Variational Approaches to Optimizations and Adaptivity in Problems Involving Visibility

涉及可见性问题的优化和自适应变分方法

• 批准号: 0513394
• 负责人: Yen-Hsi Tsai
• 金额: --
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0513394&HistoricalAwards=false
• 关键词: Variational; Approaches; Optimizations; Adaptivity; Problems

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.

可见性问题涉及到当观察者的视线受到阻碍时，确定空间中对给定观察者可见的区域。当观察者被一个光源所代替，并且视线的障碍物（遮挡物）构成非反射障碍物时，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。新的最优控制和游戏配方与probabilityobjectives将介绍和研究。介绍了激波捕捉技术和Hamilton-Jacobi方程的数值算法，严格推导了一类新的间断系数Hamilton-Jacobi方程，并相应地发展了其粘性解理论。变分微积分和高阶偏微分方程将被研究和整合。这个建议涉及开发实用的数学和计算策略，用于在许多不同的情况下优化监控。拟议项目的潜在影响包括监视和机器人路径规划，这是直接的国家利益，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。使命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可见度？如何设计最优搜索路径？如果域中放置了多个机器人，则机器人应如何协调以获得联合最佳搜索结果？在无线通信的上下文中，可以提出类似的问题，关于找到n个无线基站的最佳放置以用于城市区域中的最大平均覆盖，或者确定给定基站的位置的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使其尽可能长时间地保持在视线内？在一般的追逃问题中：什么是使某个隐藏对象变得可见的最佳方法？或者反过来：躲避移动威胁的最佳方法是什么？在拟议的项目中的计算方法将由所列的例子以及一定程度的严格的数学理论指导的实用性考虑。