Abstract
In this sequel to [7–9] we focus on the parallelization of our multilevel partition of unity method for distributed memory computers. The presented parallelization is based on a data decomposition approach which utilizes a key-based tree implementation and a weighted space filling curve ordering scheme for the load balancing problem. We present numerical results in two and three dimensions with up to 128 processors and 42 million degrees of freedom. These results show the optimal scaling behavior of our algorithm in the discretization as well as the solution phase.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. Babuška, U. Banerjee, and J. E. Osborn, Meshless and Generalized Finite Element Methods: A Survey of Some Major Results, in Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Springer, 2002
P. Bastian, Load Balancing for Adaptive Multigrid Methods, SIAM J. Sci. Comp., 19 (1998), pp. 1303–1321.
A. Brandt, Multigrid Solvers on Parallel Computers, in Elliptic Problem Solvers, M. H. Schultz, ed., Academic Press, 1981, pp. 39–83.
A. Caglar, M. Griebel, M. A. Schweitzer, and G. W. Zumbusch, Dynamic Load-Balancing of Hierarchical Tree Algorithms on a Cluster of Multiprocessor PCs and on the Cray T3E, in Proceedings 14th Supercomputer Conference, Mannheim, H. W. Meuer, ed., Mannheim, Germany, 1999, Mateo.
M. Griebel, Parallel Domain-Oriented Multilevel Methods, SIAM J. Sci. Comp., 16 (1995), pp. 1105–1125.
M. Griebel and T. Neunhoeffer, Parallel Point- and Domain-Oriented Multilevel Methods for Elliptic PDE on Workstation Networks, J. Comp. Appl. Math., 66 (1996), pp. 267–278.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic and Hyperbolic PDE, SIAM J. Sci. Comp., 22 (2000), pp. 853–890.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method—Part II: Efficient Cover Construction and Reliable Integration, SIAM J. Sci. Comp., 23 (2002), pp. 1655–1682.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method—Part III: A Multilevel Solver, SIAM J. Sci. Comp., (2002). to appear.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method—Part V: Boundary Conditions, in Geometric Analysis and Nonlinear Partial Differential Equations, S. Hildebrandt and H. Karcher, eds., Springer, 2002. to appear.
M. Griebel and G. W. Zumbusch, Hash-Storage Techniques for Adaptive Multilevel Solvers and their Domain Decomposition Parallelization, in Domain Decomposition Methods 10, The 10th International Conference, Boulder, J. Mandel, C. Farhat, and X.-C. Cai, eds., vol. 218 of Contemp. Math., AMS, 1998, pp. 271–278.
C. E. Grosch Poisson Solvers on Large Array Computers, in Proc. 1978 LANL Workshop on Vector and Parallel Processors, B. L. Buzbee and J. F. Morrison, eds., 1978.
F. C. Günther and W. K. Liu, Implementation of Boundary Conditions for Meshless Methods, Comput. Meth. Appl. Mech. Engrg., 163 (1998), pp. 205–230.
J. E. Jones and S. F. McCormick, Parallel Multigrid Methods, in Parallel Numerical Algorithms, D. E. Keyes, A. Sameh, and V. Venkatakrishnan, eds., Kluwer Academic Publishers, 1997, pp. 203–224.
O. A. Mcbryan, P. O. Frederickson, J. Linden, A. Schüller, K. Solchenbach, K. Stüben, C. A. Thole, and U., Trottenberg Multi-grid Methods on Parallel Computers — A Survey of Recent Developments, IMPACT Comput. Sci. Engrg., 3 (1991), pp. 1–75.
W. F. Mitchell, A Parallel Multigrid Method using the Full Domain Partition, Electron. Trans. Numer. Anal., 6 (1997), pp. 224–233. Special Issue for Proc. of the 8th Copper Mountain Conf. on Multigrid Methods.
J. Nitsche, ü ber ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg, 36 (1970–1971), pp. 9–15.
A. Pothen, Graph Partitioning Algorithms with Applications to Scientific Computing, in Parallel Numerical Algorithms, D. E. Keyes, A. Sameh, and V. Venkatakrishnan, eds., Kluwer Academic Publishers, 1997, pp. 323–368.
H. Sagan, Space-Filling Curves, Springer, New York, 1994.
M. A. Schweitzer Ein Partikel-Galerkin-Verfahren mit Ansatzfunktionen der Partition of Unity Method, Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, 1997.
M. A. Schweitzer, A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, Dissertation, Institut für Angewandte Mathematik, Universität Bonn, 2002.
M. A. Schweitzer, G. W. Zumbusch, and M. Griebel Parnass 2 : A Cluster of Dual-Processor PCs, in Proceedings of the 2nd Workshop Cluster-Computing, Karlsruhe, W. Rehm and T. Ungerer, eds., no. CSR-99-02 in Chemnitzer Informatik Berichte, Chemnitz, Germany, 1999, TU Chemnitz, pp. 45–54.
K. Solchenbach, C. A. Thole, and U. Trottenberg, Parallel Multigrid Methods: Implementation on SUPRENUM-like architectures and applications, in Super computing, vol. 297 of Lecture Notes in Computer Science, Springer, 1987, pp. 28–42.
L. Stals, Parallel Implementation of Multigrid Methods, PhD thesis, Department of Mathematics, Australian National University, 1995.
M. S. Warren and J. K. Salmon, A Parallel Hashed Oct-Tree N-Body Algorithm, in Supercomputing ‘93, Los Alamitos, 1993, IEEE Comp. Soc, pp. 12–21.
M. S. Warren and J. K. Salmon, A Portable Parallel Particle Program, Computer Physics Communications, 87 (1995).
M. S. Warren and J. K. Salmon, Parallel, Out-of-core Methods for N-body Simulation, in Proceedings of the 8th SIAM Conference on Parallel Processing for Scientific Computing, M. Heath, V. Torczon, G. Astfalk, P.. E. Bjørstad, A. H. Karp, C. H. Koebel, V. Kumar, R. F. Lucas, L. T. Watson, and D. E. Womble, eds., Philadelphia, 1997, SIAM.
G. W. Zumbusch, Simultanous h-p Adaptation in Multilevel Finite Elements, dissertation, Fachbereich Mathematik und Informatik, FU Berlin, 1995.
G. W. Zumbusch, Adaptive Parallel Multilevel Methods for Partial Differential Equations, Habilitation, Insitut für Angewandte Mathematik, Universität Bonn, 2001.
G. W. Zumbusch, On the Quality of Space-Filling Curve Induced Partitions, Z. Angew. Math. Mech., 81 Suppl. 1 (2001), pp. 25–28.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Griebel, M., Schweitzer, M.A. (2003). A Particle-Partition of Unity Method-Part IV: Parallelization. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56103-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-56103-0_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43891-5
Online ISBN: 978-3-642-56103-0
eBook Packages: Springer Book Archive