Rigorous analysis for efficient statistically accurate algorithms for solving fokker-planck equations in large dimensions

Nan Chen, Andrew Majda, Xin T. Tong

Research output: Contribution to journalArticle

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 the conditional covariance is completely determined by the underlying dynamics and is independent of L. This is fundamentally different from a direct application of kernel methods to solve the full PDF, where L needs to increase exponentially with the dimension of the system and the bandwidth shrinks. A detailed comparison between different methods justifies that the efficient statistically accurate algorithms are able to overcome the curse of dimensionality. It is also shown with mathematical rigor that these algorithms are robust in long time provided that the system is controllable and stochastically stable. Particularly, dynamical systems with energy-conserving quadratic nonlinearity as in most geophysical and engineering turbulence are proved to have these properties.

Original languageEnglish (US)
Pages (from-to)1198-1223
Number of pages26
JournalSIAM-ASA Journal on Uncertainty Quantification
Volume6
Issue number3
DOIs
StatePublished - Jan 1 2018

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Fokker Planck equation
Fokker-Planck Equation
Probability density function
High-dimensional
Subspace
Stochastic systems
Mean Integrated Squared Error
Nonlinear Stochastic Systems
Oils and fats
Gaussian Kernel
Kernel Density Estimation
Gaussian Mixture
Curse of Dimensionality
Kernel Methods
Intermittency
Nonlinear systems
Dynamical systems
Turbulence
Square root
Justify

Keywords

  • Fokker-Planck equation
  • High-dimensional non-Gaussian PDFs
  • Hybrid strategy
  • Long time persistence
  • Small sample size

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Rigorous analysis for efficient statistically accurate algorithms for solving fokker-planck equations in large dimensions. / Chen, Nan; Majda, Andrew; Tong, Xin T.

In: SIAM-ASA Journal on Uncertainty Quantification, Vol. 6, No. 3, 01.01.2018, p. 1198-1223.

Research output: Contribution to journalArticle

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