  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 computationally 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.
