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Parallel preconditioning for time-dependent PDEs and PDE control

By Andy Wathen

Appears in collection : Parallel Solution Methods for Systems Arising from PDEs / Méthodes parallèles pour la résolution de systèmes issus d'équations aux dérivées partielles

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation. This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated. The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method. This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)

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Citation data

  • DOI 10.24350/CIRM.V.19561603
  • Cite this video Wathen Andy (9/16/19). Parallel preconditioning for time-dependent PDEs and PDE control. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19561603
  • URL https://dx.doi.org/10.24350/CIRM.V.19561603

Bibliography

  • MCDONALD, Eleanor, PESTANA, Jennifer, et WATHEN, Andy. Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM Journal on Scientific Computing, 2018, vol. 40, no 2, p. A1012-A1033. - https://doi.org/10.1137/16M1062016
  • PESTANA, Jennifer et WATHEN, Andrew J. A preconditioned MINRES method for nonsymmetric Toeplitz matrices. SIAM Journal on Matrix Analysis and Applications, 2015, vol. 36, no 1, p. 273-288. - https://doi.org/10.1137/140974213
  • GODDARD, Anthony et WATHEN, Andrew. A note on parallel preconditioning for all-at-once evolutionary PDEs. arXiv preprint arXiv:1810.00615, 2018. - https://arxiv.org/abs/1810.00615
  • WATHEN, Andy J. Preconditioning. Acta Numerica, 2015, vol. 24, p. 329-376. - https://doi.org/10.1017/S0962492915000021

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