CAREER: Generated Jacobian Equations in Geometric Optics and Optimal Transport

设计透镜和反射镜以精确控制光束的强度图案和相位对于包括微光刻、光学数据存储、医疗、前灯设计和天文学的许多应用是重要的。透镜或反射镜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，目前还没有可用于求解这些方程的方法。该项目将引入新的数学和计算技术来解决GJE，这将导致新的软件可以解决这些具有挑战性的方程。这些新技术将用于解决几个不同的透镜设计问题。补充这一研究计划，研究人员将制作一系列全面的视频讲座，教授前沿研究背景下的核心数学主题。这些将用于改善课堂环境，吸引学生进入STEM领域。它们还将作为免费开放的课件向公众提供，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有用性的态度，该项目的目标是引入新的分析和数值技术来解决一大类生成雅可比方程（GJE）在平面和球面上。典型的GJE需要补充一个全局的，非线性约束的解决方案梯度。本计画将介绍一个等效的局部公式，并发展一个稳健的弱解理论。这将被用来建立标准，确保收敛的数值方法。研究人员将介绍广义有限差分法求解GJE的平面上。这些将被分析，实施，并用于解决几个透镜设计问题。通过利用局部坐标，研究者还将设计用于求解球面上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。

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.