CAREER: Generated Jacobian Equations in Geometric Optics and Optimal Transport

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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.