Improving the Rate of Convergence of the Quasi-Monte Carlo Method in Estimating Expectations on a Geotechnical Slope Stability Problem

Abstract

The propagation of parameter uncertainty through engineering models is a key task in uncertainty quantification. In many cases, taking into account this uncertainty involves the estimation of expected values by means of the Monte Carlo method. While the performance of the classical Monte Carlo method is independent of the number of uncertainties, its main drawback is the slow convergence rate of the root mean square error, i.e., O(N −1/2 ) where N is the number of model evaluations. Under appropriate conditions, the quasi-Monte Carlo method improves the order of convergence to O(N −1 ) by using deterministic sample points instead of random sample points. Two examples of such point sets are rank-1 lattice sequences and Sobol' sequences. However, it is possible to further improve the order of convergence by applying the so-called "tent transformation" to a rank-1 lattice sequence, and by "interlacing" a Sobol' sequence. In this work, we benchmark these two techniques on a slope stability problem from geotechnical engineering, where the uncertainty is located in the cohesion of the soil. The soil cohesion is modeled as a lognormal random field of which realizations are computed by means of the Karhunen-Loève (KL) expansion. The quasi-Monte Carlo points are mapped to the normal distribution required in the KL expansion using a novel truncation strategy. We observe an order of convergence of O(N −1.5 ) in our numerical experiments.