Spectral methods for SPDEs with applications

Abstract

This talk is devoted to spectral methods for solving linear and nonlinear stochastic partial differential equations with applications in optimal control. Particularly, the aim is to solve these equations using the polynomial chaos expansion (PCE) method, 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 evolution equations with Wick-type nonlinearities where the coefficients, initial and boundary conditions might be highly singular, i.e., they are generalized stochastic processes. The motivation comes from models arising in ecology, medicine, seismology, aerodynamics, structural acoustics and financial mathematics. 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. Numerical experiments show the performance of the proposed method.