Dimension Reduction of the FPK Equation via an Equivalence of Probability Flux for Additively Excited Systems

Abstract

Solving the Fokker-Planck-Kolmogorov (FPK) equation is one of the most important and challenging problems in high-dimensional nonlinear stochastic dynamics, which is widely encountered in various science and engineering disciplines. To date, no method available is capable of dealing with systems of dimensions higher than eight. The present paper aims at tackling this problem in a different way, i.e., reducing the dimension of the FPK equation by constructing an equivalent probability flux. In the paper, two different treatments for multidimensional stochastic dynamical systems, the FPK equation and the probability density evolution method (PDEM), are outlined. Particularly, the FPK equation is revisited in a completely new way by constructing the probability fluxes based on the embedded dynamics mechanism and then invoking the principle of preservation of probability. The FPK equation is then marginalized to reduce the dimension, resulting in a flux-form equation involving unknown probability flux caused by the drift effect. On the basis of the equivalence of the two treatments, this unknown probability flux could be replaced by an equivalent probability flux, which is available using the PDEM. An algorithm is proposed to adopt the data from the solution of the generalized density evolution equation in PDEM to construct the numerical equivalent flux. Consequently, a one-dimensional parabolic partial differential equation, i.e., the flux-equivalent probability density evolution equation, could then be solved to yield the probability density function of the response of concern. By doing so, a high-dimensional FPK equation is reduced to a one-dimensional partial differential equation, for which the numerical solution is quite easy. Several numerical examples are studied to verify and validate the proposed method. Problems to be further studied are discussed.