The use of numerical simulations for complex systems is common. However, significant uncertainties may exist for many of the involved variables, and in order to ensure the reliability of our simulation results and the safety of such complex systems, a stochastic approach providing statistics of the probability distribution of the results is of crucial importance. However, when a highly accurate result is required, the conventional Monte Carlo based probabilistic methodology inherently requires many repetitions of the deterministic analysis and in cases where that deterministic simulation is (relatively) time consuming, such probabilistic assessment can easily become computationally intractable. Hence, to reduce the computational expense of such probabilistic assessments as much as possible, an efficient quasi-Monte Carlo sampling strategy which can minimize the number of needed simulations of Monte Carlo based probabilistic analysis is introduced in this seminar.
In particular, the potential benefits of quasi-Monte Carlo (QMC) methods for uncertainty propagation are assessed via two applications: a numerical case study and a realistic and complex engineering case study. The sampling efficiency of four quasi-Monte Carlo sampling strategies — Optimized Latin hypercube, Sobol’ sequence, Niederreiter–Xing sequence and lattice sequence — are quantified and the errors of these quasi-Monte Carlo methods are estimated. In addition, for getting a better understanding of the potential factors that may influence the performance of quasi-Monte Carlo methods, the effect of the different parameters and the smoothness of the target function for the engineering case study are investigated for two quantities of interest. A sensitivity analysis is performed in which first and total order Sobol’ indices are calculated and a kernel smoother is used to show the effect on the quantity of interest for certain input parameters. The outcomes show that quasi-Monte Carlo methods perform at least as well as the Monte Carlo method and are capable of outperforming the standard Monte Carlo method if the target function is sufficiently smooth and mainly depends on a limited number of parameters.