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

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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.