Dual-primal domain decomposition method for uncertainty quantification

The spectral stochastic finite element method (SSFEM) may offer an efficient alternative to the traditional Monte Carlo simulations (MCS) for uncertainty quantification of large-scale numerical simulations. In the framework of the intrusive SSFEM, the main computational challenge involves solving a coupled set of deterministic linear systems. For large-scale numerical models, the computational efficiency of the intrusive SSFEM primarily depends on the solution techniques employed to tackle the resulting coupled linear systems. In this paper, we report a probabilistic version of the dual-primal domain decomposition method for the intrusive SSFEM in order to exploit high performance computing platforms for uncertainty quantification. In particular, we formulate a probabilistic version of the dual-primal finite element tearing and interconnect (FETI-DP) technique to solve the large-scale linear systems in the intrusive SSFEM. In the probabilistic setting, the operator of the dual interface system in the dual-primal approach contains a coarse problem. The introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual-primal iterative substructuring method for uncertainty quantification of large-scale computational models. The convergence properties, numerical and parallel scalabilities of the probabilistic FETI-DP method and the recently developed probabilistic version of the balancing domain decomposition by constraints (BDDC) method are contrasted. For numerical illustrations, we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non-Gaussian random processes. The algorithms are implemented on a Linux cluster using MPI and PETSc parallel libraries.