Abstract
In this work we benchmark the performance of a preconditioned iterative method, used in large scale computer simulations of a geophysical application, namely, the elastic Glacial Isostatic Adjustment model. The model is discretized using the finite element method that gives raise to algebraic systems of equations with matrices that are large, sparse, nonsymmetric, indefinite and with a saddle point structure. The efficiency of solving systems of the latter type is crucial as it is to be embedded in a time-evolution procedure, where systems with matrices of similar type have to be solved repeatedly many times.
The implementation is based on available open source software packages - Deal.II, Trilinos, PARALUTION and AGMG. These packages provide toolboxes with state-of-the-art implementations of iterative solution methods and preconditioners for multicore computer platforms and GPU. We present performance results in terms of numerical and the computational efficiency, number of iterations and execution time, and compare the timing results against a sparse direct solver from a commercial finite element package, that is often used by applied scientists in their simulations.
Chapter PDF
Similar content being viewed by others
Keywords
References
Abaqus FEA, http://www.3ds.com/
Valgrind, http://www.valgrind.org
Axelsson, O.: On iterative solvers in structural mechanics; separate displacement orderings and mixed variable methods. Math. Comput. Simulation 50(1-4), 11–30 (1999); Modelling 1998, Prague (1998)
Axelsson, O.: Milestones in the development of iterative solution methods. J. Electr. Comput. Eng., Art. ID 972794, 33 (2010)
Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning of boundary value problems using elementwise schur complements. SIAM J. Matrix Anal. Appl. 31(2), 767–789 (2009)
Balay, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zhang, H.: PETSc Web page (2014), http://www.mcs.anl.gov/petsc
Bangerth, W., Kanschat, G., Hartmann, R.: deal.II differential equations analysis library, http://www.dealii.org
Bängtsson, E., Lund, B.: A comparison between two solution techniques to solve the equations of glacially induced deformation of an elastic earth. International Journal for Numerical Methods in Engineering 75(4), 479–502 (2008)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14, 1–137 (2005)
Braess, D.: Finite elements, 3rd edn. Theory, fast solvers, and applications in elasticity theory. Cambridge University Press, Cambridge (2007)
Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R-2), 129–151 (1974)
Dorostkar, A., Lukarski, D., Lund, B., Neytcheva, M., Notay, Y., Schmidt, P.: Parallel performance study of block-preconditioned iterative methods on multicore computer systems. Technical Report 2014-007, Department of Information Technology, Uppsala University (March 2014)
Eisenstat, S.C., Elman, H.C., Schultz, M.H.: Variational iterative methods for nonsymmetric systems of linear equations 20, 345–357 (1983)
Heroux, M.A., Willenbring, J.M.: Trilinos Users Guide. Technical Report SAND2003-2952, Sandia National Lab. (2003), http://trilinos.sandia.gov
Karypis, G., Kumar, V.: MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0 (2009), http://www.cs.umn.edu/~metis
Kraus, J.: Additive Schur complement approximation and application to multilevel preconditioning. SIAM J. Sci. Comput. 34(6), A2872–A2895 (2012)
Lukarski, D.: Parallel Sparse Linear Algebra for Multi-core and Many-core Platforms – Parallel Solvers and Preconditioners. PhD thesis, Karlsruhe Institute of Technology (January 2012)
Lurkarski, D.: Paralution project, http://www.paralution.com
Notay, Y.: AGMG software and documentation, http://homepages.ulb.ac.be/~ynotay/AGMG
Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012)
Neytcheva, M.: On element-by-element Schur complement approximations. Linear Algebra Appl. 434(11), 2308–2324 (2011)
Neytcheva, M., Bängtsson, E.: Preconditioning of nonsymmetric saddle point systems as arising in modelling of viscoelastic problems. Electronic Transactions on Numerical Analysis 29, 193–211 (2008)
Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–146 (2010)
Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)
Notay, Y.: A new analysis of block preconditioners for saddle point problems. SIAM J. Matrix Anal. Appl. 35, 143–173 (2014)
Notay, Y., Vassilevski, P.S.: Recursive Krylov-based multigrid cycles. Numer. Lin. Alg. Appl. 15, 473–487 (2008)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7(3), 856–869 (1986)
Wu, P.: Viscoelastic versus viscous deformation and the advection of pre-stress. Geophysical Journal International 108(1), 136–142 (1992)
Wu, P.: Using commercial finite element packages for the study of earth deformations, sea levels and the state of stress. Geophysical Journal International 158(2), 401–408 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Dorostkar, A., Lukarski, D., Lund, B., Neytcheva, M., Notay, Y., Schmidt, P. (2014). CPU and GPU Performance of Large Scale Numerical Simulations in Geophysics. In: Lopes, L., et al. Euro-Par 2014: Parallel Processing Workshops. Euro-Par 2014. Lecture Notes in Computer Science, vol 8805. Springer, Cham. https://doi.org/10.1007/978-3-319-14325-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-14325-5_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14324-8
Online ISBN: 978-3-319-14325-5
eBook Packages: Computer ScienceComputer Science (R0)