首页>>学术前沿>>学术预告:汪徐家谈“Partial differential equations arising in light reflection and optimization”

学术预告:汪徐家谈“Partial differential equations arising in light reflection and optimization”

作者:沈冬杰   点击:63  发布时间:2015-01-04 12:41:43

时间:201518日上午930-1030

地点:下沙校区教学D104(理学院党员之家)

摘要:

In the design of a reflector antenna, we are given a light source and a surface which is to be illuminated.  We want to design a reflector such that the output light covers the given surface. We show that this is in fact an optimal transport problem. The general optimal transportation is to find an optimal mapping of transferring one mass density to another one such that the total cost is minimised. This problem was first introduced by Monge in 1781. Monge's cost function is propositional to the distance the mass is transferred, namely c(x,y)=|x-y|, but more general costs are allowed. The optimal transportation has found a variety of applications. In 1940s Kantorovich introduced a dual functional, by which one can determine the optimal mapping through the associated potential function, for a large class of cost functions.

The potential function satisfies a complicated partial differential equation of Monge-Ampere type, subject to a second boundary condition. This is a fully nonlinear partial differential equation which also arises in a number of geometric settings, and has been extensively studied in the last century. In this talk we will first introduce the optimal transportation and review the existence of optimal mappings. We then show that the reflector problem is an optimal transportation with a special cost function. By studying the associated Monge-Ampere equation, sharp conditions on the cost function have been found by the speaker and his collaborators for the regularity of potential functions. For Monge's cost function |x-y|, which does not satisfy the sharp conditions, we have also obtained the existence of optimal mappings, and established interesting regularity and singularity results for the mapping.

专家简介:

汪徐家19639月出生于浙江省淳安县(现千岛湖),19799月到19837月就读于浙江大学数学系本科。19839月到19907月,浙江大学数学系硕士生、博士生。师从董光昌教授,研究方向为偏微分方程/数学。19908月到19958月,浙江大学数学系任教。19959月至今,在澳大利亚国立大学从事数学研究。2003年到2007年,为南开大学数学系长江讲座教授。2002年至今,为浙江大学客座教授。2002年获澳大利亚数学会奖章。汪徐家教授以其对完全非线性椭圆议程理论及其应用的贡献而在2007年荣获第四届华人数学家大会晨兴数学金奖。2009年当选为澳大利亚科学院院士。