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Matching Schur Complement Approximations for Certain Saddle-Point Systems

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Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Abstract

The solution of many practical problems described by mathematical models requires approximation methods that give rise to linear(ized) systems of equations, solving which will determine the desired approximation. This short contribution describes a particularly effective solution approach for a certain class of so-called saddle-point linear systems which arises in different contexts.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday.

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Notes

  1. 1.

    In more detail, the multiplication of the terms \(\left ( \gamma ^{-1} A \right ) A^{-1} \left ( \gamma ^{-1} A \right )\) within \(\widehat {S}\) allows one to capture the first term of the exact Schur complement, γ −2A. In a similar way, the multiplication of the terms \(\left ( \alpha ^{-1} B \right ) A^{-1} \left ( \alpha ^{-1} B \right )^T\) within \(\widehat {S}\) leads to the second term of S, that is α −2BA −1B T.

  2. 2.

    The analysis reads the same as presented for Theorem 1, except with a := αA 1∕2x, b := γA −1∕2B x.

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Acknowledgements

Ian Sloan has been a great friend and mentor over many years. Through his mathematical insight, he has derived really useful approximations in different contexts throughout his long career. We wish to express our deep appreciation for his leadership in our field and to acknowledge his encouragement of all those who seek to contribute to it.

The authors are grateful to Shev MacNamara and an anonymous referee for their interesting and helpful comments which have left us with further avenues to consider. JWP was funded for this research by the Engineering and Physical Sciences Research Council (EPSRC) Fellowship EP/M018857/1.

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Authors and Affiliations

  1. University of Edinburgh, Edinburgh, UK

    John W. Pearson

  2. Oxford University, Oxford, UK

    Andy Wathen

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Editors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Josef Dick

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Frances Y. Kuo

  3. Department of Computer Science, Columbia University, New York, USA

    Henryk Woźniakowski

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Pearson, J.W., Wathen, A. (2018). Matching Schur Complement Approximations for Certain Saddle-Point Systems. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_44

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