Abstract
A parallel iterative algorithm is described for efficient solution of the Schur complement (interface) problem arising in the domain decomposition of stochastic partial differential equations (SPDEs) recently introduced in [1,2]. The iterative solver avoids the explicit construction of both local and global Schur complement matrices. An analog of Neumann-Neumann domain decomposition preconditioner is introduced for SPDEs. For efficient memory usage and minimum floating point operation, the numerical implementation of the algorithm exploits the multilevel sparsity structure of the coefficient matrix of the stochastic system. The algorithm is implemented using PETSc parallel libraries. Parallel graph partitioning tool ParMETIS is used for optimal decomposition of the finite element mesh for load balancing and minimum interprocessor communication. For numerical demonstration, a two dimensional elliptic SPDE with non-Gaussian random coefficients is tackled. The strong and weak scalability of the algorithm is investigated using Linux cluster.
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References
Sarkar, A., Benabbou, N., Ghanem, R.: Domain Decomposition Of Stochastic PDEs: Theoretical Formulations. International Journal for Numerical Methods in Engineering 77, 689–701 (2009)
Sarkar, A., Benabbou, N., Ghanem, R.: Domain Decomposition Of Stochastic PDEs: Performance Study. International Journal of High Performance Computing Applications (to appear, 2009)
Toselli, A., Widlund, O.: Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, New York (2004)
Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Philadelphia (1996)
Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Web page (2009), http://www.mcs.anl.gov/petsc
Karypis, G., Schloegel, K., Kumar, V.: PARMETIS Parallel Graph Partitioning and Sparse Matrix Ordering Library, University of Minnesota, Dept. of Computer Sci. and Eng. (1998)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2000)
Ghanem, R., Spanos, P.: Stochastic Finite Element: A Spectral Approach. Springer, New York (1991)
Subber, W., Monajemi, H., Khalil, M., Sarkar, A.: A Scalable Parallel Uncertainty Analysis and Data Assimilation Framework Applied to some Hydraulic Problems. In: International Symposium on Uncertainties in Hydrologic and Hydraulic Modeling (2008)
Message Passing Interface Forum, http://www.mpi-forum.org
Keyes, D.: How scalable is Domain Decomposition in Practice? In: Proc. Int. Conf. Domain Decomposition Methods, pp. 286–297 (1998)
Dohrmann, C.: A Preconditioner for Substructuring Based on Constrained Energy Minimization. SIAM Journal on Scientific Computing 25, 246–258 (2003)
Farhat, C., Lesoinne, M., Pierson, K.: A Scalable Dual-Primal Domain Decomposition Method. Numerical Linear Algebra with Application 7, 687–714 (2000)
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Subber, W., Sarkar, A. (2010). Domain Decomposition of Stochastic PDEs: A Novel Preconditioner and Its Parallel Performance. In: Mewhort, D.J.K., Cann, N.M., Slater, G.W., Naughton, T.J. (eds) High Performance Computing Systems and Applications. HPCS 2009. Lecture Notes in Computer Science, vol 5976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12659-8_19
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DOI: https://doi.org/10.1007/978-3-642-12659-8_19
Publisher Name: Springer, Berlin, Heidelberg
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