A Race in Monte Carlo Method: Numerical and Analytical Methods

Abstract

Most real mechanical problems present nonlinear characteristics and are governed by nonlinear initial value problems (IVP). These nonlinear IVP, usually, do not present a known solution, then, to allow the study of these problems, we use approximation methods. Analytical and numerical methods are approximation techniques used in the literature. Both methods are efficient in this assignment and can provide approximations with the desired precision. The analytical methods have the advantage of present analytical approximations to the problem solution. These analytical expressions can be convenient, when we are using a stochastic approach especially, Monte Carlo method. Monte Carlo method is an important tool, which permit us to construct statistical models of random objects transformations. To construct an accurate statistical model (often histograms and sample statistics), usually several samples of the transformation output are required. If each sample is obtained by a numerical integration, the computation of the Monte Carlo method becomes a task with high computational and temporal cost. An option, to reduce the costs is to use analytical approaches to perform the Monte Carlo method. Doing this, instead of to compute a numerical integration, we need to do only substitutions into the analytical expressions for each realization. A much less costly task. Therefore, this article aims to compare the computational cost of the construction of statistical models by Monte Carlo method using numerical and analytical approximations. The objective is to compare the gain in terms of CPU time when one uses analytical approximations instead of numerical approximations. For this, a classic example of the literature will be used, an IVP involving the Duffing equation, where one of the parameters of the problem will be modeled as a random variable.