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

This talk focuses on challenges that we address when designing linear solvers that aim at achieving scalability on large scale computers, while also preserving numerical robustness. We will consider preconditioned Krylov subspace solvers. Getting scalability relies on reducing global synchronizations between processors, while also increasing the arithmetic intensity on one processor. Achieving robustness relies on ensuring that the condition number of the preconditioned matrix is bounded. We will discuss two different approaches for this. The first approach relies on enlarged Krylov subspace methods that aim at computing an enlarged subspace and obtain a faster convergence of the iterative method. The second approach relies on a multilevel Schwarz preconditioner, a multilevel extension of the GenEO preconditioner, that is basedon constructing robustly a hierarchy of coarse spaces. Numerical results on large scale computers, in particular for linear systems arising from solving linear elasticity problems, will discuss the efficiency of the proposed methods.

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Citation data

  • DOI 10.24350/CIRM.V.19561203
  • Cite this video Grigori Laura (9/18/19). Challenges in achieving scalable and robust linear solvers. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19561203
  • URL https://dx.doi.org/10.24350/CIRM.V.19561203

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Bibliography

  • H. Al Daas and L. Grigori. A class of efficient locally constructed preconditioners based on coarse spaces. SIAM Journal on Matrix Analysis and Applications, 40, pp. 66–91, 2019. - https://doi.org/10.1137/18M1194365
  • H. Al Daas, L. Grigori, P. Jolivet, P. H. Tournier. A multilevel Schwarz preconditioner based on a hierarchy of robust coarse spaces. Tech report hal-02151184, 2019. - https://hal.archives-ouvertes.fr/hal-02151184/
  • L. Grigori, S. Moufawad, and F. Nataf. Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication. SIAM Journal on Scientific Computing, 37(2):744–773, 2016. - https://doi.org/10.1137/140989492
  • L. Grigori and O. Tissot. Scalable linear solvers based on enlarged Krylov subspaces with dynamic reduction of search directions. SIAM Journal on Scientific Computing, in press, 2019. - https://hal.inria.fr/hal-01828521/
  • N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numerische Mathematik, 126, pp. 741–770, 2014. - http://dx.doi.org/10.1007/s00211-013-0576-y

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