Mixed Finite Elements, Monge-Ampere equation and Optimal Transportation

该方案的目的是研究在最优运输问题中计算运输地图的高效混合有限元方法。焦点集中在运输成本是距离的二次函数的问题上。它们导致了Monge-Ampere方程。该项目的第一部分在于阐明有限元类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，将混合有限元分析技术应用于方程光滑解的逼近。最优输运在从几何和分析等理论到生物学、模式识别、图像处理、流体力学、地球物理、气象学、光学、海洋学和宇宙学等应用领域得到了越来越广泛的应用。这就产生了对高效和稳健的数值方法的迫切需求，这些方法在理论上是有依据的，以解决最优运输问题。该项目开发的高效、可靠的方法可用于解决在天气预报、交通拥堵、经济学、网格均匀分布、纹理映射等许多应用中出现的最优交通问题。它促进了对涉及Monge-Ampere型方程的分析和几何中的一些公开问题的解决的认识。

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.