Variance-reduced weak Monte Carlo simulations of stochastically driven oscillators of engineering interest

Abstract

Analytical solutions of nonlinear and higher-dimensional stochastically driven oscillators are rarely possible and this leaves the direct Monte Carlo simulation of the governing stochastic differential equations (SDEs) as the only tool to obtain the required numerical solution. Engineers, in particular, are mostly interested in weak numerical solutions, which provide a faster and simpler computational framework to obtain the statistical expectations (moments) of the response functions. The numerical integration tools considered in this study are weak versions of stochastic Euler and stochastic Newmark methods. A well-known limitation of a Monte Carlo approach is however the requirement of a large ensemble size in order to arrive at convergent estimates of the statistical quantities of interest. Presently, a simple form of a variance reduction strategy is proposed such that the ensemble size may be significantly reduced without affecting the accuracy of the predicted expectations of any function of the response vector process provided that the function can be adequately represented through a power-series approximation. The basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the stochastic forcing function (which is typically a filtered white noise process) through a set of variance-reduced functions. In the process, the additional computational cost due to system augmentation is far smaller than the accrued advantages due to a drastically reduced ensemble size. Indeed, we show that the proposed method appears to work satisfactorily even in the special case of the ensemble size being just 1. The variance reduction scheme is first illustrated through applications to a nonlinear Duffing equation driven by additive and multiplicative white noise processes—a problem for which exact stationary solutions are known. This is followed up with applications of the strategy to a few higher-dimensional systems, i.e., 2- and 3-dof nonlinear oscillators under additive white noises.