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。在计算几何光学中，本研究项目要解决的基本问题是通过可证明的收敛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。这个项目包括通过参与研究来训练研究生。设计折射器和反射器的工具是有限的，有些工具没有可靠的理论支持。有希望的方法是数值求解相关的非线性蒙日-安培型偏微分方程和将照明问题作为测度方程求解的变分方法。现有的基于偏微分方程的方法做出了特别的假设，并没有适当地处理这些问题的异常边界条件。另一方面，大多数现有的变分方法很难与问题的规模相匹配。这就需要在严格分析的基础上改进高效和稳健的数值方法来解决计算几何光学问题。本项目旨在：(1)实现并分析一种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)用一种可证明收敛且有效的有限差分方法求解相关偏微分方程的数值解；(3)基于光滑函数近似求解相关非线性混合有限元方程。该项目还将研究不同方法的收敛性。预期该项目的结果将确定哪种方法是最有效的。

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.