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CAREER: Generated Jacobian Equations in Geometric Optics and Optimal Transport

职业：生成几何光学和最优传输中的雅可比方程

基本信息

批准号：
1751996
负责人：
Brittany Froese Hamfeldt
金额：
$ 40万
依托单位：
New Jersey Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1751996&HistoricalAwards=false
关键词：
CAREER Generated Jacobian Equations Geometric

项目摘要

The design of lenses and mirrors to precisely control the intensity pattern and phase of light beams is important for many applications including microlithography, optical data storage, medical treatment, headlight design, and astronomy. The shape of the lenses or mirrors can be obtained by solving equations known as Generated Jacobian Equations (GJEs). However, there are currently no methods available for solving these equations except in the very simplest settings. This project will introduce new mathematical and computational techniques for solving GJEs, which will lead to new software that can solve these challenging equations. These new techniques will be used to solve several different lens design problems. Complementing this research plan, the investigator will produce a comprehensive series of video lectures that teach core mathematical topics within the context of cutting edge research. These will be used to enhance the classroom environment and attract students into STEM fields. They will also be made available to the public as free open courseware that can be used to facilitate self-study in non-traditional learning environments, complement course content in developing nations, change public attitudes about the nature and usefulness of mathematics, and inspire women to pursue mathematics.The goal of this project is to introduce new analytical and numerical techniques for solving a large class of Generated Jacobian Equations (GJEs) on the plane and sphere. Typical GJEs need to be supplemented with a global, nonlinear constraint on the solution gradient. This project will introduce an equivalent local formulation and develop a robust theory of weak solutions. This will be used to establish criteria that ensure convergence of numerical methods. The investigator will introduce generalized finite difference methods for solving GJEs on the plane. These will be analyzed, implemented, and used to solve several lens design problems. By exploiting local coordinates, the investigator will also design generalized finite difference methods for solving GJEs on the sphere, which will be applied to problems in geometric optics and optimal transportation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

用于精确控制光束强度模式和相位的透镜和反射镜设计对于许多应用（包括微光刻、光学数据存储、医疗、前灯设计和天文学）都很重要。透镜或镜子的形状可以通过求解称为生成雅可比方程 (GJE) 的方程来获得。然而，除了最简单的设置之外，目前没有可用的方法来求解这些方程。该项目将引入新的数学和计算技术来求解 GJE，这将带来能够求解这些具有挑战性的方程的新软件。这些新技术将用于解决几个不同的镜头设计问题。作为该研究计划的补充，研究人员将制作一系列全面的视频讲座，在前沿研究的背景下教授核心数学主题。这些将用于改善课堂环境并吸引学生进入 STEM 领域。它们还将作为免费开放课件向公众提供，可用于促进非传统学习环境中的自学，补充发展中国家的课程内容，改变公众对数学的性质和实用性的态度，并激励女性追求数学。该项目的目标是引入新的分析和数值技术，用于在飞机上求解一大类雅可比方程（GJE）和球体。典型的 GJE 需要补充对解梯度的全局非线性约束。该项目将引入等效的局部公式并开发稳健的弱解理论。这将用于建立确保数值方法收敛的标准。研究人员将引入广义有限差分方法来求解平面上的 GJE。这些将被分析、实施并用于解决多个镜头设计问题。通过利用局部坐标，研究人员还将设计用于求解球体上的 GJE 的广义有限差分方法，该方法将应用于几何光学和最佳运输中的问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。