Sparse-Grid Methods for Several Types of Stochastic Differential Equations

This work focuses on developing and analyzing novel, efficient sparse-grid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification. First, we consider linear parabolic partial differential equations with random diffusion coefficients, forcing term and initial condition. Error analysis for a stochastic collocation method is carried out in a wider range of situations than previous literatures, including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both semi-discrete and fully-discrete solutions. Second, we consider multi-dimensional backward stochastic differential equations driven by a vector of white noise. A sparse-grid scheme are proposed to discretize the target equation in the multi-dimensional time-space domain. In our scheme, the time discretization is conducted by the multi-step scheme. In the multi-dimensional spatial domain, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and adaptive hierarchical sparse-grid interpolation. Error estimates are rigorously proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Third, we investigate the propagation of input uncertainty through nonlocal diffusion models. Since the stochastic local diffusion equations, e.g. heat equations, have already been well studied, we are interested in extending the existing numerical methods to solve nonlocal diffusion problems. In this work, we use sparse-grid stochastic collocation method to solve nonlocal diffusion equations with colored noise and Monte-Carlo method to solve the ones with white noise. Our numerical experiments show that the existing methods can achieve the desired accuracy in the nonlocal setting. Moreover, in the white noise case, the nonlocal diffusion operator can reduce the variance of the solution because the nonlocal diffusion operator has "smoothing" effect on the random field. At last, stochastic inverse problem is investigated. We propose sparse-grid Bayesian algorithm to improve the efficiency of the classic Bayesian methods. Using sparse-grid interpolation and integration, we construct a surrogate posterior probability density function and determine an appropriate alternative density which can capture the main features of the true PPDF to improve the simulation efficiency in the framework of indirect sampling. By applying this method to a groundwater flow model, we demonstrate its better accuracy when compared to brute-force MCMC simulation results.