Numerical approximation of the flux identification problem for a scalar conservation law

We consider the problem of flux identification for 1-d scalar conservation laws stated as an optimal control problem.

We first prove the existence of minimizers and the convergence of discrete minima obtained by means of numerical approximation schemes. Then we address the problem of developing efficient
descent algorithms. We 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. We show that, in this case, the latter is more efficient than the
discrete one in presence of discontinuities.