The Sparse Grids Matlab Kit provides a Matlab implementation of sparse grids, and can be used for approximating high-dimensional functions and, in particular, for surrogate-model-based uncertainty quantification. It is lightweight, high-level and easy to use, good for quick prototyping and teaching; however, it is equipped with some features that allow its use also in realistic applications. The goal of this paper is to provide an overview of the data structure and of the mathematical aspects forming the basis of the software, as well as comparing the current release of our package to similar available software.

This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, and kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF). The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier–Stokes (RANS) multi-grid solver $$\chi$$χ navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementation of both methods, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBF seems to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.

Groundwater flow modeling is commonly used to calculate groundwater heads, estimate groundwater flow paths and travel times, and provide insights into solute transport processes within an aquifer. However, the values of input parameters that drive groundwater flow models are often highly uncertain due to subsurface heterogeneity and geologic complexity in combination with lack of measurements/unreliable measurements. This uncertainty affects the accuracy and reliability of model outputs. Therefore, parameters’ uncertainty must be quantified before adopting the model as an engineering tool. In this study, we model the uncertain parameters as random variables and use a Bayesian inversion approach to obtain a posterior, data-informed, probability density function (pdf) for them: in particular, the likelihood function we consider takes into account both well measurements and our prior knowledge about the extent of the springs in the domain under study. To keep the modelistic and computational complexities under control, we assume Gaussianity of the posterior pdf of the parameters. To corroborate this assumption, we run an identifiability analysis of the model: we apply the inversion procedure to several sets of synthetic data polluted by increasing levels of noise, and we determine at which levels of noise we can effectively recover the “true value” of the parameters. We then move to real well data (coming from the Ticino River basin, in northern Italy, and spanning a month in summer 2014), and use the posterior pdf of the parameters as a starting point to perform an uncertainty quantification analysis on groundwater travel-time distributions.