Geometric PDE's and Monge-Kantorovich Theory in Problems of Optics and Differential Geometry

光学和微分几何问题中的几何偏微分方程和蒙日-康托罗维奇理论

• 批准号: 0405622
• 负责人: Vladimir Oliker
• 金额: $ 10.8万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405622&HistoricalAwards=false
• 关键词: Geometric; PDE; Monge; Kantorovich; Theory

DMS-0405622P.I.: Vladimir Oliker, Emory UniversityTitle: Geometric PDE's and Monge-Kantorovich Theory in Problems ofOptics and Differential GeometryABSTRACTThe goal of this work is the development of geometricmethods for solving nonlinear partial differential equations (PDE's)arising in problems involving maps with controlled Jacobiandeterminant. Many problems in differential geometry, optics,and other areas of mathematics and engineering are inthis class. Recently discovered deep connections between such equationsand Monge-Kantorovich optimal mass transfer theory in Euclidean spaceand on manifolds will also be studied.Among the topics that will be considered are the following.(1) Development of geometrical and analytic techniques for solving problems requiring determination of reflecting and refracting interfaces with capabilities to transform intensitydistributions in a prescribed manner. (2) Investigation of geometric problems involving hypersurfaces with prescribed curvature functions and geometric inequalities with emphasis on variational methods, especially,those connected with Monge-Kantorovich theory; applications of these variational methods to problems in convexity, in particular, to the Minkowski problem andits various generalizations, will be studied as a part of this program.(3) Development of geometrically motivated, provably convergent and efficient multi-scale numerical methods for solving nonlinear second order PDE's arising in reflector/refractor problems of optics and in geometric problems involving curvature functions and maps with controlled volume. Nonlinear partial differentials equations expressing energy conservation laws as a constraint on the Jacobian of a map describing a physicalphenomenon are very common in science and engineering. For example,in optics such equations arise naturally in problemsrequiring determination of interfaces with prescribedrefractive and/or reflective properties; in astrophysics these equationshave to be solved when the shape of targets in the solarsystem must be determined from indirect and limited set of measurements;in weather prediction models based on quasi- and semi-geostrophicapproximations of atmospheric motion such equations describeenergy conservation laws; in computer science the same type ofequations arise in problems connected with radiosity estimates.Typically, the theoretical analysis and numerical solution ofthese equations is very difficult because of their highly nonlinearstructure. Fortunately, the geometric content common to all these problemsprovides important insights leading to effective methods for theirinvestigation and numerical solution. Development of such methodsis the main goal of this research.

DMS-0405622P.I.: Vladimir Oliker, Emory UniversityTitle: Geometric PDE's and Monge-Kantorovich Theory in Problems ofOptics and Differential GeometryABSTRACTThe goal of this work is the development of geometricmethods for solving nonlinear partial differential equations (PDE's)arising in problems involving maps with controlled Jacobiandeterminant.差异几何形状，光学和其他数学和工程领域的许多问题都是班级。最近还将研究这些方程式和Monge-Kantorovich在欧几里得间隔中的最佳传质理论之间的深厚联系。也将研究歧管上的间隔。将考虑以下主题。（1）开发几何和分析技术，用于解决反映和折射界面的求解，以求解与Cospability infortions informentions in tromentibs in tromentib in tromentib in coscability in coscability in coscability in tromenty in coscability of Compability in the Mentersife in tossiquals的差异和分析技术。 （2）研究涉及具有规定曲率功能和几何不平等的高曲面的几何问题，重点是变异方法，尤其是与Monge-Kantorovich理论相关的方法； applications of these variational methods to problems in convexity, in particular, to the Minkowski problem andits various generalizations, will be studied as a part of this program.(3) Development of geometrically motivated, provably convergent and efficient multi-scale numerical methods for solving nonlinear second order PDE's arising in reflector/refractor problems of optics and in geometric problems involving curvature functions and maps with受控体积。表达节能定律的非线性部分差分方程是对描述物理苯甲瘤的地图的雅各布式的限制，在科学和工程中非常普遍。例如，在光学方程式中，这种方程自然出现在确定具有处方赋予和/或反射属性的界面的问题上。在天体物理学中，当必须从间接和有限的测量集中确定太阳能系统中目标的形状；在天气预测模型中，基于大气和半神经遗传的大气运动这些方程式描述了人缘保护定律，则可以解决这些方程；在计算机科学中，同一类型的等式出现在与放射线估计相关的问题中。从典型上讲，由于它们高度非线性结构，因此这些方程的理论分析和数值解决方案非常困难。幸运的是，所有这些问题共有的几何含量表明了重要的见解，从而为他们的研究和数值解决方案提供了有效的方法。这种方法的开发是这项研究的主要目标。