OP: Variational Principles, Minimization Diagrams, and Mixed Finite Elements in Computational Geometric Optics

在许多设备中，包括投影显示器、激光武器和医疗照明器，都需要精确控制光线。 这个计算几何光学研究项目要解决的基本问题是通过可证明收敛的数值方法有效地设计透镜和反射镜。该项目的目标是为各种照明问题开发改进的高效和理论上合理的算法。该项目涉及通过参与研究来培训研究生。用于折射器和反射器设计的可用工具有限，有些没有可靠的理论支持。有前途的方法包括数值求解相关的非线性偏微分方程的Monge-安培型和变分方法，解决照明问题作为一个方程的措施。现有的方法基于偏微分方程作出特设的假设，并没有适当地解决这些问题的不寻常的边界条件。另一方面，大多数现有的变分方法的规模与问题的大小差。这就产生了一个需要改进的高效和强大的数值方法的基础上严格的分析，以解决计算几何光学问题。该项目旨在：（1）实现和分析基于最小化图的高效变分方法，用于计算各种照明问题的解;（2）用可证明收敛且高效的有限差分方法解决相关偏微分方程的数值解;以及（3）基于平滑函数的近似，用混合有限元求解相关非线性方程。该项目还将研究不同方法的收敛特性。预计该项目的结果将确定哪种方法最有效。

In many devices, including projection displays, laser weapons, and medical illuminators, it is required to accurately control light. The fundamental question to be addressed by this research project in computational geometric optics is the efficient design of lenses and mirrors through provably convergent numerical methods. The goal of the project is to develop improved efficient and theoretically sound algorithms for a variety of illumination problems. The project involves training of graduate students through involvement in the research.Available tools for the design of refractors and reflectors are limited, and some are not backed up by a sound theory. Promising approaches consist in solving numerically the associated nonlinear partial differential equations of Monge-Ampere type and variational methods that solve the illumination problem as an equation in measures. Existing methods based on partial differential equations make ad hoc assumptions and do not address appropriately the unusual boundary conditions for these problems. On the other hand, most existing variational methods scale poorly with the size of the problem. This has created a need for improved efficient and robust numerical methods based on rigorous analysis to solve computational geometric optics problems. This project aims to: (1) implement and analyze an efficient variational method, based on minimization diagrams, for computing solutions of a variety of illumination problems; (2) address the numerical resolution of the relevant partial differential equations with a provably convergent and efficient finite-difference method; and (3) solve the relevant nonlinear equations with mixed finite elements based on approximations by smooth functions. The project will also investigate the convergence properties of the different approaches. It is anticipated that the results of the project will identify which approach is the most efficient.