Mixed Finite Elements, Monge-Ampere equation and Optimal Transportation

本文的目标是研究混合有限元方法在最优运输问题中运输图的有效计算。重点是运输成本是距离的二次函数的问题。它们引出了蒙日-安培方程。项目的第一部分包括阐明有限元型方法对方程弱解的适用性。这里的关键方法是用光滑函数逼近。在第二部分中，混合有限元分析技术适用于方程光滑解的近似。最优运输在各个领域的应用越来越广泛，从理论领域，如几何和分析，到应用领域，如生物学、模式识别、图像处理、流体力学、地球物理学、气象学、光学、海洋学和宇宙学。这就产生了对高效、稳健的数值方法的迫切需求，这些方法可以从理论上支持解决最优运输问题。该项目开发的高效可靠的方法可用于解决许多其他应用中出现的最优运输问题，例如天气预报，交通拥堵，经济，网格均匀分布，纹理映射等。本文研究了最优输运的蒙日-安培方程，目的是从理论上阐明高效混合有限元方法的应用。它推进了一些涉及蒙日-安培型方程的解析和几何开放问题的解决。

The goal of the proposal is to study efficient mixed finite element methods for the computation of transport maps in optimal transportation problems. The focus is on problems in which the cost of transport is a quadratic function of the distance. They lead to Monge-Ampere equations. The first part of the project consists in clarifying the applicability of finite element type methods to weak solutions of the equation. The key approach here is approximation by smooth functions. In the second part, techniques of mixed finite element analysis are adapted to the approximation of smooth solutions of the equation.Optimal transportation has a growing application in various fields ranging from theoretical ones such as geometry and analysis to applied fields such as biology, pattern recognition, image processing, fluid mechanics, geophysics, meteorology, optics, oceanography and cosmology. This has created the critical need for efficient and robust numerical methods backed up theoretically to solve optimal transportation problems. The efficient and reliable methods developed from this project could be used to solve optimal transportation problems which appear in many other applications e.g. weather forecasting, traffic congestion, economics, mesh equidistribution, texture mapping, etc. The proposal studies the Monge-Ampere equation of optimal transportation with the goal of clarifying theoretically the use of the efficient mixed finite element methods. It advances knowledge towards the resolution of some open problems in analysis and geometry involving Monge-Ampere type equations.