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 an approximated problem 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.
Highlights
Introduction and motivationIn this paper, we present a method which describes how to deal with uncertain loads in the context of flow networks
Such a flow problem is modeled as a system of hyperbolic balance laws based on e.g. the isothermal Euler equations or the shallow water equations
We have shown two different ways to evaluate probabilistic constraints in the context of hyperbolic balance laws on graphs for both, a stationary and a dynamic setting with box constraints for the solution
Summary
We present a method which describes how to deal with uncertain loads in the context of flow networks. On we use SRD instead of spheric radial decomposition and KDE instead of kernel density estimator. A KDE approach was used in Caillau et al (2018) for solving chance constrained optimal control problems with ODE constraints It was neither used in optimal control problems with PDE constraints and random boundary data nor in the context of continuous optimization with hyperbolic balance laws on networks. We state necessary optimality conditions for probabilistic constrained optimization problems related to our dynamic model and solve a probabilistic constrained optimization problem for a realistic water contamination network setting
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