Mixed Finite Elements, Monge-Ampere equation and Optimal Transportation

该提案的目标是研究有效的混合有限元方法计算的运输地图在最佳运输问题。重点是运输成本是距离的二次函数的问题。它们导致了Monge-Ampere方程。该项目的第一部分包括在澄清有限元类型的方法弱解方程的适用性。这里的关键方法是光滑函数的近似。在第二部分中，混合有限元分析技术被用来逼近方程的光滑解，最优运输在各个领域有越来越多的应用，从理论的，如几何和分析，应用领域，如生物学，模式识别，图像处理，流体力学，物理学，气象学，光学，海洋学和宇宙学。这就迫切需要有理论支持的高效、稳健的数值方法来解决最优运输问题。从这个项目开发的有效和可靠的方法可以用来解决最优运输问题，出现在许多其他应用，如天气预报，交通拥堵，经济，网格均匀分布，纹理映射等建议研究的蒙格-安培方程的最优运输的目标是澄清理论上使用的高效混合有限元方法。它推进知识对解决一些开放的问题，在分析和几何涉及蒙日安培型方程。

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.