Stochastic Analysis Involving the Computational Cost of a Monte-Carlo Simulation

To present ideas, a model problem consisting of a moving mass-belt system with random friction showing the stick-slip phenomenon is treated. The dynamics is simulated. The objective of this work is to assess the behaviour of the computation cost in terms of the run-time, which is random, and its relationship with some of the output variables that define the dynamical behaviour of the mechanical system, such as the duration of the phases present in the simulation, sticks and slips, and the number of phases that occur in each realisation. All this is analysed from a stochastic perspective. However, the probabilistic model to analyse the distribution of a three-dimensional random vector, formed by the run-time, duration and number, belongs to \(R^4\), thus it is difficult to characterise and visualise. Hence, in this study, the use of random variable transformations to produce new independent variables is explored as an attempt to reduce the number of dimensions that need to be considered. Also, the change of variables is used to assess the link between the behaviour of the results and the chosen integration method. It is shown that the predictions obtained with the Monte Carlo method combined with a Multiple Scales analytical approximation are influenced by the number of transition phases rather than their durations.