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Challenges in achieving scalable and robust linear solvers
Post-edited
Authors : Grigori, Laura (Author of the conference)
CIRM (Publisher )
65F10 - Iterative methods for linear systems
65N55 - Multigrid methods; domain decomposition (BVP of PDE)
65F08 - Preconditioners for iterative methods
Additional resources :
https://www.cirm-math.fr/RepOrga/2064/Slides/LGrigori.pdf
CIRM (Publisher )
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| domain decomposition two level preconditioners condition number algebraic setting multilevel methods parallel performance enlarged Krylov methods |
Abstract :
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.
Keywords : preconditioned Krylov subspace solvers; domain decomposition; multilevel methods; subspace correction; condition number; parallel performance; enlarged Krylov methods
MSC Codes : Keywords : preconditioned Krylov subspace solvers; domain decomposition; multilevel methods; subspace correction; condition number; parallel performance; enlarged Krylov methods
65F10 - Iterative methods for linear systems
65N55 - Multigrid methods; domain decomposition (BVP of PDE)
65F08 - Preconditioners for iterative methods
Additional resources :
https://www.cirm-math.fr/RepOrga/2064/Slides/LGrigori.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 08/10/2019 Conference Date : 18/09/2019 Subseries : Research talks arXiv category : Numerical Analysis Mathematical Area(s) : Numerical Analysis & Scientific Computing ; Computer Science Format : MP4 (.mp4) - HD Video Time : 00:55:43 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-09-18_Grigory.mp4 
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Information on the Event
Event Title : Parallel Solution Methods for Systems Arising from PDEs / Méthodes parallèles pour la résolution de systèmes issus d'équations aux dérivées partiellesEvent Organizers : Dolean, Victorita ; Spillane, Nicole ; Szyld, Daniel Dates : 16/09/2019 - 20/09/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/2064.html
Citation Data
DOI : 10.24350/CIRM.V.19561203Cite this video as: Grigori, Laura (2019). Challenges in achieving scalable and robust linear solvers. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19561203 URI : http://dx.doi.org/10.24350/CIRM.V.19561203 |
See Also
- [Multi angle] Multigrid and domain decomposition: similarities and differences / Author of the conference Gander, Martin.
- [Multi angle] Generalised finite elements: domain decomposition, optimal local approximation, reduced order modelling / Author of the conference Scheichl, Robert.
- [Multi angle] Parallel preconditioning for time-dependent PDEs and PDE control / Author of the conference Wathen, Andy.
- [Multi angle] Computational methods for large-scale matrix equations and application to PDEs / Author of the conference Simoncini, Valeria.
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
