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.
Mots-clés : Example, Applied mathematics, Journal
@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 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/ 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://msia.centre-mersenne.org/articles/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, Special issue Maths and Industry, Volume 11 (2022) no. 1, pp. 61-71. doi : 10.5802/msia.17. https://msia.centre-mersenne.org/articles/10.5802/msia.17/
[1] PETSc Users Manual (2014) no. ANL-95/11 - Revision 3.5 http://www.mcs.anl.gov/petsc (Technical report)
[2] Efficient Management of Parallelism in Object Oriented Numerical Software Libraries, Modern Software Tools in Scientific Computing (1997), pp. 163-202 | DOI | Zbl
[3] 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] A comparison of coarse spaces for Helmholtz problems in the high frequency regime, 2021 (https://arxiv.org/abs/2012.02678) | Zbl
[5] Two-level DDM preconditioners for positive Maxwell equations, 2020 (https://arxiv.org/abs/2012.02388)
[6] 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] A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., Volume 21 (1999), pp. 239-247 | MR | Zbl
[8] Deflated and augmented Krylov subspace techniques, Numer. Linear Algebra Appl., Volume 4 (1997) no. 1, pp. 43-66 | DOI | MR | Zbl
[9] A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations (2021) (https://arxiv.org/abs/2107.09006)
[10] An Introduction to Domain Decomposition Methods: algorithms, theory and parallel implementation, Society for Industrial and Applied Mathematics, 2015 | DOI
[11] 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] Why restricted additive Schwarz converges faster than additive Schwarz, BIT, Volume 43 (2003), pp. 945-959 | DOI | MR | Zbl
[13] 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] 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] Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255 | MR | Zbl
[16] 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] Fully algebraic domain decomposition preconditioners with adaptive spectral bounds (2021) (https://arxiv.org/abs/2106.10913)
[18] 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] On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., Volume 70 (1995) no. 2, pp. 163-180 | DOI | MR | Zbl
[20] 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] New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-265 | MR | Zbl
[22] HPDDM: High-Performance Unified framework for Domain Decomposition methods, MPI-C++ library, 2014 (https://github.com/hpddm/hpddm)
[23] 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] 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] On the Schwarz alternating method. II., Domain Decomposition Methods (1989), pp. 47-70
[26] 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] Novel design and analysis of generalized FE methods based on locally optimal spectral approximations (2021) (https://arxiv.org/abs/2103.09545)
[28] Balancing domain decomposition, Commun. Numer. Methods Eng., Volume 9 (1992), pp. 233-241 | DOI | MR | Zbl
[29] 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] Mathematical Analysis of Robustness of Two-Level Domain Decomposition Methods with respect to Inexact Coarse Solves, Numer. Math. (2020) | MR | Zbl
[31] A GenEO Domain Decomposition method for Saddle Point problems (2021) (https://arxiv.org/abs/1911.01858)
[32] 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] 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] 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] Recycling Krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput., Volume 28 (2006) no. 5, pp. 1651-1674 | DOI | MR | Zbl
[36] 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] Analysis of augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., Volume 18 (1997) no. 2, pp. 435-449 | DOI | MR | Zbl
[38] Über einen Grenzübergang durch alternierendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Volume 15 (1870), pp. 272-286
[39] 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] 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] 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] 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] FFDDM: FreeFem Domain Decomposition Methd, 2019 (https://doc.freefem.org/documentation/ffddm/index.html)
Cited by Sources: