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Convex Analysisの二,三の進展について

Convex bodies: the brunn–minkowski theory

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

Optical Waveguide Theory

Full waveform inversion using the conventional L2 distance to measure the misfit between seismograms is known to suffer from cycle skipping. An alternative strategy is proposed in this study, based on a measure of the misfit computed with an optimal transport distance. This measure allows to account for the lateral coherency of events within the seismograms, instead of considering each seismic trace independently, as is done generally in full waveform inversion. The computation of this optimal transport distance relies on a particular mathematical formulation allowing for the non-conservation of the total energy between seismograms. The numerical solution of the optimal transport problem is performed using proximal splitting techniques. Three synthetic case studies are investigated using this strategy: the Marmousi 2 model, the BP 2004 salt model, and the Chevron 2014 benchmark data. The results emphasize interesting properties of the optimal transport distance. The associated misfit function is less prone to cycle skipping. A workflow is designed to reconstruct accurately the salt structures in the BP 2004 model, starting from an initial model containing no information about these structures. A high-resolution P-wave velocity estimation is built from the Chevron 2014 benchmark data, following a frequency continuation strategy. This estimation explains accurately the data. Using the same workflow, full waveform inversion based on the L2 distance converges towards a local minimum. These results yield encouraging perspectives regarding the use of the optimal transport distance for full waveform inversion: the sensitivity to the accuracy of the initial model is reduced, the reconstruction of complex salt structure is made possible, the method is robust to noise, and the interpretation of seismic data dominated by reflections is enhanced.

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Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the $L^2$ optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.

https://hal.archives-ouvertes.fr/hal-00816211/document

Optimal Transport with Proximal Splitting

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.

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

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.

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Numerical methods for matching for teams and wasserstein barycenters

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: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a subdomain minimizing the ratio surface area on volume; Newton's 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.

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Edouard Oudet

Papers

Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion

Optimal Transport with Proximal Splitting

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

Numerical methods for matching for teams and wasserstein barycenters

Minimizing within Convex Bodies Using a Convex Hull Method