Rigorous Analysis for Efficient Statistically Accurate Algorithms for Solving Fokker--Planck Equations in Large Dimensions

This article presents a rigorous analysis for efficient statistically accurate algorithms for solving the Fokker--Planck equations associated with high-dimensional nonlinear stochastic systems with conditional Gaussian structures. Despite the conditional Gaussianity, these nonlinear systems can contain strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy that requires only a small number of samples $L$ to capture both the transient and the equilibrium non-Gaussian PDFs with high accuracy. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. Rigorous analysis shows that the mean integrated squared error in the recovered PDFs in the high-dimensional subspace is bounded by the inverse square root of the determinant of the conditional covariance, where th...