A scalable solver for sampling-free uncertainty quantification of time-dependent partial differential equations

Abstract

This research addresses the uncertainty quantification of time-dependent partial differential equations (PDEs) with random parameters. The stochastic Galerkin method, a sampling-free intrusive approach, is employed instead of sampling- or quadrature-based methods to overcome the slow convergence and high computational cost associated with high-resolution models. An acoustic wave propagation problem with a log-normal random field approximation for wave speed is illustrated. The stochastic partial differential equation with the inputs and outputs expanded using polynomial chaos expansion (PCE) is transformed into a set of coupled deterministic PDEs and discretized to yield a system of linear equations. To handle the increased memory requirements with increasing mesh size, time step and number of random parameters, domain decomposition-based (DD-based) solvers are utilized. A conjugate gradient iterative solver with a two-level Neumann-Neumann preconditioner is applied to the symmetric positive
definite system matrix showing their efficient scalability. This combination of the stochastic Galerkin method and DD-based solvers enables large-scale real-time applications involving acoustic waves.