A spectral method approach for solving SPDEs and the related control problems

Abstract

We present a spectral method approach for solving linear and nonlinear stochastic partial differential equations where the coefficients, initial and boundary conditions might be highly singular, i.e., they are generalized stochastic processes. We consider nonlinearities of Wick-type and a quadratic cost functional for the related optimal control problem. Particularly, we apply a 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. Using the PCE method combined with operator theory, semigroup theory, and theory of deterministic partial differential equations, we prove that the stochastic equations under consideration have unique solutions in appropriate (weighted) spaces of stochastic processes. In an analogous way we prove the existence and uniqueness of the related optimal control. The solutions are given in explicit forms and provide a novel theoretical and numerical framework for treating these problems. The novelty also relies on the application of splitting methods to approximate the so-called optimal state. Numerical experiments show the potential of our method.