Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Dolean, Victorita; Hecht, Frédéric; Jolivet, Pierre; Nataf, Frédéric; Tournier, Pierre-Henri

doi:10.5802/msia.17

Dolean, Victorita ¹ ; Hecht, Frédéric ² ; Jolivet, Pierre ³ ; Nataf, Frédéric ² ; Tournier, Pierre-Henri ²

¹ Université Côte d’Azur-CNRS- Labo Dieudonné, Nice, France
² Sorbonne Université, Université de Paris, Inria Equipe Alpines, Laboratoire Jacques-Louis Lions, F-75005 Paris, France
³ IRIT, CNRS, F-31071 Toulouse, France

MathematicS In Action, Tome 11 (2022) no. 1, pp. 61-71.

Résumé

Scalability of parallel solvers for problems with high heterogeneities relies on adaptive coarse spaces built from generalized eigenvalue problems in the subdomains. The corresponding theory is powerful and flexible but the development of an efficient parallel implementation is challenging. We report here on recent advances in adaptive coarse spaces and on their open source implementations.

Publié le : 2022-09-27

DOI : 10.5802/msia.17

Classification : 00X99
Mots clés : Example, Applied mathematics, Journal

Affiliations des auteurs :

Dolean, Victorita ¹ ; Hecht, Frédéric ² ; Jolivet, Pierre ³ ; Nataf, Frédéric ² ; Tournier, Pierre-Henri ²

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Dolean, Victorita; Hecht, Frédéric; Jolivet, Pierre; Nataf, Frédéric; Tournier, Pierre-Henri. Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries. MathematicS In Action, Tome 11 (2022) no. 1, pp. 61-71. doi : 10.5802/msia.17. http://www.numdam.org/articles/10.5802/msia.17/

Bibliographie
Cité par

[1] Balay, Satish; Abhyankar, Shrirang; Adams, Mark F.; Brown, Jed; Brune, Peter; Buschelman, Kris; Eijkhout, Victor; Gropp, William D.; Kaushik, Dinesh; Knepley, Matthew G.; McInnes, Lois Curfman; Rupp, Karl; Smith, Barry F.; Zhang, Hong PETSc Users Manual (2014) no. ANL-95/11 - Revision 3.5 http://www.mcs.anl.gov/petsc (Technical report)

[2] Balay, Satish; Gropp, William D.; McInnes, Lois Curfman; Smith, Barry F. Efficient Management of Parallelism in Object Oriented Numerical Software Libraries, Modern Software Tools in Scientific Computing (Arge, E.; Bruaset, A. M.; Langtangen, H. P., eds.), Birkhäuser (1997), pp. 163-202 | DOI | Zbl

[3] Bastian, Peter; Heimann, Felix; Marnach, Sven Generic implementation of finite element methods in the distributed and unified numerics environment (DUNE), Kybernetika, Volume 46 (2010) no. 2, pp. 294-315 | MR | Zbl

[4] Bootland, Niall; Dolean, Victorita; Jolivet, Pierre; Tournier, Pierre-Henri A comparison of coarse spaces for Helmholtz problems in the high frequency regime, 2021 (https://arxiv.org/abs/2012.02678) | Zbl

[5] Bootland, Niall; Dolean, Victorita; Nataf, Frédéric; Tournier, Pierre-Henri Two-level DDM preconditioners for positive Maxwell equations, 2020 (https://arxiv.org/abs/2012.02388)

[6] Butler, Richard; Dodwell, Tim; Reinarz, Anne; Sandhu, Anhad; Scheichl, Robert; Seelinger, Linus High-performance Dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites, Comput. Phys. Commun., Volume 249 (2020), p. 106997 | DOI | MR

[7] Cai, Xiao-Chuan; Sarkis, Marcus A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., Volume 21 (1999), pp. 239-247 | MR | Zbl

[8] Chapman, Andrew; Saad, Yousef Deflated and augmented Krylov subspace techniques, Numer. Linear Algebra Appl., Volume 4 (1997) no. 1, pp. 43-66 | DOI | MR | Zbl

[9] Daas, Hussam Al; Jolivet, Pierre; Scott, Jennifer A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations (2021) (https://arxiv.org/abs/2107.09006)

[10] Dolean, Victorita; Jolivet, Pierre; Nataf, Frédéric An Introduction to Domain Decomposition Methods: algorithms, theory and parallel implementation, Society for Industrial and Applied Mathematics, 2015 | DOI

[11] Dolean, Victorita; Tournier, Pierre-Henri; Jolivet, Pierre; Operto, Stéphane Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners, SEG Technical Program Expanded Abstracts 2020, Society of Exploration Geophysicists, 2020, pp. 2683-2688 | DOI

[12] Efstathiou, Evridiki; Gander, Martin J. Why restricted additive Schwarz converges faster than additive Schwarz, BIT, Volume 43 (2003), pp. 945-959 | DOI | MR | Zbl

[13] Erhel, Jocelyne; Guyomarc’h, Frédéric An augmented conjugate gradient method for solving consecutive symmetric positive definite linear systems, SIAM J. Matrix Anal. Appl., Volume 21 (2000) no. 4, pp. 1279-1299 | DOI | MR | Zbl

[14] Erlangga, Yogi A.; Nabben, Reinhard Deflation and balancing preconditioners for Krylov subspace methods applied to nonsymmetric matrices, SIAM J. Matrix Anal. Appl., Volume 30 (2008) no. 2, pp. 684-699 | DOI | MR | Zbl

[15] Gander, Martin J. Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255 | MR | Zbl

[16] Gosselet, Pierre; Rixen, Daniel; Roux, François-Xavier; Spillane, Nicole Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions, Int. J. Numer. Meth. Engng., Volume 104 (2015) no. 10, pp. 905-927 | DOI | MR | Zbl

[17] Gouarin, Loïc; Spillane, Nicole Fully algebraic domain decomposition preconditioners with adaptive spectral bounds (2021) (https://arxiv.org/abs/2106.10913)

[18] Graham, Ivan G.; Spence, Euan A.; Zou, Jun Domain Decomposition with local impedance conditions for the Helmholtz equation with absorption, SIAM J. Numer. Anal., Volume 58 (2020) no. 5, pp. 2515-2543 | DOI | MR | Zbl

[19] Griebel, Michael; Oswald, Peter On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., Volume 70 (1995) no. 2, pp. 163-180 | DOI | MR | Zbl

[20] Haferssas, Ryadh; Jolivet, Pierre; Nataf, Frédéric An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method, SIAM J. Sci. Comput., Volume 39 (2017) no. 4, p. A1345-A1365 | DOI | MR

[21] Hecht, Frédéric New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-265 | MR | Zbl

[22] Jolivet, Pierre; Nataf, Frédéric HPDDM: High-Performance Unified framework for Domain Decomposition methods, MPI-C++ library, 2014 (https://github.com/hpddm/hpddm)

[23] Jolivet, Pierre; Roman, Jose E.; Zampini, Stefano KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners, Comput. Math. Appl., Volume 84 (2021), pp. 277-295 | DOI | MR | Zbl

[24] Knyazev, Andrew V.; Argentati, Merico E.; Lashuk, Ilya; Ovtchinnikov, Evgueni E. Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc, SIAM J. Sci. Comput., Volume 29 (2007) no. 5, pp. 2224-2239 | DOI | MR | Zbl

[25] Lions, Pierre-Louis On the Schwarz alternating method. II., Domain Decomposition Methods (Chan, Tony; Glowinski, Roland; Périaux, Jacques; Widlund, Olof, eds.), Society for Industrial and Applied Mathematics (1989), pp. 47-70

[26] Lions, Pierre-Louis On the Schwarz alternating method. III: A variant for nonoverlapping subdomains, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Chan, Tony F.; Glowinski, Roland; Périaux, Jacques; Widlund, Olof, eds.), Society for Industrial and Applied Mathematics (1990) | Zbl

[27] Ma, Chupeng; Scheichl, Robert; Dodwell, Tim Novel design and analysis of generalized FE methods based on locally optimal spectral approximations (2021) (https://arxiv.org/abs/2103.09545)

[28] Mandel, Jan Balancing domain decomposition, Commun. Numer. Methods Eng., Volume 9 (1992), pp. 233-241 | DOI | MR | Zbl

[29] Marchand, Pierre; Claeys, Xavier; Jolivet, Pierre; Nataf, Frédéric; Tournier, Pierre-Henri Two-level preconditioning for the h-version boundary element approximation of hypersingular operator with GenEO, Numer. Math., Volume 146 (2020) no. 3, pp. 597-628 | DOI | MR | Zbl

[30] Nataf, Frédéric Mathematical Analysis of Robustness of Two-Level Domain Decomposition Methods with respect to Inexact Coarse Solves, Numer. Math. (2020) | MR | Zbl

[31] Nataf, Frédéric; Tournier, Pierre-Henri A GenEO Domain Decomposition method for Saddle Point problems (2021) (https://arxiv.org/abs/1911.01858)

[32] Nataf, Frédéric; Xiang, Hua; Dolean, Victorita; Spillane, Nicole A coarse space construction based on local Dirichlet to Neumann maps, SIAM J. Sci. Comput., Volume 33 (2011) no. 4, pp. 1623-1642 | DOI | MR | Zbl

[33] Nepomnyaschikh, Sergey V. Mesh theorems of traces, normalizations of function traces and their inversions, Sov. J. Numer. Anal. Math. Model., Volume 6 (1991), pp. 1-25 | MR | Zbl

[34] Nicolaides, Roy A. Deflation of conjugate gradients with applications to boundary value problems, SIAM J. Numer. Anal., Volume 24 (1987) no. 2, pp. 355-365 | DOI | MR | Zbl

[35] Parks, Michael L.; de Sturler, Eric; Mackey, Greg; Johnson, Duane D.; Maiti, Spandan Recycling Krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput., Volume 28 (2006) no. 5, pp. 1651-1674 | DOI | MR | Zbl

[36] Rathgeber, Florian; Ham, David A.; Mitchell, Lawrence; Lange, Michael; Luporini, Fabio; McRae, Andrew T. T.; Bercea, Gheorghe-Teodor; Markall, Graham R.; Kelly, Paul H. J. Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., Volume 43 (2016) no. 3, pp. 1-27 | DOI | MR | Zbl

[37] Saad, Yousef Analysis of augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., Volume 18 (1997) no. 2, pp. 435-449 | DOI | MR | Zbl

[38] Schwarz, Hermann A. Über einen Grenzübergang durch alternierendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Volume 15 (1870), pp. 272-286

[39] Spillane, Nicole; Dolean, Victorita; Hauret, Patrice; Nataf, Frédéric; Pechstein, Clemens; Scheichl, Robert Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer. Math., Volume 126 (2014) no. 4, pp. 741-770 | DOI | MR

[40] Spillane, Nicole; Rixen, Daniel Automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms, Int. J. Numer. Meth. Engng., Volume 95 (2013) no. 11, pp. 953-990 | DOI | MR | Zbl

[41] Tang, Jok M.; Nabben, Reinhard; Vuik, Cornelis; Erlangga, Yogi A. Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods, J. Sci. Comput., Volume 39 (2009) no. 3, pp. 340-370 | DOI | MR | Zbl

[42] Tournier, Pierre-Henri; Bonazzoli, Marcella; Dolean, Victorita; Rapetti, Francesca; Hecht, Frédéric; Nataf, Frédéric; Aliferis, Iannis; El Kanfoud, Ibtissam; Migliaccio, Claire; De Buhan, Maya et al. Numerical Modeling and High-Speed Parallel Computing: New Perspectives on Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring., IEEE Trans. Antennas Propag., Volume 59 (2017) no. 5, pp. 98-110 | DOI

[43] Tournier, Pierre-Henri; Nataf, Frédéric FFDDM: FreeFem Domain Decomposition Methd, 2019 (https://doc.freefem.org/documentation/ffddm/index.html)

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