×

Machine learning in adaptive domain decomposition methods – predicting the geometric location of constraints. (English) Zbl 1429.65065

Summary: Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial differential equations by finite elements. The convergence rate of these methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. A remedy can be obtained by enhancing the coarse space with elements, which are often called constraints, that are computed by solving small eigenvalue problems on portions of the interface of the domain decomposition, i.e., edges in two dimensions or faces and edges in three dimensions. In the present work, without restriction of generality, the focus is on two dimensions. In general, it is difficult to predict where these constraints have to be added, and therefore the corresponding local eigenvalue problems have to be computed, i.e., on which edges. Here, a machine learning based strategy using neural networks is suggested to predict the geometric location of these edges in a preprocessing step. This reduces the number of eigenvalue problems that have to be solved before the iteration. Numerical experiments for model problems and realistic microsections using regular decompositions as well as decompositions from graph partitioners are provided, showing very promising results.

MSC:

65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
68T05 Learning and adaptive systems in artificial intelligence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, https://www.tensorflow.org/, 2015; software available from https://tensorflow.org/.
[2] S. Badia, A. F. Martín, and J. Principe, On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections, Parallel Comput., 50 (2015), pp. 1-24.
[3] S. Badia, A. F. Martín, and J. Principe, Multilevel balancing domain decomposition at extreme scales, SIAM J. Sci. Comput., 38 (2016), pp. C22-C52, https://doi.org/10.1137/15M1013511. · Zbl 1334.65217
[4] L. Beira͂o da Veiga, L. F. Pavarino, S. Scacchi, O. B. Widlund, and S. Zampini, Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners, SIAM J. Sci. Comput., 39 (2017), pp. A281-A302, https://doi.org/10.1137/15M1054675. · Zbl 1360.65090
[5] P. E. Bjørstad, J. Koster, and P. Krzyżanowski, Domain decomposition solvers for large scale industrial finite element problems, in PARA2000 Workshop on Applied Parallel Computing, Lecture Notes in Comput. Sci. 1947, Springer-Verlag, Berlin, Heidelberg, 2000, pp. 373-383.
[6] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd ed., Cambridge University Press, Cambridge, UK, 2001. · Zbl 0976.65099
[7] S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, Berlin, 2002. · Zbl 1012.65115
[8] J. G. Calvo and O. B. Widlund, An adaptive choice of primal constraints for BDDC domain decomposition algorithms, Electron. Trans. Numer. Anal., 45 (2016), pp. 524-544. · Zbl 1357.65295
[9] C. R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25 (2003), pp. 246-258, https://doi.org/10.1137/S1064827502412887. · Zbl 1038.65039
[10] C. R. Dohrmann, A. Klawonn, and O. B. Widlund, Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions, SIAM J. Numer. Anal., 46 (2008), pp. 2153-2168, https://doi.org/10.1137/070685841. · Zbl 1183.65160
[11] C. R. Dohrmann, A. Klawonn, and O. B. Widlund, A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners, in Domain Decomposition Methods in Science and Engineering XVII, Lect. Notes Comput. Sci. Eng. 60, Springer, Berlin, 2008, pp. 247-254. · Zbl 1359.65289
[12] C. R. Dohrmann and O. B. Widlund, An overlapping Schwarz algorithm for almost incompressible elasticity, SIAM J. Numer. Anal., 47 (2009), pp. 2897-2923, https://doi.org/10.1137/080724320. · Zbl 1410.74064
[13] V. Dolean, F. Nataf, R. Scheichl, and N. Spillane, Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps, Comput. Methods Appl. Math., 12 (2012), pp. 391-414. · Zbl 1284.65050
[14] J. Duchi, E. Hazan, and Y. Singer, Adaptive subgradient methods for online learning and stochastic optimization, J. Mach. Learn. Res., 12 (2011), pp. 2121-2159. · Zbl 1280.68164
[15] Y. Efendiev, J. Galvis, R. Lazarov, and J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms, ESAIM Math. Model. Numer. Anal., 46 (2012), pp. 1175-1199. · Zbl 1272.65098
[16] C. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, FETI-DP: A dual-primal unified FETI method. I. A faster alternative to the two-level FETI method, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523-1544. · Zbl 1008.74076
[17] C. Farhat, M. Lesoinne, and K. Pierson, A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7 (2000), pp. 687-714. · Zbl 1051.65119
[18] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), pp. 1461-1483, https://doi.org/10.1137/090751190. · Zbl 1206.76042
[19] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: Reduced dimension coarse spaces, Multiscale Model. Simul., 8 (2010), pp. 1621-1644, https://doi.org/10.1137/100790112. · Zbl 1381.65029
[20] M. J. Gander, A. Loneland, and T. Rahman, Analysis of a New Harmonically Enriched Multiscale Coarse Space for Domain Decomposition Methods, preprint, https://arxiv.org/abs/1512.05285, 2015.
[21] X. Glorot, A. Bordes, and Y. Bengio, Deep sparse rectifier neural networks, in Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, 2011, pp. 315-323.
[22] I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning, Vol. 1, MIT Press, Cambridge, MA, 2016. · Zbl 1373.68009
[23] A. Heinlein, Parallel Overlapping Schwarz Preconditioners and Multiscale Discretizations with Applications to Fluid-Structure Interaction and Highly Heterogeneous Problems, PhD thesis, Universität zu Köln, Cologne, Germany, 2016.
[24] A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach, An adaptive GDSW coarse space for two-level overlapping Schwarz methods in two dimensions, in Domain Decomposition Methods in Science and Engineering XXIV, Springer, Cham, pp. 373-382. · Zbl 1425.65045
[25] A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach, Adaptive GDSW coarse spaces for overlapping Schwarz methods in three dimensions, SIAM J. Sci. Comput., 41 (2019), pp. A3045-A3072, https://doi.org/10.1137/18M1220613. · Zbl 1425.65045
[26] A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach, Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D, Electron. Trans. Numer. Anal., 48 (2018), pp. 156-182. · Zbl 1448.65263
[27] A. Heinlein, A. Klawonn, M. Lanser, and J. Weber, A Generalized Robust FETI-DP Coarse Space for Heterogeneous Problems and Arbitrary Scalings, manuscript. · Zbl 1460.65136
[28] A. Heinlein, A. Klawonn, and O. Rheinbach, A parallel implementation of a two-level overlapping Schwarz method with energy-minimizing coarse space based on Trilinos, SIAM J. Sci. Comput., 38 (2016), pp. C713-C747, https://doi.org/10.1137/16M1062843. · Zbl 1355.65168
[29] A. Heinlein, A. Klawonn, O. Rheinbach, and O. Widlund, Improving the parallel performance of overlapping Schwarz methods by using a smaller energy minimizing coarse space, in Domain Decomposition Methods in Science and Engineering XXIV, Springer, Cham, 2018, pp. 383-392. · Zbl 1450.65165
[30] M. Jarošová, A. Klawonn, and O. Rheinbach, Projector preconditioning and transformation of basis in FETI-DP algorithms for contact problems, Math. Comput. Simulation, 82 (2012), pp. 1894-1907. · Zbl 1254.90147
[31] K. Jarrett, K. Kavukcuoglu, M. Ranzato, and Y. LeCun, What is the best multi-stage architecture for object recognition?, in Proceedings of the 12th International IEEE Conference on Computer Vision, 2009, pp. 2146-2153.
[32] G. Karypis and V. Kumar, MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0.
[33] H. H. Kim and E. T. Chung, A BDDC algorithm with enriched coarse spaces for two-dimensional elliptic problems with oscillatory and high contrast coefficients, Multiscale Model. Simul., 13 (2015), pp. 571-593, https://doi.org/10.1137/140970598. · Zbl 1317.65090
[34] D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, preprint, https://arxiv.org/abs/1412.6980, 2014.
[35] A. Klawonn, M. Kühn, and O. Rheinbach, Adaptive coarse spaces for FETI-DP in three dimensions, SIAM J. Sci. Comput., 38 (2016), pp. A2880-A2911, https://doi.org/10.1137/15M1049610. · Zbl 1346.74168
[36] A. Klawonn, M. Kühn, and O. Rheinbach, Adaptive coarse spaces for FETI-DP in three dimensions with applications to heterogeneous diffusion problems, in Domain Decomposition Methods in Science and Engineering XXIII, Lect. Notes Comput. Sci. Eng. 116, Springer, Cham, 2017, pp. 187-196. · Zbl 1367.65186
[37] A. Klawonn, M. Kühn, and O. Rheinbach, Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems, Electron. Trans. Numer. Anal., 49 (2018), pp. 1-27. · Zbl 1448.65236
[38] A. Klawonn, M. Lanser, and O. Rheinbach, Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations, SIAM J. Sci. Comput., 37 (2015), pp. C667-C696, https://doi.org/10.1137/140997907. · Zbl 1329.65294
[39] A. Klawonn, M. Lanser, and O. Rheinbach, A highly scalable implementation of inexact nonlinear FETI-DP without sparse direct solvers, in Numerical Mathematics and Advanced Applications–ENUMATH 2015, Lect. Notes Comput. Sci. Eng. 112, Springer, Cham, 2016, pp. 255-264. · Zbl 1352.65627
[40] A. Klawonn, M. Lanser, and O. Rheinbach, Nonlinear BDDC methods with approximate solvers, Electron. Trans. Numer. Anal., 49 (2018), pp. 244-273, https://doi.org/10.1553/etna_vol49s244. · Zbl 1410.65473
[41] A. Klawonn, M. Lanser, O. Rheinbach, and M. Uran, Nonlinear FETI-DP and BDDC methods: A unified framework and parallel results, SIAM J. Sci. Comput., 39 (2017), pp. C417-C451, https://doi.org/10.1137/16M1102495. · Zbl 1377.65163
[42] A. Klawonn, P. Radtke, and O. Rheinbach, FETI-DP methods with an adaptive coarse space, SIAM J. Numer. Anal., 53 (2015), pp. 297-320, https://doi.org/10.1137/130939675. · Zbl 1327.65063
[43] A. Klawonn, P. Radtke, and O. Rheinbach, A comparison of adaptive coarse spaces for iterative substructuring in two dimensions, Electron. Trans. Numer. Anal., 45 (2016), pp. 75-106. · Zbl 1338.65084
[44] A. Klawonn and O. Rheinbach, Robust FETI-DP methods for heterogeneous three dimensional elasticity problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1400-1414. · Zbl 1173.74428
[45] A. Klawonn and O. Rheinbach, Highly scalable parallel domain decomposition methods with an application to biomechanics, ZAMM Z. Angew. Math. Mech., 90 (2010), pp. 5-32. · Zbl 1355.65169
[46] A. Klawonn and O. Rheinbach, Deflation, projector preconditioning, and balancing in iterative substructuring methods: Connections and new results, SIAM J. Sci. Comput., 34 (2012), pp. A459-A484, https://doi.org/10.1137/100811118. · Zbl 1248.65129
[47] A. Klawonn, O. Rheinbach, and O. B. Widlund, An analysis of a FETI-DP algorithm on irregular subdomains in the plane, SIAM J. Numer. Anal., 46 (2008), pp. 2484-2504, https://doi.org/10.1137/070688675. · Zbl 1176.65135
[48] A. Klawonn and O. B. Widlund, Dual-primal FETI methods for linear elasticity, Comm. Pure Appl. Math., 59 (2006), pp. 1523-1572. · Zbl 1110.74053
[49] A. Klawonn, O. B. Widlund, and M. Dryja, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40 (2002), pp. 159-179, https://doi.org/10.1137/S0036142901388081. · Zbl 1032.65031
[50] J. Li and O. B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg., 66 (2006), pp. 250-271.
[51] J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10 (2003), pp. 639-659. · Zbl 1071.65558
[52] J. Mandel, C. R. Dohrmann, and R. Tezaur, An algebraic theory for primal and dual substructuring methods by constraints, Appl. Numer. Math., 54 (2005), pp. 167-193. · Zbl 1076.65100
[53] J. Mandel and B. Sousedík, Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1389-1399. · Zbl 1173.74435
[54] J. Mandel, B. Sousedík, and J. Sístek, Adaptive BDDC in three dimensions, Math. Comput. Simulation, 82 (2012), pp. 1812-1831. · Zbl 1255.65225
[55] J. Mandel and R. Tezaur, On the convergence of a dual-primal substructuring method, Numer. Math., 88 (2001), pp. 543-558. · Zbl 1003.65126
[56] A. Müller and S. Guido, Introduction to Machine Learning with Python: A Guide for Data Scientists, O’Reilly Media, Sebastopol, CA, 2016.
[57] V. Nair and G. E. Hinton, Rectified linear units improve restricted Boltzmann machines, in Proceedings of the 27th International Conference on Machine Learning (ICML-10), 2010, pp. 807-814.
[58] D.-S. Oh, O. B. Widlund, S. Zampini, and C. R. Dohrmann, BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields, Math. Comp., 87 (2018), pp. 659-692. · Zbl 1380.65065
[59] C. Pechstein and C. R. Dohrmann, A unified framework for adaptive BDDC, Electron. Trans. Numer. Anal., 46 (2017), pp. 273-336. · Zbl 1368.65043
[60] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12 (2011), pp. 2825-2830. · Zbl 1280.68189
[61] S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning, Cambridge University Press, Cambridge, UK, 2014. · Zbl 1305.68005
[62] 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, Numer. Math., 126 (2014), pp. 741-770. · Zbl 1291.65109
[63] N. Spillane and D. J. Rixen, Automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms, Internat. J. Numer. Methods Engrg., 95 (2013), pp. 953-990. · Zbl 1352.65553
[64] A. Toselli and O. Widlund, Domain decomposition methods–algorithms and theory, Springer Ser. Comput. Math. 34, Springer-Verlag, Berlin, 2005. · Zbl 1069.65138
[65] J. Watt, R. Borhani, and A. K. Katsaggelos, Machine Learning Refined: Foundations, Algorithms, and Applications, Cambridge University Press, Cambridge, UK, 2016. · Zbl 1376.68004
[66] S. Zampini, PCBDDC: A class of robust dual-primal methods in PETSc, SIAM J. Sci. Comput., 38 (2016), pp. S282-S306, https://doi.org/10.1137/15M1025785. · Zbl 1352.65632
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.