Spectral methods for stochastic partial differential equations and the related optimal control problem

Abstract

This talk is devoted to spectral methods for solving linear and nonlinear stochastic partial differential equations with applications. The aim is to solve these equations using the polynomial chaos expansion (PCE) method. This is a spectral method based on the tensor product of deterministic orthogonal polynomials as a basis in the space of square integrable stochastic processes. We focus on stochastic (non)linear evolution equations where the coefficients, initial and boundary conditions might be highly singular, i.e., they are generalized stochastic processes. Examples include reaction-diffusion equations, nonlinear heat equations, time dependent Schrödinger equations, motion of an elastic string in a viscous random environment, etc. Using the PCE method combined with operator theory, semigroup theory, and theory of deterministic partial differential equations, we prove that the stochastic evolution equations under consideration have unique solutions in appropriate (weighted) spaces of stochastic processes. The obtained solutions are given in explicit forms, which allows an efficient numerical approximation. In addition, we consider the related stochastic optimal control problem where the cost functional is quadratic. We provide a novel framework for solving these optimal control problems using the PCE approach. Finally, we briefly discuss an splitting-PCE method for computing the optimal state, i.e., the state equation including the optimal control. Numerical experiments show the performance of the proposed method.