Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

Abstract

Uncertainty quantification of practical engineering applications using the intrusive spectral stochastic finite element methods (SSFEM) may involve solving a system of linear equations in the order of billions of unknowns. Therefore, in this thesis the intrusive polynomial chaos expansion (PCE) based two-level domain decomposition (DD) algorithms for stochastic partial differential equations (PDEs) are extended to handle high resolution numerical models using an in-house scalable parallel solvers toolkit. First, attention is given to facilitate the numerical simulation of the elliptic stochastic PDEs with a large number of random variables to address the so-called curse of dimensionality issue. Second, for three-dimensional coupled stochastic PDE systems such as equations of linear elasticity, the extended wirebasket-based coarse grid is developed to improve the performance and overcome the scalability issues of the DD based iterative solvers with a vertex-based coarse grid. Third, the developed DD solvers for the SSFEM are coupled with FEniCS deterministic finite element assembly routines in order to reduce the coding required for the implementation and generalize the application of these solvers to a variety of PDEs using FEniCS. Fourth, the intrusive SSFEM with scalable DD solver is shown to outperform the non-intrusive SSFEM with the sparse grid quadrature for a stochastic PDE with the non-Gaussian random variables. This highlights the advantages of the intrusive approach and demonstrates the necessity of scalable parallel solvers for uncertainty quantification. This thesis also elaborates on the HPC implementational aspects of the DD solvers for SSFEM. Three-level nested sparse iterative solvers, which employ an efficient DD based preconditioners are used to simulate two and three-dimensional scalar and vector-valued stochastic PDEs. The random system parameters and the solution process are modeled as a non-Gaussian II III stochastic process characterized by using up to 25 random variables with a third-order expansion resulting in 3276 PCE coefficients. Numerical and parallel scalabilities of these algorithms are investigated employing large-scale high-performance computing clusters with MPI, PETSc and FEniCS libraries. For all these scalability measurements, we report the results for different numbers of random variables and different orders of PCE to measure the sensitivity of the algorithms to the stochastic parameters.Dedicated to my wife, Pragati, and my son, Prajit. IV

Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems

“…However, it is noted that only linear static problems are considered for these studies and the application of the solvers to time-dependent and nonlinear problems is an open area of research. More details on these algorithms can be found in [32,106]. An additive Schwarz preconditioner is used for solving the decoupled stochastic elliptic PDEs with recycled Krylov subspace and preconditioners in [107].…”

“…The two-level Neumann-Neumann preconditioner used for the acoustic wave propagation problem in a two-dimensional domain does not scale well to three-dimensional problems of vector-valued PDEs [106]. This is due to the vertex-based coarse grid utilized for two-dimensional problems.…”

“…With the enriched coarse grid (having more nodes), the wirebasket is capable of better global propagation of error and hence 2.1. INTRODUCTION exhibiting better scalabilities than that of the vertex-based methods (for static problems with random parameters [106]). This is demonstrated for a three-dimensional stochastic Poisson problem and elasticity problem in [106].…”

“…INTRODUCTION exhibiting better scalabilities than that of the vertex-based methods (for static problems with random parameters [106]). This is demonstrated for a three-dimensional stochastic Poisson problem and elasticity problem in [106]. However, the wirebasket grid was not used to solve time-dependent problems such as wave propagation in random media.…”

“…As the number of random variables increases, this difference in computational cost also increases. It is also shown in [106] that for a fixed accuracy and large number of random variables, the time to solution for the non-intrusive sparse quadrature method is much higher than the intrusive SSFEM.…”