ALEX 2018 Workshop: Abstracts

Entropic regularization of optimal transport and mean field planning

Giuseppe Savaré

University of Pavia, Department of Mathematics “F. Casorati” (Italy)

First order mean field planning problems can be characterized by a nonlinear system of a Hamilton-Jacobi equation coupled with a continuity equation for the nonnegative density distribution $m$

\left\{\begin{aligned} \displaystyle-\partial_{t}u+H(x,Du)&\displaystyle=f(x,m% )&&\displaystyle\text{in }(0,1)\times\mathbb{R}^{d},\\ \displaystyle\partial_{t}m-\nabla\cdot(m\,H_{\bm{p}}(x,Du))&\displaystyle=0&&% \displaystyle\text{in }(0,1)\times\mathbb{R}^{d},\\ \displaystyle m(0,\cdot)=m_{0},\;m(1,\cdot)&\displaystyle=m_{1}&&\displaystyle% \text{in }\mathbb{R}^{d},\end{aligned}\right.{}

(MFPP)

with prescribed initial and final boundary condition at $t=0,1$ . In the framework of mean field game theory [3], the planning problem was suggested and developed by P.-L. Lions in his courses at Collège de France.

((MFPP)) can be interpreted as the first order optimality condition of the following minimization problem in $(0,1)\times\mathbb{R}^{d}$

\min\iint\big{[}L(x,\bm{v})\,m+F(x,m)\big{]}\,\,\mathrm{d}x\,\mathrm{d}t\,:\ % \bm{v}\in L^{2}(m\,\mathrm{d}x\mathrm{d}t),\quad\left\{\begin{aligned} % \displaystyle\partial_{t}m+\nabla\cdot(m\,\bm{v})&\displaystyle=0\\ \displaystyle m(0,\cdot)=m_{0}\,,m(1,\cdot)&\displaystyle=m_{1},\end{aligned}\right.

which is the entropic regularization (by the primitive $F$ of $f$ ) of the dynamic optimal transportation problem [2, 1, 4], whose Lagrangian cost $L$ is the Fenchel conjugate of the Hamiltonian $H$ . The structure of ((MFPP)) naturally arises from the coupling with the dual problem

\begin{gathered}\displaystyle\max\int u_{0}m_{0}\,\mathrm{d}x-\int u_{1}m_{1}% \,\mathrm{d}x-\iint F^{*}(\alpha)\,\mathrm{d}x\,\mathrm{d}t,\\ \displaystyle\text{under the constraint}\quad-\partial_{t}u+H(x,Du)\leq\alpha% \quad\text{in }(0,1)\times\mathbb{R}^{d}.\end{gathered}

By using some ideas and techniques of optimal transport theory, minimax duality, and dynamic superposition principles, we will discuss the well posedness of both the variational problems in a suitable functional setting, their strong duality, and their link with an appropriate measure-theoretic formulation of (MFPP).
(In collaboration with Carlo Orrieri and Alessio Porretta).

References

1 Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.
2 Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84(3):375–393, 2000.
3 Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Jpn. J. Math., 2(1):229–260, 2007.
4 Cédric Villani. Optimal transport. Old and new, volume 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2009.