zbMATH — the first resource for mathematics

A massively parallel geometric multigrid solver on hierarchically distributed grids. (English) Zbl 1380.65463
Summary: A parallel geometric multigrid solver on hierarchically distributed grids is presented. Using a tree-structure for grid distribution onto the processing entities, the multigrid cycle is performed similarly to the serial algorithm, using additional vertical communication during transfer operations. The workload is gathered to fewer processes on coarser levels. Involved parallel structures are described in detail and the multigrid algorithm is formulated, discussing parallelization details. A performance study is presented that shows close to optimal efficiency for weak scaling up to 262k processes in 2 and 3 space dimensions.

65Y05 Parallel numerical computation
Full Text: DOI
[1] Baker, A; Falgout, R; Kolev, T; Yang, U, Multigrid smoothers for ultra-parallel computing, SIAM J. Sci. Comput., 33, 2864-2887, (2011) · Zbl 1237.65032
[2] Baker, A., Falgout, R., Kolev, T., Yang, U.: Scaling hypre’s multigrid solvers to 100,000 cores. In: M.W. Berry, K.A. Gallivan, E. Gallopoulos, A. Grama, B. Philippe, Y. Saad, F. Saied (eds.) High-Performance Scientific Computing, pp. 261-279. Springer, London (2012)
[3] Bastian, P; Birken, K; Johannsen, K; Lang, S; Reichenberger, V; Wieners, C; Wittum, G; Wrobel, C, A parallel software-platform for solving problems of partial differential equations using unstructured grids and adaptive multigrid methods, High Perform. Comput. Sci. Eng., 98, 326-339, (1999) · Zbl 0928.65120
[4] Bastian, P., Birken, K., Johannsen, K., Lang, S., Reichenberger, V., Wieners, C., Wittum, G., Wrobel, C.: Parallel solution of partial differential equations with adaptive multigrid methods on unstructured grids. In: E. Krause, W. Jäger (eds.) High Performance Computing in Science and Engineering ’99, pp. 496-508. Springer, Berlin Heidelberg (2000) · Zbl 0945.65139
[5] Bastian, P; Blatt, M; Scheichl, R, Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems, Numer. Linear Algebra Appl., 19, 367-388, (2012) · Zbl 1274.65313
[6] Bergen, B; Gradl, T; Rude, U; Hulsemann, F, A massively parallel multigrid method for finite elements, Comput. Sci. Eng., 8, 56-62, (2006)
[7] Birken, K.: Dynamic Distributed Data in a Parallel Programming Environment, DDD, Reference Manual. Rechenzentrum Univ, Stuttgart (1994)
[8] Blatt, M.: A parallel algebraic multigrid method for elliptic problems with highly discontinuous coefficients. Ph.D. thesis, Ruprecht-Karls-University Heidelberg (2010) · Zbl 1194.65002
[9] Blatt, M; Bastian, P, On the generic parallelisation of iterative solvers for the finite element method, Int. J. Comput. Sci. Eng., 4, 56-69, (2008)
[10] Falgout, R, An introduction to algebraic multigrid computing, Comput. Sci. Eng., 8, 24-33, (2006)
[11] Gmeiner, B., Gradl, T., Kostler, H., Rude, U.: Highly parallel geometric multigrid algorithm for hierarchical hybrid grids. In: NIC Symposium 2012-Proceedings, p. 323. Forschungszentrum Jülich (2012) · Zbl 1274.65313
[12] Gropp, W., Lusk, E., Skjellum, A.: Using MPI: portable parallel programming with the message-passing interface, vol. 1. MIT press (1999) · Zbl 0875.68206
[13] Hackbusch, W.: Multi-grid Methods and Applications, vol. 4. Springer, Berlin (1985) · Zbl 0595.65106
[14] Lang, S., Wittum, G.: Large-scale density-driven flow simulations using parallel unstructured grid adaptation and local multigrid methods. Concurr. Comput.: Pract. Exp. 17(11), 1415-1440 (2005) · Zbl 0928.65120
[15] Sampath, R; Biros, G, A parallel geometric multigrid method for finite elements on octree meshes, SIAM J. Sci. Comput., 32, 1361-1392, (2010) · Zbl 1213.65144
[16] Stroustrup, B.: The C++ Programming Language, 3rd edn. Addison-Wesley Longman Publishing Co., Boston (1997) · Zbl 0825.68056
[17] Sundar, H., Biros, G., Burstedde, C., Rudi, J., Ghattas, O., Stadler, G.: Parallel geometric-algebraic multigrid on unstructured forests of octrees. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC ’12, pp. 43:1-43:11. IEEE Computer Society Press, Los Alamitos, CA, USA (2012)
[18] Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: Ug 4: a novel flexible software system for simulating pde based models on high performance computers. Comput. Vis. Sci. pp. 1-15 (2014) · Zbl 1375.35003
[19] Wieners, C.: Distributed point objects. A new concept for parallel finite elements. In: T. Barth, M. Griebel, D. Keyes, R. Nieminen, D. Roose, T. Schlick, R. Kornhuber, R. Hoppe, J. Priaux, O. Pironneau, O. Widlund, J. Xu (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40, pp. 175-182. Springer, Berlin Heidelberg (2005) · Zbl 1274.65313
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.