Variational Approaches to Optimizations and Adaptivity in Problems Involving Visibility

能见度问题涉及到当观察者的视线存在障碍时，确定给定观察者在空间中可见的区域。当观察者被光源取代，视线的障碍物(遮光物)构成非反射障碍物时，问题就转化为找到照明区域的问题。PI建议研究涉及能见度优化的计算和数学方面的问题。将介绍和研究具有可见性目标的新的最优控制和博弈公式。将介绍激波捕捉技术和求解Hamilton-Jacobi方程的数值算法，严格地推导出一类新的间断系数的Hamilton-Jacobi方程，并相应地发展其粘性解理论。将研究变分和高阶偏微分方程组。这项建议涉及开发实用的数学和计算策略，以优化许多不同的背景下的监视。拟议项目的潜在影响包括与国家利益直接相关的监视和机器人路径规划，以及涉及高频波传播计算的应用，例如隐形战斗机设计中的雷达截面计算。作为潜在应用的一个例子，考虑放置在火星表面等地形上的机器人。任务是使用各种设备探索地形，包括视频记录设备。如何计算这个机器人的能见度？如何设计最优的搜索路径？如果一个以上的机器人被放置在域中，这些机器人应该如何协调以获得共同最优的搜索结果？在无线通信的上下文中，可以提出类似的问题，即在城市区域中寻找用于最大平均覆盖的n个无线基站的最佳布置，或者在给定基站位置的情况下确定具有最大平均信号覆盖的路径。在设计无人机(无人机)时，应该如何跟踪目标以尽可能长时间地保持其在视线中？一般而言，追逃问题：让某个隐藏的物体变得可见的最好方法是什么？或者反过来：躲避移动威胁的最佳方式是什么？拟议项目中的计算方法将以所列实例指导的实用性考虑以及一定程度的严格数学理论为指导。

The problem of visibility involves the determination of regions in space visible to a given observer whenobstruction of the observer's line-of-sight is present. When the observer is replaced by a light source, andthe obstruction to sight (the occluders) constitute non-reflecting obstacles, the problem translates to thatof finding the illuminated regions. The PI proposes to study computational and mathematical aspects ofproblems involving visibility optimization. Novel optimal control and game formulations with visibilityobjectives will be introduced and investigated. Shock-capturing techniques as well as numerical algorithmsfor Hamilton-Jacobi equations will be introduced and a new type of Hamilton-Jacobi equation withdiscontinuous coefficients will be rigorously derived, and whose viscosity solution theory willcorrespondingly be developed. Variational calculus and higher order PDEs will be investigated and incoporated.This proposal concerns developing practical mathematical and computational strategies for optimizing surveillance in many different contexts. The potential impacts for the proposed project include surveillance and robotic path planning, which are of immediate national interest, and applications that involve computations of high frequency wave propagation such as radar cross section computations in stealth fighter jet design. As an example of a potential application, consider a robot placed on a terrain such as the surface of Mars. The mission is the explore the terrain using various devices, including a video recording device. How does one compute the visibility of this robot? How should an optimal search path be designed? If more than one robot is placed in the domain, how should the robots coordinate for a jointly optimal search result? In the context of wireless communication, a similar question can be raised as to finding an optimal placement of n wireless base stations for maximal averaged coverage in an urban region, or determining a path with maximal averaged signal coverage given the locations of base stations. In designing UAV's (Unmanned Aviation Vehicles), how should one track a target to keep it in sight for as long time as possible? In general pursuit-evasion problems: What is the best way to make a certain hidden object become visible? Or the reverse: What is the best way to hide from a moving threat? The computational approaches in the proposed project will be guided by the practicality considerations guided by the listed examples as well as by certain level of rigorous mathematical theory.