Abstract
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...
Highlights
The Fokker--Planck equation is a partial differential equation (PDE) that governs the time evolution of the probability density function (PDF) of a stochastic differential equation [28, 67]
If a direct kernel density method is applied to recover the PDF of u\bfI \bfI, the bandwidth of the kernel H is scaled as the reciprocal of L to a certain power in order to minimize the mean integrated square error (MISE) and the resulting MISE is proportional to L . - 1/N\bfI \bfI In order to reach a fixed accuracy, say L - 1/N\bfI \bfI = \epsilon, the sample size L = \epsilon - N\bfI \bfI has to increase exponentially with N\bfI \bfI
This article presents a rigorous analysis for the efficient statistically accurate algorithms developed in [16], which succeed in solving both the transient and the equilibrium solutions of Fokker--Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures
Summary
The Fokker--Planck equation is a partial differential equation (PDE) that governs the time evolution of the probability density function (PDF) of a stochastic differential equation [28, 67]. Note that the kernel density estimation is essentially the Monte Carlo simulation when L is large and it suffers from the curse of dimensionality Such comparison facilitates the understanding of the advantages of the efficient algorithm (9) in recovering the high-dimensional subspace of u\bfI \bfI using only a small number of samples. H , and - one needs to increase sample size exponentially with the dimension in order to have a small bandwidth that guarantees the accuracy of the recovered PDFs. when H is small, the kernel density method approximates the standard Monte Carlo simulation, which suffers from the curse of dimensionality.
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