Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations

Abstract

Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. In this thesis, we study a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing. WCE represents a stochastic solution as a spectral expansion with respect to a set of random basis. By deriving a governing equation for the expansion coefficients, we can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation. All the statistical information of the solution can be recovered from the deterministic coefficients using very simple formulae. We apply the WCE-based method to solve stochastic Burgers equations, Navier-Stokes equations and nonlinear reaction-diffusion equations with either additive or multiplicative random forcing. Our numerical results demonstrate convincingly that the new method is much more efficient and accurate than MC simulations for solutions in short to moderate time. For a class of model equations, we prove the convergence rate of the WCE method. The analysis also reveals precisely how the convergence constants depend on the size of the time intervals and the variability of the random forcing. Based on the error analysis, we design a sparse truncation strategy for the Wiener chaos expansion. The sparse truncation can reduce the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rates. For long time solutions, we propose a new computational strategy where MC simulations are used to correct the unresolved small scales in the sparse Wiener chaos solutions. Numerical experiments demonstrate that the WCE-MC hybrid method can handle SPDEs in much longer time intervals than the direct WCE method can. The new method is shown to be much more efficient than the WCE method or the MC simulation alone in relatively long time intervals. However, the limitation of this method is also pointed out. Using the sparse WCE truncation, we can resolve the probability distributions of a stochastic Burgers equation numerically and provide direct evidence for the existence of a unique stationary measure. Using the WCE-MC hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows. Our numerical results confirm the conjecture by Jack Xin that the front propagation speed obeys a quadratic enhancing law. Using the machinery we have developed for the Wiener chaos method, we resolve a few technical difficulties in solving stochastic elliptic equations by Karhunen-Loeve-based polynomial chaos method. We further derive an upscaling formulation for the elliptic system of the Wiener chaos coefficients. Eventually, we apply the upscaled Wiener chaos method for uncertainty quantification in subsurface modeling, combined with a two-stage Markov chain Monte Carlo sampling method we have developed recently.