Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

Abstract

In this paper, a new methodology is formulated for solving the reduced Fokker-Planck (FP) equations in high dimensions based on the idea that the state space of large-scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high-dimensional state space is reduced to some FP equations in low-dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.

Approximate Probability Density Function Solution of Multi-Degree-of-Freedom Coupled Systems Under Poisson Impulses

“…According to the state-space-split (SSS) method (Er 2011;Er and Iu 2011), the state variables x are divided into two subspaces x 1 2 R n x 1 and x 2 2 R n x 2 . x D fx 1 , x 2 g. In analysis of a second-order dynamical stochastic system, for instance, x 1 contains pairs of displacement and its first derivative (i.e., the corresponding velocity).…”

“…First Sect. 2 introduces the state-spacesplit (SSS) method to reduce the high-dimensional generalized FPK equation to a low-dimensional equation (Er 2011;Er and Iu 2011). After that, the exponentialpolynomial closure (EPC) method is further adopted to solve the reduced FPK equation for the PDF solution in Sect.…”

Numerical Methods for Reliability and Safety Assessment

“…According to the state-space-split (SSS) method (Er 2011;Er and Iu 2011), the state variables x are divided into two subspaces x 1 2 R n x 1 and x 2 2 R n x 2 .…”

Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

“…It is seen that the problem of obtaining the PDF solutions of large-scale NSD systems has been a challenge in this area for decades without effective method available for obtaining the acceptable approximate PDF solutions of large-scale NSD systems with high nonlinearity. Recently, a new method named subspace method or state-space-split method was reported for the approximate PDF solutions of large-scale nonlinear stochastic dynamic systems [30][31]. In this paper, the subspace method is further employed to analyze the NSD systems with various polynomial types of system nonlinearities in displacements to validate the effectiveness of subspace method in these cases.…”