Rigorous analysis for efficient statistically accurate algorithms for solving Fokker-Planck equations in large dimensions. (English) Zbl 1410.35244
The purpose of this article is to solve the Fokker-Planck equations associated with high-dimensional nonlinear stochastic systems with conditional Gaussian structures in large dimensions numerically. First, the efficient statistically accurate algoritms for solving these PDEs are reviewed. The state variabes are decomposed into two groups, the conditional Gaussian system becomes a Gaussian process.
Let MISE denote the mean integrated square error. The first theorem states that the two parts of MISE for the hybrid method are bounded. The second theorem states that under certain assumptions, the joint density converges geometrically to an ergodic measure with a rate \(>0\). The third theorem states certain properties for the stochastic flow with energy conserving quadratic nonlinearity. Numerical examples are illustrated by a three-dimensional coupled nonlinear system and beautiful pictures. The proofs use Taylor expansion, variance-bias decomposition, block diagonal covariance, Lyapunov function, Gronvall and Cauchy-Schwarz inequalities.
Let MISE denote the mean integrated square error. The first theorem states that the two parts of MISE for the hybrid method are bounded. The second theorem states that under certain assumptions, the joint density converges geometrically to an ergodic measure with a rate \(>0\). The third theorem states certain properties for the stochastic flow with energy conserving quadratic nonlinearity. Numerical examples are illustrated by a three-dimensional coupled nonlinear system and beautiful pictures. The proofs use Taylor expansion, variance-bias decomposition, block diagonal covariance, Lyapunov function, Gronvall and Cauchy-Schwarz inequalities.
Reviewer: Thomas Ernst (Uppsala)
MSC:
| 35Q84 | Fokker-Planck equations |
| 76F55 | Statistical turbulence modeling |
| 65C05 | Monte Carlo methods |
| 37C75 | Stability theory for smooth dynamical systems |
| 93B05 | Controllability |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
Fokker-Planck equation; high-dimensional non-Gaussian PDFs; hybrid strategy; small sample size; long time persistenceReferences:
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