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、激光武器和医用照明器，都需要精确地控制光。计算几何光学的这个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来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、高效的、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。可用来设计折射镜和反射镜的工具有限，有些工具没有可靠的理论支持。有希望的方法包括数值求解相关的Monge-Ampere型非线性偏微分方程组，以及将照明问题作为测量方程来求解的变分方法。现有的基于偏微分方程组的方法作了特殊的假设，没有适当地处理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都不能很好地适应问题的规模。这就产生了对基于严格分析的改进的高效和健壮的数值方法来解决计算几何光学问题的需求。该项目的目标是：(1)实现和分析基于极小化图的高效变分方法，用于计算各种照明问题的解；(2)利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组的数值解；以及(3)基于光滑函数逼近的混合有限元求解相关的非线性方程。该项目还将调查不同方法的收敛特性。预计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.