Variational data assimilation for discrete Burgers equation

We present an optimal control formulation of the data assimilation problem for the Burgers' equation, with the initial condition as the control. First the convergence of the implicit Lax-Friedrichs numerical discretization scheme is presented. Then we study the dependence of the convergence of the associated minimization problem on different terms in the cost function, specifically, the weight for the regularization and the number of observations, as well as the a priori approximation of the initial condition. We present numerical evidence for multiple minima of the cost function without regularization, while only a single minimum is seen for the regularized problem.