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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Applied Mathematics Seminars: Phase-Field Approximation of Branched Transport Problems

Presented by Filippo Santambrogio, Université Paris-Sud

17 March 2017 14:00 in CM219

The branched transport problem is the minimization of a concave functional on vector measures with prescribed divergence which translates into a continuous framework a classical problem over graphs. The only admissible measures are those concentrated on 1-rectifiable sets (say, infinite graphs) and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$ (for an exponent $\alpha<1$ ,which favors concentration and joint transportation). I'll present an approximation by Gamma-convergence, through elliptic functionals defined on more regular vector fields, in the same spirit of the well-known Modica-Mortola approximation for the perimeter functional. In such a case, the energies $\frac 1 \varepsilon \int W(v)+ \varepsilon |Dv|^2$ (where $W$ is a double-well potential $W$, minimal on $0$ and $1$), converge to the perimeter of the interface between $\{v=0\}$ and $\{v=1\}$. Here the double-well is replaced with a concave power, so that there is a sort of double-well at $0$ and $\infty$. In this case as well, the energy at the limit concentrates on a lower dimensional structure. Besides the link with the theory of elliptic approximations, the interest of this convergence lies in its applications for numerics. Actually, we built some years ago (in collaboration with E. Oudet) a quite efficient method, which allows to find reasonable local minima of the limit problem in 2D, avoiding the NP complications of the usual combinatorial approaches. The Steiner problem of minimal connection may be approached in this way as well as a limit $\alpha\to 0$. I will present the general picture of this problem and its approximation, also mentioning some recent results by my student A. Monteil, who generalized the convergence proof in higher dimension, and inserted the divergence constraint in the Gamma-convergence result.

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