Domain Decomposition of Stochastic PDEs: A Novel Preconditioner and Its Parallel Performance

摘要

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.