Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme (Communications in Computational Physics, 2020)
We study an identification problem which estimates the parameters of theunderlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkinmethod, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertaininitial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus onidentifying the correct endpoints of this interval. The first-order optimality conditionsfrom the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backwardschemes with the Runge-Kutta method. To illustrate the feasibility of the approach,we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parametersof the random variable in the uncertain differential equation even if discontinuities arepresent.