Édouard Oudet

Minimizing within Convex Bodies Using a Convex Hull Method

We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex. To deal with this, our method mixes geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrovs problem of finding a convex body of prescribed surface function; Cheegers problem of a subdomain minimizing the ratio surface area on volume; Newtons problem of the body of minimal resistance. In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes.

An Optimization Problem for Mass Transportation with Congested Dynamics

Starting from the work by Brenier [Extended Monge-Kantorovich theory, in Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin (2003), pp. 91-121], where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kinds of cost functions are allowed. This seems relevant in some problems presenting congestion effects as, for instance, traffic on a highway, crowds moving in domains with obstacles, and, in general, in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in Benamou and Brenier [A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393].

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

An optimal transport approach for seismic tomography: application to 3D full waveform inversion

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Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint

⩾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.

Regularization of Discrete Contour by Willmore Energy

A numerical process to approximate optimal partitions in any dimension is reported. The key idea of the method is to relax the problem into a functional framework based on the famous result of Γ-convergence obtained by Modica and Mortolla.

Discrete Optimal Transport: Complexity, Geometry and Applications

The use of optimal transport distance has recently yielded significant progress in image processing for pattern recognition, shape identification, and histograms matching. In this study, the use of this distance is investigated for a seismic tomography problem exploiting the complete waveform; the full waveform inversion. In its conventional formulation, this high resolution seismic imaging method is based on the minimization of the L2 distance between predicted and observed data. Application of this method is generally hampered by the local minima of the associated L2 misfit function, which correspond to velocity models matching the data up to one or several phase shifts. Conversely, the optimal transport distance appears as a more suitable tool to compare the misfit between oscillatory signals, for its ability to detect shifted patterns. However, its application to the full waveform inversion is not straightforward, as the mass conservation between the compared data cannot be guaranteed, a crucial assumption for optimal transport. In this study, the use of a distance based on the Kantorovich–Rubinstein norm is introduced to overcome this difficulty. Its mathematical link with the optimal transport distance is made clear. An efficient numerical strategy for its computation, based on a proximal splitting technique, is introduced. We demonstrate that each iteration of the corresponding algorithm requires solving the Poisson equation, for which fast solvers can be used, relying either on the fast Fourier transform or on multigrid techniques. The development of this numerical method make possible applications to industrial scale data, involving tenths of millions of discrete unknowns. The results we obtain on such large scale synthetic data illustrate the potentialities of the optimal transport for seismic imaging. Starting from crude initial velocity models, optimal transport based inversion yields significantly better velocity reconstructions than those based on the L2 distance, in 2D and 3D contexts.

Phase-field approximations of the Willmore functional and flow

We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of