Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries
MathematicS In Action, Volume 11 (2022) no. 1, pp. 61-71.

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.

Published online:
DOI: 10.5802/msia.17
Classification: 00X99
Keywords: Example, Applied mathematics, Journal
Victorita Dolean 1; Frédéric Hecht 2; Pierre Jolivet 3; Frédéric Nataf 2; Pierre-Henri Tournier 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{MSIA_2022__11_1_61_0,
     author = {Victorita Dolean and Fr\'ed\'eric Hecht and Pierre Jolivet and Fr\'ed\'eric Nataf and Pierre-Henri Tournier},
     title = {Recent {Advances} in {Adaptive} {Coarse} {Spaces} and {Availability} in {Open} {Source} {Libraries}},
     journal = {MathematicS In Action},
     pages = {61--71},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {11},
     number = {1},
     year = {2022},
     doi = {10.5802/msia.17},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.17/}
}
TY  - JOUR
AU  - Victorita Dolean
AU  - Frédéric Hecht
AU  - Pierre Jolivet
AU  - Frédéric Nataf
AU  - Pierre-Henri Tournier
TI  - Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries
JO  - MathematicS In Action
PY  - 2022
DA  - 2022///
SP  - 61
EP  - 71
VL  - 11
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.17/
UR  - https://doi.org/10.5802/msia.17
DO  - 10.5802/msia.17
LA  - en
ID  - MSIA_2022__11_1_61_0
ER  - 
%0 Journal Article
%A Victorita Dolean
%A Frédéric Hecht
%A Pierre Jolivet
%A Frédéric Nataf
%A Pierre-Henri Tournier
%T Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries
%J MathematicS In Action
%D 2022
%P 61-71
%V 11
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://doi.org/10.5802/msia.17
%R 10.5802/msia.17
%G en
%F MSIA_2022__11_1_61_0
Victorita Dolean; Frédéric Hecht; Pierre Jolivet; Frédéric Nataf; Pierre-Henri Tournier. Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries. MathematicS In Action, Volume 11 (2022) no. 1, pp. 61-71. doi : 10.5802/msia.17. https://msia.centre-mersenne.org/articles/10.5802/msia.17/

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

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

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

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

[6] Richard Butler; Tim Dodwell; Anne Reinarz; Anhad Sandhu; Robert Scheichl; Linus Seelinger 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] Xiao-Chuan Cai; Marcus Sarkis A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., Volume 21 (1999), pp. 239-247 | MR | Zbl

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

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

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

[11] Victorita Dolean; Pierre-Henri Tournier; Pierre Jolivet; Stéphane Operto 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] Evridiki Efstathiou; Martin J. Gander Why restricted additive Schwarz converges faster than additive Schwarz, BIT, Volume 43 (2003), pp. 945-959 | DOI | MR | Zbl

[13] Jocelyne Erhel; Frédéric Guyomarc’h 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] Yogi A. Erlangga; Reinhard Nabben 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] Martin J. Gander Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255 | MR | Zbl

[16] Pierre Gosselet; Daniel Rixen; François-Xavier Roux; Nicole Spillane 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] Loïc Gouarin; Nicole Spillane Fully algebraic domain decomposition preconditioners with adaptive spectral bounds (2021) (https://arxiv.org/abs/2106.10913)

[18] Ivan G. Graham; Euan A. Spence; Jun Zou 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] Michael Griebel; Peter Oswald On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., Volume 70 (1995) no. 2, pp. 163-180 | DOI | MR | Zbl

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

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

[23] Pierre Jolivet; Jose E. Roman; Stefano Zampini 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] Andrew V. Knyazev; Merico E. Argentati; Ilya Lashuk; Evgueni E. Ovtchinnikov 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] Pierre-Louis Lions On the Schwarz alternating method. II., Domain Decomposition Methods (1989), pp. 47-70

[26] Pierre-Louis Lions On the Schwarz alternating method. III: A variant for nonoverlapping subdomains, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (1990) | Zbl

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

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

[29] Pierre Marchand; Xavier Claeys; Pierre Jolivet; Frédéric Nataf; Pierre-Henri Tournier 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] Frédéric Nataf Mathematical Analysis of Robustness of Two-Level Domain Decomposition Methods with respect to Inexact Coarse Solves, Numer. Math. (2020) | MR | Zbl

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

[32] Frédéric Nataf; Hua Xiang; Victorita Dolean; Nicole Spillane 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] Sergey V. Nepomnyaschikh 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] Roy A. Nicolaides 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] Michael L. Parks; Eric de Sturler; Greg Mackey; Duane D. Johnson; Spandan Maiti Recycling Krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput., Volume 28 (2006) no. 5, pp. 1651-1674 | DOI | MR | Zbl

[36] Florian Rathgeber; David A. Ham; Lawrence Mitchell; Michael Lange; Fabio Luporini; Andrew T. T. McRae; Gheorghe-Teodor Bercea; Graham R. Markall; Paul H. J. Kelly 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] Yousef Saad Analysis of augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., Volume 18 (1997) no. 2, pp. 435-449 | DOI | MR | Zbl

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

[39] Nicole Spillane; Victorita Dolean; Patrice Hauret; Frédéric Nataf; Clemens Pechstein; Robert Scheichl 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] Nicole Spillane; Daniel Rixen 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] Jok M. Tang; Reinhard Nabben; Cornelis Vuik; Yogi A. Erlangga 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] Pierre-Henri Tournier; Marcella Bonazzoli; Victorita Dolean; Francesca Rapetti; Frédéric Hecht; Frédéric Nataf; Iannis Aliferis; Ibtissam El Kanfoud; Claire Migliaccio; Maya De Buhan 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] Pierre-Henri Tournier; Frédéric Nataf FFDDM: FreeFem Domain Decomposition Methd, 2019 (https://doc.freefem.org/documentation/ffddm/index.html)

Cited by Sources: