Optimal Transport: Theory and Applications to cosmological Reconstruction and Image processing – OTARIE

Optimal transport is a mathematical subject connecting the fields of optimization, partial differential equations, and dynamical systems in both finite and infinite dimensions. The basic problem of optimal transport is to find a transport plan that connects two given distributions of mass and minimizes a certain transportation cost; such plans turn out to be generated by solutions to suitable PDEs, notably the Monge-Ampere and Hamilton-Jacobi equations. The recent vigorous theoretical development in this area, which has been particularly strong in France, led to realization that many important fundamental and practical problems, arising in subjects as diverse as cosmology and computer science, may be rendered into optimal transport in suitable spaces. These formulations provide new competitive approaches, or sometimes even open the possibility of solution for the first time. However in practice optimal transport strategies offer serious challenges for numerical analysis due to the large size of typical problems. The primary aim of the present project is to develop a toolbox of efficient numerical methods and codes for optimal transport problems. The participating teams involve mathematicians as well as specialists in cosmology, image processing, and hydrodynamics. The motivation comes from actual applications to these subject fields, already explored in pilot studies of the prospective participants, but the tools developed in this project will have wider applicability to numerical transport optimization. Reconstruction of peculiar velocities of galaxies from large-scale redshift catalogues provides an exemplary application of optimal transport. Here the cosmological Zeldovich approximation leads to either Monge-Ampère ot Hamilton-Jacobi (Bernouilli) equation, which is solved by optimization of transport cost for data sets that include up to millions of objects, using continuous as well as combinatorial strategies; the resulting reconstructed velocities may further be used to constrain various quantities of interest for cosmologists. Three of the four participating teams (Poncelet, IAP, Doeblin) will contribute substantially to developing, enhancing and applying optimal transport methods in cosmological reconstruction. Using these tools we expect to significantly improve the accuracy of reconstructed peculiar velocities and refine scales of reconstruction. In image analysis and processing, morphing, colour adjustment, and image restoration can be tackled as McCann interpolation (transport) in physical or colour space. For image registration and pattern recognition, it is very useful to know the Monge-Kantorovich (transport) distance. At Cérémade new efficient parallelizable numerical transport algorithms will be developed, able to cope with the ever increasing resolution of images and size of their collections. On the numerical and algorithmic side we will pursue both combinatorial approaches (efficient Euclidean matching in low dimensions, semidefinite programming) and continuum approaches (the Benamou-Brenier-Uzawa method, optimal bilinear control, multiscale methods for the Monge-Ampère equation and Hamilton-Jacobi equations). Here we expect to improve performance and achieve much larger problem sizes with respect to already efficient pilot solvers that were developed in participating teams. This work will be coupled with theoretical study of several aspects of optimal transport (regularity of solutions, measure spaces with distances generated by Lagrangian dynamical systems) that are of practical importance for numerical transport optimization. We will also explore theoretically the relevant models of hydrodynamics (zero-pressure gas dynamics, caustic formation in multistream flows), which are connected to optimal transport and provide mathematical concepts and tools for the description of dynamics and the search for dark matter in the Universe. To our knowledge this is a first coordinated project of developing a suite of optimal transport solvers based on a wide spectrum of promising numerical approaches. An important feature of the project is the close collaboration of mathematicians and specialists in applied fields (cosmology, image processing, hydrodynamics) at both team and inter-team levels. Integration of different generation is another priority (participation of 3 PhD students and 4.5 postdoc-years are envisaged). The project will also help achieve a tighter integration between Paris (Ceremade, IAP) and Nice (Institut Wolfgang Doeblin) research centres. Last but not least, participation of the J. V. Poncelet laboratory (UMI 2615) as coordinating partner will increase integration between the French and Russian research communities.