, we propose a simple yet effective preconditioning approach for solving the original system using the preconditioned minimal residual (PMINRES) method that can achieve convergence guarantees depending only on eigenvalues. Namely, for a large class of ill-conditioned Toeplitz systems, we propose a number of preconditioners that attain the overall \({\mathcal {O}}(n\log {n})\) complexity. We first symmetrize the given Toeplitz system by using a permutation matrix and construct a band-Toeplitz plus circulant preconditioner for the modified system. Then, under certain assumptions, we show that the eigenvalues of the preconditioned system are clustered around \(\pm 1\) except a number of outliers and hence superlinear convergence rate of PMINRES can be achieved. Particularly, we indicate that our solver can be applied to solve certain fractional diffusion equations. An extension of this work to the block Toeplitz case is also included. Numerical examples are provided to demonstrate the effectiveness of our proposed method.

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Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments that improved the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Sean Hon

  2. Department of Science and High Technology, University of Insubria, 22100, Como, Italy

    Stefano Serra-Capizzano

  3. Division of Scientific Computing, Department of Information Technology, ITC, Uppsala University, 751 05, Uppsala, Sweden

    Stefano Serra-Capizzano

  4. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK

    Andy Wathen

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  1. Sean Hon
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Correspondence to Sean Hon.

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Communicated by Lothar Reichel.

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The work of Hon was supported in part by the Hong Kong RGC under Grant 22300921 and a start-up allowance from the Croucher Foundation.

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Hon, S., Serra-Capizzano, S. & Wathen, A. Band-Toeplitz preconditioners for ill-conditioned Toeplitz systems. Bit Numer Math 62, 465–491 (2022). https://doi.org/10.1007/s10543-021-00889-6

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