@article{SchusterStrauchGugatetal.2020, author = {Schuster, Michael and Strauch, Elisa and Gugat, Martin and Lang, Jens}, title = {Probabilistic Constrained Optimization on Flow Networks}, volume = {Optimization and Engineering}, doi = {https://doi.org/10.1007/s11081-021-09619-x}, pages = {50}, year = {2020}, abstract = {Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.}, language = {en} } @article{LangDomschkeStrauch2020, author = {Lang, Jens and Domschke, Pia and Strauch, Elisa}, title = {Adaptive Single- and Multilevel Stochastic Collocation Methods for Uncertain Gas Transport in Large-Scale Networks}, volume = {In: Mesh Generation and Adaptation, Cutting-Edge Techniques. R. Sevilla, S. Perotto, K. Morgan (eds.), SEMA-SIMAI Springer Series}, number = {Vol. 30}, pages = {113 -- 135}, year = {2020}, abstract = {In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.}, language = {en} }