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. 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.Īs an application-relevant example, we consider the gas transport in pipeline networks which can be described by the isothermal Euler equations and their simplifications. To this end, the usually unknown probability density function (PDF) of the output quantity is required. Examples of stochastic processes will be taken from both classical and quantum processes. The course Stochastic Methods will introduce students to different random processes, their theoretical description and the numerical methods employed to study them. We approximate the probability that the quantity takes values between a given lower and upper bound on the whole time horizon. In other words, stochastic processes are the norm, not the exception, in everyday life. Galton-Watson tree is a branching stochastic process arising from Fracis Galton’s statistical investigation of the extinction of family names. Moreover, we propose and analyze a sampling-based approach to validate the feasibility of relevant uncertain output quantities based on kernel density estimators. In addition, we analyze the convergence, the computational cost and the complexity of our developed methods. 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. 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. 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. In this thesis, we develop reliable and fully error-controlled uncertainty quantification methods for hyperbolic partial differential equations with random data on networks.
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