Thibault Liard, Enrique Zuazua. Analysis and numerics solvability of backward-forward conservation laws. (2020)
Abstract. In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce a non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L2(R) norm. The two main contributions of this work are as follows.
• We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem.
• A wave-front tracking method is implemented to construct numerically all of them.
One of minimizers is the backward entropy solution, constructed using a backward-forward method.