Avec cedram.org  english version
Rechercher un article
Table des matières de ce volume | Article précédent | Article suivant
Stéphane Descombes; Max Duarte; Thierry Dumont; Thomas Guillet; Violaine Louvet; Marc Massot
Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures
SMAI-Journal of computational mathematics, 3 (2017), p. 29-51, doi: 10.5802/smai-jcm.19
Article PDF
Class. Math.: 65Y05, 65T60, 65M50, 65L04, 35K57
Mots clés: Task-based parallelism, multi-core architectures, multiresolution, adaptive grid, stiff reaction-diffusion equations


A new solver featuring time-space adaptation and error control has been recently introduced to tackle the numerical solution of stiff reaction-diffusion systems. Based on operator splitting, finite volume adaptive multiresolution and high order time integrators with specific stability properties for each operator, this strategy yields high computational efficiency for large multidimensional computations on standard architectures such as powerful workstations. However, the data structure of the original implementation, based on trees of pointers, provides limited opportunities for efficiency enhancements, while posing serious challenges in terms of parallel programming and load balancing. The present contribution proposes a new implementation of the whole set of numerical methods including Radau5 and ROCK4, relying on a fully different data structure together with the use of a specific library, TBB, for shared-memory, task-based parallelism with work-stealing. The performance of our implementation is assessed in a series of test-cases of increasing difficulty in two and three dimensions on multi-core and many-core architectures, demonstrating high scalability.


[1] A. Abdulle, “Fourth order Chebyshev methods with recurrence relation”, SIAM J. Sci. Comput. 23 (2002) no. 6, p. 2041-2054 (electronic) Article |  MR 1923724
[2] S.B. Baden, N.P. Chrisochoides, D.B. Gannon & M.L. Norman (éd.), Structured adaptive mesh refinement (SAMR) grid methods, The IMA Volumes in Mathematics and its Applications 117, Springer-Verlag, New York, 2000, Papers from the IMA Workshop held at the University of Minnesota, Minneapolis, MN, March 12–13, 1997 Article
[3] D.S. Balsara & C.D. Norton, “Highly parallel structured adaptive mesh refinement using parallel language-based approaches”, Parallel Comput. 27 (2001) no. 1–2, p. 37-70 Article
[4] J. Bell, M. Berger, J. Saltzman & M. Welcome, “Three-dimensional adaptive mesh refinement for hyperbolic conservation laws”, SIAM J. Sci. Comput. 15 (1994) no. 1, p. 127-138 Article
[5] R.D. Blumofe, C.F. Joerg, B.C. Kuszmaul, C.E. Leiserson, K.H. Randall & Y. Zhou, “Cilk: An efficient multithreaded runtime system”, J. Parallel Distr. Com. 37 (1996) no. 1, p. 55-69
[6] R.D. Blumofe & C.E. Leiserson, “Scheduling Multithreaded Computations by Work Stealing”, J. ACM 46 (1999) no. 5, p. 720-748 Article
[7] K. Brix, S. Melian, S. Müller & M. Bachmann, Adaptive multiresolution methods: Practical issues on data structures, implementation and parallelization, Summer School on Multiresolution and Adaptive Mesh Refinement Methods, ESAIM Proc. 34, EDP Sci., Les Ulis, 2011, p. 151–183
[8] K. Brix, S.S. Melian, S. Müller & G. Schieffer, Parallelisation of multiscale-based grid adaptation using space-filling curves, in ESAIM: Proc., EDP Sciences, 2009, p. 108-129
[9] C. Burstedde, L. Wilcox & O. Ghattas, “p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees”, SIAM J. Sci. Comput. 33 (2011) no. 3, p. 1103-1133 Article
[10] A. Cohen, S. M. Kaber, S. Müller & M. Postel, “Fully Adaptive Multiresolution Finite Volume Schemes for Conservation Laws”, Math. Comput. 72 (2003), p. 183-225
[11] W. Crutchfield & M.L. Welcome, “Object-oriented implementation of adaptive mesh refinement algorithms”, J. Sci. Program. (1993), p. 145-156
[12] W. Dahmen, T. Gotzen, S.S. Melian–Flamand & S. Müller, Numerical Simulation of Cooling Gas Injection using Adaptive Multiscale Techniques, Inst. für Geometrie und Praktische Mathematik, 2011
[13] R. Deiterding, A generic framework for block-structured Adaptive Mesh Refinement in Object-oriented C++, Technical report, available at http://amroc.sourceforge.net, 2003
[14] R. Deiterding, Block-structured adaptive mesh refinement - Theory, implementation and application, Summer School on Multiresolution and Adaptive Mesh Refinement Methods, ESAIM Proc. 34, EDP Sci., Les Ulis, 2011, p. 97–150 Article
[15] S. Descombes, M. Duarte, T. Dumont, F. Laurent, V. Louvet & M. Massot, “Analysis of operator splitting in the nonasymptotic regime for Nonlinear Reaction-Diffusion Equations. Application to the Dynamics of Premixed Flames”, SIAM J. Numer. Anal. 52 (2014) no. 3, p. 1311-1334 Article
[16] S. Descombes & T. Dumont, “Numerical simulation of a stroke: Computational problems and methodology”, Prog. Biophys. Mol. Bio. 97 (2008) no. 1, p. 40-53
[17] S. Descombes, T. Dumont, V. Louvet & M. Massot, “On the local and global errors of splitting approximations of reaction-diffusion equations with high spatial gradients”, Int. J. Comput. Math. 84 (2007) no. 6, p. 749-765 Article
[18] S. Descombes, T. Dumont & M. Massot, Operator splitting for stiff nonlinear reaction-diffusion systems: Order reduction and application to spiral waves, Patterns and waves (Saint Petersburg, 2002), AkademPrint, St. Petersburg, 2003, p. 386–482
[19] S. Descombes & M. Massot, “Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction”, Numer. Math. 97 (2004) no. 4, p. 667-698
[20] J. Dreher & R. Grauer, “Racoon: A parallel mesh-adaptive framework for hyperbolic conservation laws”, Parallel Comput. 31 (2005) no. 8–9, p. 913-932 Article
[21] M.-A. Dronne, J.-P. Boissel & E. Grenier, “A mathematical model of ion movements in grey matter during a stroke”, J. Theor. Biol. 240 (2006) no. 4, p. 599-615
[22] F. Drui, A. Fikl, P. Kestener, S. Kokh, A. Larat, V. Le Chenadec & M. Massot, “Experimenting with the p4est library for AMR simulations of two-phase flows”, ESAIM: Proc. Surv. 53 (2016), p. 232-247
[23] M. Duarte, Adaptive numerical methods in time and space for the simulation of multi-scale reaction fronts, Ph. D. Thesis, Ecole Centrale Paris, 2011
[24] M. Duarte, Z. Bonaventura, M. Massot, A. Bourdon, S. Descombes & T. Dumont, “A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations”, J. Comput. Phys. 231 (2012) no. 3, p. 1002-1019 Article
[25] M. Duarte, S. Descombes, C. Tenaud, S. Candel & M. Massot, “Time-space adaptive numerical methods for the simulation of combustion fronts”, Combust. Flame 160 (2013) no. 6, p. 1083-1101
[26] M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvet & F. Laurent, New resolution strategy for multi-scale reaction waves using time operator splitting and space adaptive multiresolution: Application to human ischemic stroke, Summer School on Multiresolution and Adaptive Mesh Refinement Methods, ESAIM Proc. 34, EDP Sci., Les Ulis, 2011, p. 277–290 Article
[27] M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvet & F. Laurent, “New resolution strategy for multiscale reaction waves using time operator splitting, space adaptive multiresolution, and dedicated high order implicit/explicit time integrators”, SIAM J. Sci. Comput. 34 (2012) no. 1, p. A76-A104 Article
[28] T. Dumont, M. Duarte, S. Descombes, M.-A. Dronne, M. Massot & V. Louvet, “Simulation of human ischemic stroke in realistic 3D geometry”, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) no. 6, p. 1539-1557
[29] W. Eckhardt & T. Weinzierl, A Blocking Strategy on Multicore Architectures for Dynamically Adaptive PDE Solvers, in D. Hutchison et al., éd., Parallel Processing and Applied Mathematics, Springer Berlin Heidelberg, 2010, p. 567–575
[30] M. Essadki, S. de Chaisemartin, M. Massot, F. Laurent & A. Larat, “Adaptive mesh refinement and high order geometrical moment method for the simulation of polydisperse evaporating sprays”, Oil Gas Sci. Technol. 71 (2016), p. 1-25
[31] C.J. Forster, Parallel wavelet-adaptive direct numerical simulation of multiphase flows with phase change, Ph. D. Thesis, Georgia Institute of Technology, 2016
[32] T. Gautier, J.V.F. Lima, N. Maillard & B. Raffin, XKaapi: A Runtime System for Data-Flow Task Programming on Heterogeneous Architectures, in Parallel Distributed Processing (IPDPS), 2013 IEEE 27th International Symposium on, 2013, p. 1299-1308 Article
[33] P. Gray & S.K. Scott, Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics, International Series of Monographs on Chemistry 21, Oxford University Press, 1994
[34] E. Hairer, “Fortran and Matlab Codes”, http://www.unige.ch/~hairer/software.html
[35] E. Hairer, S.P. Nørsett & G. Wanner, Solving Ordinary Differential Equations. I, Springer Series in Computational Mathematics 8, Springer-Verlag, Berlin, 1993, Nonstiff problems
[36] E. Hairer & G. Wanner, Solving Ordinary Differential Equations. II, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin, 1996, Stiff and differential-algebraic problems
[37] A. Harten, “Multiresolution algorithms for the numerical solution of hyperbolic conservation laws”, Comm. Pure Applied Math. 48 (1995), p. 1305-1342
[38] B. Hejazialhosseini, D. Rossinelli, M. Bergdorf & P. Koumoutsakos, “High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions”, J. Comput. Phys. 229 (2010) no. 22, p. 8364-8383
[39] Intel Corporation, “Intel Threading Building Blocks”, https://www.threadingbuildingblocks.org/
[40] W. Jahnke, W.E. Skaggs & A.T. Winfree, “Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model”, J. Phys. Chem. 93 (1989) no. 2, p. 740-749
[41] H. Ji, F.-S. Lien & E. Yee, “A new adaptive mesh refinement data structure with an application to detonation”, J. Comput. Phys. 229 (2010) no. 23, p. 8981-8993 Article
[42] S. Jones & A. Lichtl, GPUs to Mars, Full Scale Simulation of SpaceX’s Mars Rocket Engine, in Proceedings of the GPU Technology Conference, GPU Tech Conferences on demand, http://on-demand.gputechconf.com/gtc/2015/video/S5398.html, 2015
[43] R. Keppens, Z. Meliani, A. J. van Marle, P. Delmont, A. Vlasis & B. van der Holst, “Parallel, grid-adaptive approaches for relativistic hydro and magnetohydrodynamics”, J. Comput. Phys. 231 (2012) no. 3, p. 718-744 Article |  MR 2869387
[44] P. MacNeice, K.M. Olson, C. Mobarry, R. de Fainchtein & C. Packer, “PARAMESH: A parallel adaptive mesh refinement community toolkit”, Comput. Phys. Commun. 126 (2000) no. 3, p. 330-354 Article
[45] G.M. Morton, A computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, Technical report, IBM, 1966
[46] S. Müller, Adaptive Multiscale Schemes for Conservation Laws 27, Springer-Verlag, Heidelberg, 2003
[47] A. Nejadmalayeri, A. Vezolainen, E. Brown-Dymkoski & O.V. Vasilyev, “Parallel adaptive wavelet collocation method for PDEs”, J. Comput. Phys. 298 (2015), p. 237-253
[48] Netlib, “Compressed Row Storage (CRS)”, http://netlib.org/linalg/html_templates/node91.html
[49] R.M. Noyes, R. Field & E. Koros, “Oscillations in chemical systems. I. Detailed mechanism in a system showing temporal oscillations”, J. Ame. Chem. Soc. 94 (1972) no. 4, p. 1394-1395
[50] L. Oliker & R. Biswas, “PLUM: Parallel Load Balancing for Adaptive Unstructured Meshes”, J. Parallel Distr. Com. 52 (1998) no. 2, p. 150-177 Article
[51] OpenMP Architecture Review Board, “OpenMP Application Program Interface. Version 3.1.”, July 2011, Available at: http://www.openmp.org
[52] J. Reinders, Intel Threading Building Blocks, O’Reilly & Associates, Inc., Sebastopol, CA, USA, 2007
[53] C.A. Rendleman, V.E. Beckner, M. Lijewski, W. Crutchfield & J.B. Bell, “Parallelization of structured, hierarchical adaptive mesh refinement algorithms”, Comput. Vis. Sci. 3 (2000) no. 3, p. 147-157 Article
[54] D. Rossinelli, B. Hejazialhosseini, M. Bergdorf & P. Koumoutsakos, “Wavelet-adaptive solvers on multi-core architectures for the simulation of complex systems”, Concurr. Comput.: Pract. Exper. 23 (2011) no. 2, p. 172-186
[55] O. Roussel, K. Schneider, A. Tsigulin & H. Bockhorn, “A conservative Fully Adaptive Multiresolution algorithm for parabolic PDEs”, J. Comput. Phys. 188 (2003) no. 2, p. 493-523
[56] H. Sagan, Space-filling curves, Universitext, Springer-Verlag, New York, 1994 Article
[57] K. Schneider & O.V. Vasilyev, “Wavelet methods in computational fluid dynamics”, Annu. Rev. Fluid Mech. 42 (2010) no. 1, p. 473-503
[58] M. Schreiber, T. Weinzierl & H.-J. Bungartz, Cluster Optimization and Parallelization of Simulations with Dynamically Adaptive Grids, in D. Hutchison et al., éd., Euro-Par 2013 Parallel Processing, Springer Berlin Heidelberg, 2013, p. 484–496
[59] G. Strang, “On the construction and comparison of difference schemes”, SIAM J. Numer. Anal. 5 (1968), p. 506-517
[60] R. Teyssier, “Cosmological hydrodynamics with adaptive mesh refinement: A new high resolution code called RAMSES”, Astron. Astrophys. 385 (2002) no. 1, p. 337-364 Article
[61] A.I. Volpert, V.A. Volpert & V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, RI, 1994, Translated from the Russian manuscript by James F. Heyda
[62] T. Weinzierl & M. Mehl, “Peano-A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids”, SIAM J. Sci. Comput. 33 (2011) no. 5, p. 2732-2760 Article
[63] S. Williams, A. Waterman & D. Patterson, “Roofline: An Insightful Visual Performance Model for Multicore Architectures”, Commun. ACM 52 (2009) no. 4, p. 65-76 Article
[64] A.M. Wissink, R.D. Hornung, S.R. Kohn, S.S. Smith & N. Elliott, Large Scale Parallel Structured AMR Calculations Using the SAMRAI Framework, in Supercomputing, ACM/IEEE 2001 Conference, 2001, p. 22-22 Article
[65] Ya.B. Zelʼdovich, G.I. Barenblatt, V.B. Librovich & G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau [Plenum], New York, 1985, Translated from the Russian by Donald H. McNeill