Adaptive Multi-Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks
Elisa Strauch
In this thesis, we develop reliable and fully error-controlled uncertainty quantification methods for hyperbolic partial differential equations with random data on networks. For the study of the influence of uncertainties, we focus on two sampling-based approaches: the widely used Monte Carlo (MC) method and the stochastic collocation (SC) method. The goal is to combine adaptive strategies in the stochastic and physical spaces with a multi-level structure in such a way that a prescribed accuracy of the simulation is achieved while the computational effort is reduced. 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. In addition, we analyze the convergence, the computational cost and the complexity of our developed methods.
Moreover, we propose and analyze a sampling-based approach to validate the feasibility of relevant uncertain output quantities based on kernel density estimators. We approximate 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 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.