Efficient sampling and solver enhancement for uncertainty quantification

The focus of this thesis is on the development of efficient and robust uncertainty quantiÞcation algorithms that can be used for a wide range of applications. The goal is to determine how likely certain outcomes are if some aspects of the application are not known exactly, i.e., in case of limited prior knowledge or uncertainties in the parameter values. We focus in particular on quantifying uncertainties in the output of a mathematical model induced by uncertainties in the input parameters, often referred to as forward propagation of parametric uncertainties. The first part of this thesis discusses how to signiÞcantly reduce the required number of samples needed in the forward propagation of uncertainties. In particular, if the quantity of interest is highly non-linear or discontinuous depending on the input parameters, Gibbs phenomena may occur, which deteriorate the accuracy globally. As we often do not know beforehand if a quantity of interest is smooth or discontinuous, we propose a new algorithm that works in both cases. In the second part of this thesis, instead of placing samples optimally in order to reduce the overall computational cost of forward propagation, we opt to increase the accuracy of the quantity of interest that is outputted by the solver. To clarify, when using numerical discretisation to compute outcomes of the mathematical model, the resolution of the computational grid determines the accuracy of the quantity of interest, but it also determines the computational time it takes to compute this quantity of interest. In general, a quantity of interest that is computed using a coarse computational grid is fast to compute but also inaccurate. As a remedy, we opt to use machine learning to increase the accuracy of a quantity of interest that is computed on a coarse grid, as it is able to deal with highly non-linear behaviour and high-dimensional input/ output relations. The main topics of this thesis are uncertainty quantiÞcation and machine learning. We assume that the reader possesses basic prior knowledge on these two subjects. Furthermore, the test cases discussed in both Part I and II of this thesis assume prior knowledge on basic numerical discretisation techniques, in particular Þnite volume methods.

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