Abstract
Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPerMeshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters, is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster.
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References
Arndt, D., et al.: The deal.II library, version 9.1. J. Numer. Math. 27(4), 203–213 (2019)
Balay, S., et al.: PETSc (2019). https://www.mcs.anl.gov/petsc
Bastian, P., et al.: A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE. Computing 82(2), 121–138 (2008). https://doi.org/10.1007/s00607-008-0004-9
Dauby, P., Desaive, T., Croisier, H., Kolh, P.: Standing waves in the FitzHugh-Nagumo model of cardiac electrical activity. Phys. Rev. E 73(2), 021908 (2006)
Deuflhard, P., Weiser, M.: Adaptive Numerical Solution of PDEs. Walter de Gruyter, Berlin (2012)
DeVito, Z., et al.: Liszt: a domain specific language for building portable mesh-based PDE solvers. In: Proceedings of Conference on High Performance Computing Networking, Storage and Analysis (SC 2011), p. paper 9. ACM (2011)
Götschel, S., Schiela, A., Weiser, M.: Kaskade 7 - a flexible finite element toolbox. Comp. Math. Appl. 81, 444–458 (2020)
Groth, S., Grünewald, D., Teich, J., Hannig, F.: A runtime system for finite element methods in a partitioned global address space. In: CF 2020: Proceedings of the 17th ACM International Conference on Computing Frontiers. ACM (2020). https://doi.org/10.1145/3387902.3392628
Grynko, Y., Förstner, J.: Simulation of second harmonic generation from photonic nanostructures using the discontinuous Galerkin time domain method. In: Agrawal, A., Benson, T., De La Rue, R.M., Wurtz, G.A. (eds.) Recent Trends in Computational Photonics. SSOS, vol. 204, pp. 261–284. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-55438-9_9
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012). https://freefem.org/
Hestenes, M.R., Stiefel, E., et al.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409–436 (1952)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008). https://doi.org/10.1007/978-0-387-72067-8
Hughes, T.J.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation (2012)
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. (SISC) 20(1), 359–392 (1998)
Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)
Logg, A., Mardal, K.A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-23099-8
Mudalige, G.R., Giles, M.B., Reguly, I., Bertolli, C., Kelly, P.H.J.: OP2: an active library framework for solving unstructured mesh-based applications on multi-core and many-core architectures. In: Proceedings of Innovative Parallel Computing (InPar), pp. 1–12 (2012)
Puwal, S., Roth, B.J.: Forward Euler stability of the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng. 54(5), 951–953 (2007)
Rathgeber, F., et al.: Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. (TOMS) 43(3), 24:1–24:27 (2016)
Rathgeber, F., et al.: PyOP2: a high-level framework for performance-portable simulations on unstructured meshes. In: Proceedings of the 2nd International Workshop on Domain-Specific Languages and High-Level Frameworks for High Performance Computing (WOLFHPC), pp. 1116–1123. ACM, November 2012
Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM (2003)
Schiesser, W.E., Griffiths, G.W.: A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge (2009)
Sermesant, M., Coudière, Y., Delingette, H., Ayache, N., Désidéri, J.A.: An electro-mechanical model of the heart for cardiac image analysis. In: Niessen, W.J., Viergever, M.A. (eds.) MICCAI 2001. LNCS, vol. 2208, pp. 224–231. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45468-3_27
Acknowledgments
This work was partially funded by the German Federal Ministry of Education and Research (BMBF) within the collaborative research project “HighPerMeshes” (01IH16005). The authors gratefully acknowledge the funding of this project by computing time provided by the Paderborn Center for Parallel Computing (PC2).
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Alhaddad, S. et al. (2021). HighPerMeshes – A Domain-Specific Language for Numerical Algorithms on Unstructured Grids. In: Balis, B., et al. Euro-Par 2020: Parallel Processing Workshops. Euro-Par 2020. Lecture Notes in Computer Science(), vol 12480. Springer, Cham. https://doi.org/10.1007/978-3-030-71593-9_15
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