Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin Scheme

TL;DR

This paper presents a method to identify the unknown parameters of the random distribution in uncertain scalar conservation laws using a discontinuous stochastic Galerkin scheme, demonstrating effectiveness even with discontinuities.

Contribution

It introduces a novel approach combining stochastic Galerkin discretization with optimality conditions for parameter identification in uncertain hyperbolic PDEs.

Findings

Successfully identifies distribution endpoints in stochastic advection and Burgers' equations.

Method handles discontinuities effectively.

Accurately estimates parameters from observed solution data.

Abstract

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial 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 on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes 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 parameters of the random variable in the uncertain differential equation even if discontinuities are present.