Flux identification for 1-${\mathbf d}$ scalar conservation laws in the presence of shocks

Abstract:

We consider the problem of flux identification for 1-d scalar conservation laws formulating it as an optimal control problem. We introduce a new optimization strategy to compute numerical approximations of minimizing fluxes.

We first prove the existence of minimizers. We also prove the convergence of discrete minima obtained by means of monotone numerical approximation schemes, by a $\Gamma$-convergence argument. Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches. The first one, the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a remedy we adapt the method of alternating descent directions that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions, introduced by the authors, in collaboration with F. Palacios, in the case where the control is the initial datum. This method distinguishes descent directions that move the shock and those that perturb the profile of the solution away from it. As we shall see, a suitable alternating combination of these two classes of descent directions allows building more efficient and faster descent algorithms.