The problem I'll present is a transport problem where costs depend on how crowded each path is. One can look either for equilibrium situations (known as Wardrop equilibria, which correspond to Nash equilibria, where nobody wants to change his mind) or for optimal configuration under a total displacement cost criterium. Equivalence results do exist, both in the discrete case (a finite network of roads) and in the continuous one (where every location and every path are admitted), and hence one can compute equilibria by variational methods. We propose a suitable discretization, via the well-known fast maching algoriths for solving the Eikonal equation, and we compute the gradient of the discrete (convex) functional via a similar algorithm, thus obtaining an efficient method for solving these problems.
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