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Nonlinear BDDC methods with approximate solvers. (English) Zbl 1410.65473

Summary: New nonlinear BDDC (balancing domain decomposition by constraints) domain decomposition methods using inexact solvers for the subdomains and the coarse problem are proposed. In nonlinear domain decomposition methods, the nonlinear problem is decomposed before linearization to improve concurrency and robustness. For linear problems, the new methods are equivalent to known inexact BDDC methods. The new approaches are therefore discussed in the context of other known inexact BDDC methods for linear problems. Relations are pointed out, and the advantages of the approaches chosen here are highlighted. For the new approaches, using an algebraic multigrid method as a building block, parallel scalability is shown for more than half a million (\(524\,288\)) MPI ranks on the JUQUEEN IBM BG/Q supercomputer (JSC Jülich, Germany) and on up to \(193\,600\) cores of the Theta Xeon Phi supercomputer (ALCF, Argonne National Laboratory, USA), which is based on the recent Intel Knights Landing (KNL) many-core architecture. One of our nonlinear inexact BDDC domain decomposition methods is also applied to three-dimensional plasticity problems. Comparisons to standard Newton-Krylov-BDDC methods are provided.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65Y05 Parallel numerical computation
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
65H10 Numerical computation of solutions to systems of equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

PETSc; BoomerAMG; PCBDDC
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] S. BADIA, A. F. MARTÍN,ANDJ. PRINCIPE, On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections, Parallel Comput., 50 (2015), pp. 1-24.
[2] , Multilevel balancing domain decomposition at extreme scales, SIAM J. Sci. Comput., 38 (2016), pp. C22-C52.
[3] A. H. BAKER, A. KLAWONN, T. KOLEV, M. LANSER, O. RHEINBACH,ANDU. M. YANG, Scalability of classical algebraic multigrid for elasticity to half a million parallel tasks, in Software for Exascale Computing—SPPEXA 2013-2015, H.-J. Bungartz, P. Neumann, and E. W. Nagel, eds., vol. 113 of Lect. Notes Comput. Sci. Eng., Springer, Cham, 2016, pp. 113-140.
[4] A. H. BAKER, T. V. KOLEV,ANDU. M. YANG, Improving algebraic multigrid interpolation operators for linear elasticity problems, Numer. Linear Algebra Appl., 17 (2010), pp. 495-517. · Zbl 1240.74027
[5] S. BALAY, W. D. GROPP, L. C. MCINNES,ANDB. F. SMITH, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser, Boston, 1997, pp. 163-202. · Zbl 0882.65154
[6] F. BORDEU, P.-A. BOUCARD,ANDP. GOSSELET, Balancing domain decomposition with nonlinear relocalETNA Kent State University and Johann Radon Institute (RICAM) 272A. KLAWONN, M. LANSER, AND O. RHEINBACH ization: parallel implementation for laminates, in Proc. First Int. Conf. on Parallel, Distributed and Grid Computing for Engineering, B. H. V. Topping and P. Iványi, eds., Civil-Comp Press, Stirlingshire, 2009, Paper 4.
[7] D. BRANDS, D. BALZANI, L. SCHEUNEMANN, J. SCHRÖDER, H. RICHTER,ANDD. RAABE, Computational modeling of dual-phase steels based on representative three-dimensional microstructures obtained from ebsd data, Arch. Appl. Mech., 86 (2016), pp. 575-598.
[8] P. R. BRUNE, M. G. KNEPLEY, B. F. SMITH,ANDX. TU, Composing scalable nonlinear algebraic solvers, SIAM Rev., 57 (2015), pp. 535-565. · Zbl 1336.65030
[9] X.-C. CAI ANDD. E. KEYES, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183-200. · Zbl 1015.65058
[10] X.-C. CAI, D. E. KEYES,ANDL. MARCINKOWSKI, Non-linear additive Schwarz preconditioners and application in computational fluid dynamics, Internat. J. Numer. Methods Fluids, 40 (2002), pp. 1463– 1470. · Zbl 1025.76040
[11] J.-M. CROS, A preconditioner for the Schur complement domain decomposition method, in Domain Decomposition Methods in Science and Engineering, I. Herrera, D. Keyes, O. B. Widlund, and R. Yates, eds., National Autonomous University of Mexico (UNAM), Mexico City, 2003, pp. 373-380.
[12] T. A. DAVIS, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), pp. 167-195. · Zbl 1072.65036
[13] H. DESTERCK, R. D. FALGOUT, J. W. NOLTING,ANDU. M. YANG, Distance-two interpolation for parallel algebraic multigrid, Numer. Linear Algebra Appl., 15 (2008), pp. 115-139. · Zbl 1212.65139
[14] H. DESTERCK, U. M. YANG,ANDJ. J. HEYS, Reducing complexity in parallel algebraic multigrid preconditioners, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 1019-1039. · Zbl 1102.65034
[15] C. R. DOHRMANN, A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25 (2003), pp. 246-258. · Zbl 1038.65039
[16] , An approximate BDDC preconditioner, Numer. Linear Algebra Appl., 14 (2007), pp. 149-168. · Zbl 1199.65088
[17] V. DOLEAN, M. J. GANDER, W. KHERIJI, F. KWOK,ANDR. MASSON, Nonlinear preconditioning: how to use a nonlinear Schwarz method to precondition Newton’s method, SIAM J. Sci. Comput., 38 (2016), pp. A3357-A3380. · Zbl 1352.65326
[18] C. FARHAT, M. LESOINNE,ANDK. PIERSON, A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7 (2000), pp. 687-714. · Zbl 1051.65119
[19] C. GROSS, A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies: Convergence Results and Examples From the field of Nonlinear Elastostatics and Elastodynamics, PhD. Thesis, Math-Nat. Fak., RFW University Bonn, Bonn, 2009.
[20] C. GROSS ANDR. KRAUSE, On the globalization of ASPIN employing trust-region control strategies convergence analysis and numerical examples, Tech. Rep. 2011-03, Inst. Comp. Sci., Universita della Svizzera italiana, Lugano, 2011.
[21] V. E. HENSON ANDU. M. YANG, BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155-177. · Zbl 0995.65128
[22] F.-N. HWANG ANDX.-C. CAI, Improving robustness and parallel scalability of Newton method through nonlinear preconditioning, in Domain Decomposition Methods in Science and Engineering, R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund, and J. Xu, eds., vol. 40 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005, pp. 201-208.
[23] , A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1603-1611. · Zbl 1173.76385
[24] A. KLAWONN, M. LANSER,ANDO. RHEINBACH, Nonlinear FETI-DP and BDDC methods, SIAM J. Sci. Comput., 36 (2014), pp. A737-A765.
[25] , FE2TI (ex_nl/fe2) EXASTEEL - Bridging scales for multiphase steels, Software, 2015. http://www.fz-juelich.de/ias/jsc/EN/Expertise/High-Q-Club/FE2TI/_node.html
[26] , Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations, SIAM J. Sci. Comput., 37 (2015), pp. C667-C696.
[27] , FE2TI: computational scale bridging for dual-phase steels, in Parallel Computing: On the Road to Exascale; Proceedings of ParCo2015, G. R. Joubert, H. Leather, M. Parsons, F. Peters, and M. Sawyer, eds., IOS Series Advances in Parallel Computing, IOS Press, Amsterdam, 2016, pp. 797-806. Also: TUBAF Preprint: 2015-12, TU Bergakademie Freiberg, Freiberg http://tu-freiberg.de/fakult1/forschung/preprints.
[28] , A highly scalable implementation of inexact nonlinear FETI-DP without sparse direct solvers, in Numerical mathematics and advanced applications—ENUMATH 2015, B. Karasözen, M. Manguo˘glu, M. Tezer-Sezgin, S. Göktepe, and Ö. U˘gur, eds., vol. 112 of Lect. Notes Comput. Sci. Eng., Springer, Cham, 2016, pp. 255-264. · Zbl 1352.65627
[29] , Using algebraic multigrid in inexact BDDC domain decomposition methods, in Domain Decomposition Methods in Science and Exngineering XXIV, P. E. Bjørstad, S. C. Brenner, L. Halpern, R. Kornhuber, ETNA Kent State University and Johann Radon Institute (RICAM) NONLINEAR BDDC METHODS WITH INEXACT SOLVERS273 H. H. Kim, T. Rahmann, O. B. Widlund, eds., in vol. 125 of Lect. Notes Comput. Sci. Eng., Springer, Basel, 2017, in press.
[30] A. KLAWONN, M. LANSER, O. RHEINBACH,ANDM. URAN, Nonlinear FETI-DP and BDDC methods: a unified framework and parallel results, SIAM J. Sci. Comput., 39 (2017), pp. C417-C451.
[31] A. KLAWONN, L. F. PAVARINO,ANDO. RHEINBACH, Spectral element FETI-DP and BDDC preconditioners with multi-element subdomains, Comput. Methods Appl. Mech. Engrg., 198 (2008), pp. 511-523. · Zbl 1228.74084
[32] A. KLAWONN ANDO. RHEINBACH, Inexact FETI-DP methods, Internat. J. Numer. Methods Engrg., 69 (2007), pp. 284-307. · Zbl 1194.74420
[33] , Highly scalable parallel domain decomposition methods with an application to biomechanics, ZAMM Z. Angew. Math. Mech., 90 (2010), pp. 5-32. · Zbl 1355.65169
[34] J. LI ANDO. B. WIDLUND, On the use of inexact subdomain solvers for BDDC algorithms, Tech. Rep. TR2005-871, Department of Computer Science, Courant Institute, New York, 2005. · Zbl 1173.65365
[35] , FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg., 66 (2006), pp. 250-271.
[36] , On the use of inexact subdomain solvers for BDDC algorithms, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1415-1428. · Zbl 1173.65365
[37] J. MANDEL ANDC. 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
[38] J. MANDEL, B. SOUSEDÍK,ANDC. R. DOHRMANN, Multispace and multilevel BDDC, Computing, 83 (2008), pp. 55-85.
[39] J. PEBREL, C. REY,ANDP. GOSSELET, A nonlinear dual-domain decomposition method: application to structural problems with damage, Inter. J. Multiscal Comp. Eng., 6 (2008), pp. 251-262.
[40] M. STEPHAN ANDJ. DOCTER, JUQUEEN: IBM Blue Gene/QRSupercomputer System at the Jülich Supercomputing Centre, Journal Large-Scale Res. Facilities, 1 (2015), Article A1, 18 pages.
[41] A. TOSELLI ANDO. B. WIDLUND, Domain Decomposition Methods—Algorithms and Theory, Springer, Berlin, 2005. · Zbl 1069.65138
[42] X. TU, Three-level BDDC in three dimensions, SIAM J. Sci. Comput., 29 (2007), pp. 1759-1780. · Zbl 1163.65094
[43] U. M. YANG, On long-range interpolation operators for aggressive coarsening, Numer. Linear Algebra Appl., 17 (2010), pp. 453-472. · Zbl 1240.65286
[44] S. ZAMPINI, PCBDDC: a class of robust dual-primal methods in PETSc, SIAM J. Sci. Comput., 38 (2016), pp. S282-S306. · Zbl 1352.65632
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