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Description:
We consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen-Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen-Loève spectral decomposition of the random flux into principal components with respect to the state variables, we introduce a family of parametric, deterministic entropy solutions on high-dimensional parameter spaces. We prove bounds on the sensitivity of the parametric and of the random entropy solutions on the Karhunen-Loève parameters. We also outline the convergence analysis for two classes of discretization schemes, the Multi-Level Monte-Carlo Finite-Volume Method (MLMCFVM) developed in [22, 24, 23], and the stochastic collocation Finite Volume Method (SCFVM) of [25].