Pages: 1039 - 1040 Abstract: The mathematical theory of Optimal Transport (OT) is now one of the most visible and also most active fields in nonlinear analysis, with two Fieldsmedal winners Cedric Villani and Alessio Figalli among its protagonists. The origins of OT date back to the works of GaspardMonge in the 18th century and those of Leonid Kantorovich in the ’1940s. However, what is considered as the foundations of the modern theory have mostly been established in several ground-breaking works at the end of the 20th century: among the most prominent contributors, there is Yann Brenier, who characterized optimal transport in terms of connections with PDEs and hydrodynamics in the 1980’s; there is RobertMcCann, who introduced the powerful concept of displacement convexity in the 1990’s; there is Felix Otto, who demonstrated the implications of OT to nonlinear evolution equations around the millennium. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792519000147 Issue No:Vol. 30, No. 6 (2019)

Authors:JEAN-DAVID BENAMOU; VINCENT DUVAL Pages: 1041 - 1078 Abstract: We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000451 Issue No:Vol. 30, No. 6 (2019)

Authors:GABRIEL PEYRÉ; LÉNAÏC CHIZAT, FRANÇOIS-XAVIER VIALARD, JUSTIN SOLOMON Pages: 1079 - 1102 Abstract: This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This “quantum” formulation of optimal transport (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycentres within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792517000274 Issue No:Vol. 30, No. 6 (2019)

Authors:JOSÉ A. CARRILLO; ANSGAR JÜNGEL, MATHEUS C. SANTOS Pages: 1103 - 1122 Abstract: The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792517000389 Issue No:Vol. 30, No. 6 (2019)

Authors:CLÉMENT CANCÈS; THOMAS GALLOUËT, MAXIME LABORDE, LÉONARD MONSAINGEON Pages: 1123 - 1152 Abstract: The Wasserstein gradient flow structure of the partial differential equation system governing multiphase flows in porous media was recently highlighted in Cancès et al. [Anal. PDE10(8), 1845–1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [SIAM J. Math. Anal.29(1), 1–17]) scheme that we solve thanks to the ALG2-JKO scheme proposed in Benamou et al. [ESAIM Proc. Surv.57, 1–17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000633 Issue No:Vol. 30, No. 6 (2019)

Authors:M. FORNASIER; S. LISINI, C. ORRIERI, G. SAVARÉ Pages: 1153 - 1186 Abstract: This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ -convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792519000044 Issue No:Vol. 30, No. 6 (2019)

Authors:SIMONE CACACE; FABIO CAMILLI, RAUL DE MAIO, ANDREA TOSIN Pages: 1187 - 1209 Abstract: We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000621 Issue No:Vol. 30, No. 6 (2019)

c-cyclical+monotonicity+for+optimal+transport+problem+with+Coulomb+cost&rft.title=European+Journal+of+Applied+Mathematics&rft.issn=0956-7925&rft.date=2019&rft.volume=30&rft.spage=1210&rft.epage=1219&rft.aulast=PASCALE&rft.aufirst=LUIGI&rft.au=LUIGI+DE+PASCALE&rft_id=info:doi/10.1017/S0956792519000111">On c-cyclical monotonicity for optimal transport problem with Coulomb cost

Authors:LUIGI DE PASCALE Pages: 1210 - 1219 Abstract: It is proved that c-cyclical monotonicity is a sufficient condition for optimality in the multi-marginal optimal transport problem with Coulomb repulsive cost. The notion of c-splitting set and some mild regularity property are the tools. The result may be extended to Coulomb like costs. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792519000111 Issue No:Vol. 30, No. 6 (2019)

Authors:PIERRE-ANDRÉ CHIAPPORI; ROBERT MCCANN, BRENDAN PASS Pages: 1220 - 1228 Abstract: We study a one-parameter class of examples of optimal transport problems between a two-dimensional source and a one-dimensional target. Our earlier work identified a nestedness condition on the surplus function and marginals, under which it is possible to solve the problem semi-explicitly. In the family of examples we consider, we classify the values of parameters which lead to nestedness. In those cases, we derive an almost explicit characterisation of the solution. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000578 Issue No:Vol. 30, No. 6 (2019)

Authors:J.-J. ALIBERT; G. BOUCHITTÉ, T. CHAMPION Pages: 1229 - 1263 Abstract: We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ( $G(x,p)=\int c(x,y)dp$ ) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000669 Issue No:Vol. 30, No. 6 (2019)

Authors:ALISTAIR BARTON; NASSIF GHOUSSOUB Pages: 1264 - 1299 Abstract: Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L, $$b_T(v, x):=\inf\left\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T], M); \gamma(T)=x\right\}\!,$$ where $M = \mathbb{R}^d$ , and T >0. We also consider the stochastic counterpart: \begin{align*}\underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,\beta(t,X))\,dt\right]\!; X\in \mathcal{A}, V\sim\mu,X_T\sim \nu\right\}\!,\end{align*} where $\mathcal{A}$ is the set of stochastic processes satisfying dX = βX (t, X) dt + dWt, for some drift βX (t, X), and where Wt is σ(Xs: 0 ≤ s ≤ t)-Brownian motion. Both cases lead to Lax–Oleinik-type formulas on Wasserstein space that relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard–Buffoni and Fathi–Figalli in the deterministic case, and by Mikami–Thieullen in the stochastic setting. While inf-convolution easily covers cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian L is jointly convex on phase space, Bolza-type dualities – well known in the deterministic case but novel in the stochastic case – transform sup-inf problems to sup–sup settings. We also write Eulerian formulations and point to links with the theory of mean-field games. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792519000032 Issue No:Vol. 30, No. 6 (2019)

Authors:JEAN BÉRARD; NICOLAS JUILLET Pages: 1300 - 1310 Abstract: We discuss the reconciliation problem between probability measures: given n⩾2 probability spaces $(\Omega,{\mathcal{F}}_1,{\mathbb{P}}_1),\ldots,(\Omega,{\mathcal{F}}_n,{\mathbb{P}}_n)$ with a common sample space, does there exist an overall probability measure ${\mathbb{P}} \ \text{on} \ {\mathcal{F}} = \sigma({\mathcal{F}}_1,\ldots,{\mathcal{F}}_n)$ such that, for all i, the restriction of ${\mathbb{P}} \ \text{to} \ {\mathcal{F}}_i$ coincides with ${\mathbb{P}}_i$ ? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0956792518000682 Issue No:Vol. 30, No. 6 (2019)