Adaptive Multi-Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks

In this thesis, we develop reliable and fully error-controlled uncertainty quantification methods for hyperbolic partial differential equations (PDEs) with random data on networks. The goal is to combine adaptive strategies in the stochastic and physical space with a multi-level structure in such a way that a prescribed accuracy of the simulation is achieved while the computational effort is reduced. First, we consider hyperbolic PDEs on networks excluding any type of uncertainty. We introduce a model hierarchy with decreasing fidelity which can be obtained by simplifications of complex model equations. This hierarchy allows to apply more accurate models in regions of the network of complex dynamics and simplified models in regions of low dynamics. Next, we extend the network problem by uncertain initial data and uncertain conditions posed at the boundary and at inner network components. In order to predict the behavior of the considered system despite the uncertainties, we want to approximate relevant output quantities and their statistical properties, like the expected value and variance. For the study of the influence of the uncertainties, we focus on two sampling-based approaches: the widely used Monte Carlo (MC) method and the stochastic collocation (SC) method which is a promising alternative and therefore of main interest in this work. These approaches allow to reuse existing numerical solvers of the deterministic problem such that the implementation is simplified. We develop an adaptive single-level (SL) approach for both methods where we efficiently combine adaptive strategies in the stochastic space with adaptive physical approximations. The physical approximations are computed with a sample-dependent resolution in space, time and model hierarchy. The extension to a multi-level (ML) structure is realized by coupling physical approximations with different accuracies such that the computational cost is minimized. Due to a posteriori error indicators, we can control the discretization of the physical and stochastic approximations in such a way that a user-prescribed accuracy of the simulation is ensured. For the SC methods, we realize the adaptive stochastic strategy by adaptive sparse grids which are able to exploit any smoothness or special structure in the stochastic space, in contrast to MC methods. In addition, we analyze the convergence, the computational cost and the complexity of our SL and ML methods. In order to validate the feasibility of relevant uncertain output quantities, we propose and analyze a sample-based method which approximates the probability that the quantity takes values between a given lower and upper bound on the whole time horizon. To this end, the usually unknown probability density function (PDF) of the output quantity is required. Therefore, we introduce and analyze a kernel density estimator (KDE) which provides an approximation of the PDF of the output quantity and can be computed cost-efficiently in a post-processing step of SC methods. As an application-relevant example, we consider the gas transport in pipeline networks which can be described by the isothermal Euler equations and their simplifications. We present numerical results for two gas network instances with uncertain gas demands and demonstrate the reliability of the error control of our methods approximating the expected value of a random output quantity. The numerical examples show that the MC methods are not competitive due to high computational costs and that the multi-level SC approach outperforms the single-level SC method. Based on the SC methods, we successfully apply the KDE approach to the minimum and maximum pressures at the outflow nodes of the network