Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

Engineering problems are often characterized by significant uncertainty in their material parameters. A typical example coming from geotechnical engineering is the slope stability problem where the soil's cohesion is modeled as a random field. An efficient manner to account for this uncertainty is the novel sampling method called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). The p-MLQMC method uses a hierarchy of p-refined Finite Element meshes combined with a deterministic Quasi-Monte Carlo sampling rule. This combination yields a significant computational cost reduction with respect to classic Multilevel Monte Carlo. However, in previous work, not enough consideration was given to how to incorporate the uncertainty, modeled as a random field, in the Finite Element model with the p-MLQMC method. In the present work we investigate how this can be adequately achieved by means of the integration point method. We therefore investigate how the evaluation points of the random field are to be selected in order to obtain a variance reduction over the levels. We consider three different approaches. These approaches will be benchmarked on a slope stability problem in terms of computational runtime. We find that for a given tolerance the Local Nested Approach yields a speedup up to a factor five with respect to the Non-Nested approach.

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.