Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Abstract

Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computation-ally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from the traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has a significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

“…First of all, the theory may become the starting point of analytical approaches beyond the existing ones that are limited to IF models with weak or shortcorrelated Ornstein-Uhlenbeck noise. Second, with more efficient algorithms (efficient tools for the solution of multidimensional FPEs [102,103] might be useful here) also higher-dimensional situations, e.g., an adapting neuron with narrow-band noise input (corresponding to a system of four stochastic differential equations) might be tractable; possible candidates are eigenfunction expansions [104,105] and the matrix-continued-fraction method [13]. Third, similar to the one-dimensional white-noise case [30,31], the calculation of the firing-rate modulation in response to a time-dependent stimulus (other than noise) will follow a very similar mathematical framework as presented here for the calculation of the power spectrum.…”

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

“…In addition, two important conceptual models for turbulent dynamical systems [37,38] and a conceptual model of the coupled atmosphere and ocean [39] also fit perfectly into the physics-constrained nonlinear stochastic modeling framework with conditional Gaussian structures. These models are extremely useful for testing various new multiscale data assimilation and prediction schemes [1, 38,40]. Other physics-constrained nonlinear stochastic models with conditional Gaussian structures include a low-order model of Charney-DeVore flows [29] and a paradigm model for topographic mean flow interaction [32].…”

“…Many of the physics-constrained nonlinear stochastic models belong to the conditional Gaussian framework, including the noisy version of the famous Lorenz 63 and 84 models as well as a two-layer Lorenz 96 model [34][35][36]. In addition, two important conceptual models for turbulent dynamical systems [37,38] and a conceptual model of the coupled atmosphere and ocean [39] also fit perfectly into the physics-constrained nonlinear stochastic modeling framework with conditional Gaussian structures. These models are extremely useful for testing various new multiscale data assimilation and prediction schemes [1, 38,40].…”

“…Nevertheless, for the conditional Gaussian systems, efficient statistically accurate algorithms can be developed for solving the Fokker-Planck equation in high dimensions and thus beat the curse of dimension. The algorithms involve a hybrid strategy that requires only a small number of ensembles [38]. Specifically, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace.…”

Simple nonlinear models with rigorous extreme events and heavy tails

“…But knowing that X t is conditionally Gaussian, it suffices to compute the conditional mean and covariance, of which the computational cost only scale cubically with d X . This feature can be exploited for high dimensional prediction and data assimilation [10][11][12][13][14]. Conditional Gaussian model is known to be a good tool for studying extreme events and heavy tail phenomena.…”