TY - JOUR

T1 - Beating the curse of dimension with accurate statistics for the Fokker-Planck equation in complex turbulent systems

AU - Chen, Nan

AU - Majda, Andrew J.

N1 - Funding Information:
ACKNOWLEDGMENTS. The research of A.J.M. is partially supported by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) Grant N0001416-1-2161 and the New York University Abu Dhabi Research Institute. N.C. is supported as a postdoctoral fellow through A.J.M.’s ONR MURI Grant.
Publisher Copyright:
© 2017, National Academy of Sciences. All rights reserved.

PY - 2017/12/5

Y1 - 2017/12/5

N2 - Solving the Fokker-Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples L, where 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. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of L. The resulting algorithms can efficiently solve the Fokker-Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a 1,000-dimensional stochastic coupled FitzHugh-Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only L = 1 samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a 240-dimensional two-layer inhomogeneous Lorenz 96 model, using only L = 500 samples.

AB - Solving the Fokker-Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples L, where 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. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of L. The resulting algorithms can efficiently solve the Fokker-Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a 1,000-dimensional stochastic coupled FitzHugh-Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only L = 1 samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a 240-dimensional two-layer inhomogeneous Lorenz 96 model, using only L = 500 samples.

KW - Block decomposition

KW - High-dimensional non-Gaussian PDFs

KW - Hybrid strategy

KW - Small sample size

KW - Statistical symmetry

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U2 - 10.1073/pnas.1717017114

DO - 10.1073/pnas.1717017114

M3 - Article

C2 - 29158403

AN - SCOPUS:85033385839

SN - 0027-8424

VL - 114

SP - 12864

EP - 12869

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 49

ER -