CAREER: Generated Jacobian Equations in Geometric Optics and Optimal Transport

精确控制光束强度图案和相位的透镜和反射镜的设计在许多应用中都很重要，包括微光刻、光学数据存储、医疗、前照灯设计和天文学。透镜或反射镜的形状可以通过求解称为生成雅克比方程(GJE)的方程来获得。然而，除了在最简单的情况下，目前还没有可用的方法来求解这些方程。这个项目将引入新的数学和计算技术来求解GJE，这将导致新的软件可以解决这些具有挑战性的方程。这些新技术将被用来解决几个不同的透镜设计问题。作为对这一研究计划的补充，研究人员将制作一系列全面的视频讲座，在尖端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和吸引学生进入STEM领域。它们还将作为免费开放课程向公众提供，可用于促进在非传统学习环境中的自学，补充发展中国家的课程内容，改变公众对数学的本质和用处的态度，并激励妇女追求数学。该项目的目标是引入新的分析和数值技术，在平面和球面上解决一大类生成的雅可比方程(GJE)。典型的GJE需要补充对解梯度的全局、非线性约束。这个项目将引入一个等价的局部公式，并发展一个关于弱解的稳健理论。这将被用来建立确保数值方法收敛的标准。研究人员将介绍求解平面上广义随机微分方程的广义有限差分方法。这些将被分析、实施，并用于解决几个镜头设计问题。通过利用局部坐标，研究人员还将设计广义有限差分方法来求解球面上的GJE，这将应用于几何光学和最优交通问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.