Variational Approaches to Optimizations and Adaptivity in Problems Involving Visibility

University Of Texas At Austin, Austin TX

Investigators

Abstract

The problem of visibility involves the determination of regions in space visible to a given observer when obstruction of the observer's line-of-sight is present. When the observer is replaced by a light source, and the obstruction to sight (the occluders) constitute non-reflecting obstacles, the problem translates to that of finding the illuminated regions. The PI proposes to study computational and mathematical aspects of problems involving visibility optimization. Novel optimal control and game formulations with visibility objectives will be introduced and investigated. Shock-capturing techniques as well as numerical algorithms for Hamilton-Jacobi equations will be introduced and a new type of Hamilton-Jacobi equation with discontinuous coefficients will be rigorously derived, and whose viscosity solution theory will correspondingly 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.