Fourier-based maximum entropy method in stochastic dynamics

Abstract

As the recent research by Trybicki and Sobczyk has demonstrated [1-3] the principle of maximum entropy is a powerful tool for solving stochastic differential equations. In particular, its use in connection with the moment equations generated by the Ito formula provides accurate estimations of the probability density evolution of some oscillators for which conventional methods such as the diverse closure schemes are not applicable. A major computational requirement of the method, however, lies in the need of calculating a large number of multidimensional integrals at each time step -a numerical task for which both accurate and economic algorithms are required. In this paper it is shown that conventional economic integration techniques often lead to numerical collapse of the solution, especially when dealing with higly nonlinear oscillators. A strategy that overcomes this difficulty is proposed. In essence, the integrals are reformulated in terms of multidimensional Fourier transforms, which are solved by an ad hoe FFT algorithm aimed at obtaining only one single "frequency" point. It is demonstrated that the numerical stability and the accuracy of the proposed algorithm are superior to those afforded by other integration schemes.