Multiscale Method: A Powerful Tool to Reduce the Computational Cost of Big Data Problems Involving Stick-Slip Oscillations

Nonlinear initial values problems are often used to model the dynamics of many different physical phenomena, for example, systems with dry friction. Usually, these nonlinear IVP do not present a known analytical solution. Then, in order to study these problems, a possible approach is to use approximation methods. The literature dealing with different types of approximation techniques is extensive. Usually, the methods are classified as numerical or analytical. Both can be accurate and provide approximations with any desired precision. However, their efficiencies in terms of computational cost can be very different when they are applied in problems involving big data, for example, stochastic simulations. With analytical methods it is possible to obtain an analytical expression as an approximation to the solution to the IVP, which may be very useful. For example, these analytical expressions can applied to speed up Monte Carlo simulations. The Monte Carlo method is an important tool, which permits to construct statistical models for random object transformations. To build an accurate statistical model (often histograms and sample statistics), several realizations of the transformation output are usually required, a big data problem. If each realization is obtained by a numerical integration, the computation of the Monte Carlo simulations can become a task with high computational and temporal costs. This paper shows that an option to reduce those costs is to use analytical approximations instead of numerical approximations. By doing this, instead of to perform a numerical integration for each realization, which is time consuming task, a simple substitution of values in the analytical expressions can be done. This article aims to compare the computational costs of the construction of statistical models by Monte Carlo simulations, using numerical and analytical approximations. The objective is to show the gain in terms of CPU time when analytical approximations are used instead of numerical ones. To exemplify the gain, an interesting big data problem involving stick-slip phenomenon is developed. The system analyzed has a mass moving over a belt involving random dry friction.