A spectral-based numerical method for Kolmogorov equations in Hilbert spaces

TL;DR

This paper introduces a spectral-based numerical method for solving Kolmogorov equations in Hilbert spaces, leveraging spectral decomposition and Wiener-Chaos expansion to approximate solutions of complex stochastic PDEs.

Contribution

It presents a novel spectral decomposition approach combined with Wiener-Chaos expansion for efficient numerical solutions of Kolmogorov equations in infinite-dimensional spaces.

Findings

Effective approximation of solutions for stochastic PDEs

Validated method on diffusion, Fisher-KPP, and Burgers equations

Demonstrated convergence and accuracy of the spectral approach

Abstract

We propose a numerical solution for the solution of the Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein-Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as a infinite system of ordinary differential equations, and by truncation it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher-KPP stochastic equation and a stochastic Burgers Eq. in dimension 1.