arXiv subjects

Jean-David Benamou

Publications and source records attributed to Jean-David Benamou.

15 recordsLinked to original sources

arXiv2025

Entropic approximations of the semigeostrophic shallow water equations

We develop a discretisation of the semigeostrophic rotating shallow water equations, based upon their optimal transport formulation. This takes the form of a Moreau-Yoshida regularisation of the Wasserstein metric. Solutions of the optimal transport formulation provide the shallow water layer depth represented as a measure, which is itself the push forward of an evolving measure under the semigeostrophic coordinate transformation. First, we propose and study an entropy regularised version of the rotating shallow water equations. Second, we discretise the regularised problem by replacing both measures with weighted sums of Dirac measures, and approximate the (squared) L2 norm of the layer depth, which defines the potential energy. We propose an iterative method to solve the discrete optimisation problem relating the two measures, and analyse its convergence. The iterative method is demonstrated numerically and applied to the solution of the time-dependent shallow water problem in numerical examples.

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arXiv2024

Entropic Semi-Martingale Optimal Transport

Entropic Optimal Transport (EOT), also referred to as the Schrödinger problem, seeks to find a random processes with prescribed initial/final marginals and with minimal relative entropy with respect to a reference measure. The relative entropy forces the two measures to share the same support and only the drift of the controlled process can be adjusted, the diffusion being imposed by the reference measure. Therefore, at first sight, Semi-Martingale Optimal Transport (SMOT) problems (see [1]) seem out of the scope of applications of Entropic regularization techniques, which are otherwise very attractive from a computational point of view. However, when the process is observed only at discrete times, and become therefore a Markov chain, its relative entropy can remain finite even with variable diffusion coefficients, and discrete semi-martingales can be obtained as solutions of (multi-marginal) EOT problems.Given a (smooth) semi-martingale, the limit of the relative entropy of its time discretizations, scaled by the time step converges to the so-called ``specific relative entropy'', a convex functional of its variance process, similar to those used in SMOT.In this paper we use this observation to build an entropic time discretization of continuous SMOT problems. This allows to compute discrete approximations of solutions to continuous SMOT problems by a multi-marginal Sinkhorn algorithm, without the need of solving the non-linear Hamilton-Jacobi-Bellman pde's associated to the dual problem, as done for example in [1, 2]. We prove a convergence result of the time discrete entropic problem to the continuous time problem, we propose an implementation and provide numerical experiments supporting the theoretical convergence.

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arXiv2024

From entropic transport to martingale transport, and applications to model calibration

We propose a discrete time formulation of the semi-martingale optimal transport problem based on multi-marginal entropic transport. This approach offers a new way to formulate and solve numerically the calibration problem proposed by [17], using a multi-marginal extension of Sinkhorn algorithm as in [6, 10, 7]. When the time step goes to zero we recover, as detailed in the companion paper [8], a continuous semi-martingale process, solution to a semi-martingale optimal transport problem, with a cost function involving the so-called 'specific entropy' , introduced in [13], see also [12] and [2].

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arXiv2024

From entropic transport to martingale transport, and applications to model calibration

We propose a discrete time formulation of the semi martingale optimal transport problembased on multi-marginal entropic transport. This approach offers a new way to formulate and solve numerically the calibration problem proposed by Guo et al. 2022, using a multi-marginal extension of Sinkhorn algorithm as in Benamou, Carlier, and Nenna 2019; Carlier et al. 2017; Benamou et al. 2019. In the limit when the time step goes to zero we recover, as detailed in the companion paper Benamou et al. 2024, a semi-martingale process, solution to a semi-martingale optimal transport problem, with a cost function involving the so-called specific entropy introduced in Gantert 1991, see also F{ö}llmer 2022 and Backhoff-Veraguas and Unterberger 2023.

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arXiv2019

An entropy minimization approach to second-order variational mean-field games

We propose a new viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We show the equivalence of such mean-field games with a relative entropy minimization at the level of probabilities on curves. We also address the time-discretization of such problems, establish $Γ$-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

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arXiv2018

Second order models for optimal transport and cubic splines on the Wasserstein space

On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multi-marginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.

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arXiv2018

Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem

We present an adaptation of the MA-LBR scheme to the Monge-Amp{è}re 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 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 stepsize vanishes and we show with numerical experiments that it is able to reproduce subtle properties of the Optimal Transport problem.

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arXiv2018

Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

Starting from Brenier's relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem. Numerical results in dimension one and two illustrate the feasibility of the method.

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arXiv2015

A Numerical Method to solve Optimal Transport Problems with Coulomb Cost

In this paper, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is to introduce an entropic regularization of the Kantorovich formulation of the Optimal Transport problem. The regularized problem then corresponds to the projection of a vector on the intersection of the constraints with respect to the Kullback-Leibler distance. Iterative Bregman projections on each marginal constraint are explicit which enables us to approximate the optimal transport plan. We validate the numerical method against analytical test cases.

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arXiv2014

Iterative Bregman Projections for Regularized Transportation Problems

This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.

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arXiv2014

Monotone and Consistent discretization of the Monge-Ampere operator

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

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arXiv2014

Discretization of functionals involving the Monge-Ampère operator

Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension >= 2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge-Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.

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arXiv2014

A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation

We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an elliptic partial differential equation known as the Monge-Ampere equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We introduce a new formulation that couples the viscosity and Aleksandrov solution definitions and show that it is equivalent to the original problem. Moreover, we describe a local reformulation of the subgradient measure at the Diracs, which makes use of one-sided directional derivatives. This leads to a consistent, monotone discretisation of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail.

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arXiv2013

A viscosity solution approach to the Monge-Ampere formulation of the Optimal Transportation Problem

In this work we present a numerical method for the Optimal Mass Transportation problem. Optimal Mass Transportation (OT) is an active research field in mathematics.It has recently led to significant theoretical results as well as applications in diverse areas. Numerical solution techniques for the OT problem remain underdeveloped. The solution is obtained by solving the second boundary value problem for the MA equation, a fully nonlinear elliptic partial differential equation (PDE). Instead of standard boundary conditions the problem has global state constraints. These are reformulated as a tractable local PDE. We give a proof of convergence of the numerical method, using the theory of viscosity solutions. Details of the implementation and a fast solution method are provided in the companion paper arXiv:1208.4870.

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arXiv2012

Numerical solution of the Optimal Transportation problem using the Monge-Ampere equation

A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary conditions is combined with a finite difference scheme for the Monge-Ampere equation. Newton's method is implemented leading to a fast solver, comparable to solving the Laplace equation on the same grid several times. Theoretical justification for the method is given by a convergence proof in the companion paper (Benamou et al., 2012). In this paper, the algorithm is modified to a simpler compact stencil implementation and details of the implementation are given. Solutions are computed with densities supported on non-convex and disconnected domains. Computational examples demonstrate robust performance on singular solutions and fast computational times.

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