Observations with regard to massively parallel computation for Monte Carlo simulation of stochastic dynamical systems

The evolution of stochastic dynamical systems is governed by Fokker-Planck equations if the response process is Markovian. Analytical solutions for the transient response of multidimensional systems exist only for the simplest dynamical systems. The evolution of the transition probability density function over the phase space has been solved numerically for various low dimensional systems subjected to additive and multiplicative white noise excitations using the finite element method. Systems of higher order, however, pose difficulty when using standard finite element formulations due to memory requirements and computational expense. Direct Monte Carlo simulation (MCS), while often regarded as less elegant than other methods, can be used to solve problems of significantly higher complexity. The number of realizations required to accurately produce the transition probability density function over the entire phase space, especially in the tails, is large, but since each realization is entirely independent of the others, the Monte Carlo simulation is easily and efficiently adapted to parallel computation. The advent of high-speed, massively-parallel computers permits a large number of realizations of a complex dynamical system to be simultaneously determined. Consequently, Monte Carlo simulation may be more efficient for higher-dimensional systems than other solution methods currently in use. This investigation will examine some of these observations and compare the performance of MCS on various platforms, in the context of a four-dimensional linear oscillator and a Duffing oscillator subjected to band-limited white noise.