Variational Approaches to Optimizations and Adaptivity in Problems Involving Visibility

可见性问题涉及到当观察者的视线受到阻碍时，对给定观察者可见的空间区域的确定。当观察者被光源取代时，视线上的障碍物（遮挡物）构成了非反射障碍物，问题就转化为寻找被照亮的区域。PI建议研究涉及可见性优化问题的计算和数学方面。将介绍和研究具有可见性目标的新型最优控制和博弈公式。本文将介绍Hamilton-Jacobi方程的激波捕获技术和数值算法，并严格推导出一种具有不连续系数的新型Hamilton-Jacobi方程，并相应发展其粘度解理论。变分微积分和高阶偏微分方程将被研究和结合。该建议涉及开发实用的数学和计算策略，以在许多不同的环境中优化监测。拟议项目的潜在影响包括监视和机器人路径规划，这直接关系到国家利益，以及涉及高频波传播计算的应用，如隐形战斗机设计中的雷达横截面计算。作为潜在应用程序的一个示例，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，包括一个视频记录设备。如何计算这个机器人的可见度？如何设计最优搜索路径？如果域内放置了多个机器人，机器人如何协调以获得联合最优的搜索结果？在无线通信的背景下，可以提出类似的问题，如在城市区域中寻找n个无线基站的最大平均覆盖的最佳位置，或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（UAV）时，如何尽可能长时间地跟踪目标，使其保持在视线内？在一般的追捕逃避问题中：让某个隐藏对象变得可见的最佳方法是什么？或者相反：躲避移动威胁的最佳方法是什么？所建议项目中的计算方法将以所列出的示例以及一定程度的严格数学理论指导的实用性考虑为指导。

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.