CAREER: Variational Problems on Arbitrary Sets

职业：任意集上的变分问题

• 批准号: 1554733
• 负责人: Garving Luli
• 金额: $ 48.08万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1554733&HistoricalAwards=false
• 关键词: CAREER; Variational; Problems; Arbitrary; Sets

The rapid developments in technology and the widespread use of mobile devices have led to an explosive amount of data whose complexity is such that it becomes very challenging to extract any useful information in a reasonable amount of time and resources. The data are complex in size, dimension, and structure. The technology for rapidly processing such massive amounts of data is still in its infancy. One of the main goals of this project is to take advantage of some recent mathematical advances to develop and implement fast algorithms for fitting data by smooth functions. This could have many practical applications, for example in the design of automated systems that guide airplanes and unmanned aerial vehicles through complex terrains avoiding obstacles of arbitrary shapes. The proposed research carries the hope of advancing obstacle avoidance algorithms -- fast algorithms that can plan an optimal path under various constraints. The project will open new research directions for students and researchers; and it also will enhance our understanding of mathematical theories such as harmonic analysis, differential geometry, partial differential equations, and nonlinear programming.This 5-year integrated research and education project focuses on solving variational problems with boundary and/or obstacle constraints on arbitrary subsets in Euclidean space. The approach that the PI proposes is unorthodox: Rather than seeking exact minimizers of energy functionals, the PI will look for functions that minimize the energy functionals up to universal constant multiples. In contrast to the conventional approach to solving variational problems, which involves the solution of a partial differential equation, the PI will construct the solutions directly. The ultimate goal is to develop a complete theory revolving around the belief that any variational problem that can be solved using PDE theory can also be dealt with using extension theory. The special problems the PI will focus on are the obstacle problem, which requires the minimizers to lie above a certain barrier; and the Plateau problem, which seeks a surface of least area with prescribed boundary. Solving variational problems using PDE methods often requires the boundary of the set to have some kind of smoothness. In practice, this is a severe limitation. The novelty of the proposed approach is that there is no need to make any assumption on the geometry or the regularity of the set on which the PI prescribes the boundary conditions or constraints.

技术的快速发展和移动设备的广泛使用导致了数据量的爆炸性增长，其复杂性使得在合理的时间和资源内提取任何有用的信息变得非常具有挑战性。数据的大小、维度和结构都很复杂。快速处理如此大量数据的技术仍处于起步阶段。该项目的主要目标之一是利用最近的一些数学进展来开发和实现通过平滑函数拟合数据的快速算法。这可能有许多实际应用，例如在自动化系统的设计中，引导飞机和无人机穿过复杂的地形，避开任意形状的障碍物。所提出的研究有望推进避障算法——可以在各种约束下规划最佳路径的快速算法。该项目将为学生和研究人员开辟新的研究方向；它还将增强我们对调和分析、微分几何、偏微分方程和非线性规划等数学理论的理解。这个为期 5 年的综合研究和教育项目侧重于解决任意子集上具有边界和/或障碍约束的变分问题在欧几里得空间中。 PI 提出的方法是非正统的：PI 不会寻找能量泛函的精确最小化器，而是寻找将能量泛函最小化到通用常数倍数的函数。与解决变分问题的传统方法（涉及偏微分方程的求解）相比，PI 将直接构造解。最终目标是开发一个完整的理论，围绕这样的信念：任何可以使用偏微分方程理论解决的变分问题也可以使用可拓理论来处理。 PI 将关注的特殊问题是障碍问题，它要求最小化器位于某个障碍之上；以及高原问题，寻求具有指定边界的最小面积的表面。使用 PDE 方法解决变分问题通常要求集合的边界具有某种平滑度。实际上，这是一个严重的限制。所提出方法的新颖之处在于，不需要对 PI 规定边界条件或约束的集合的几何形状或规律性做出任何假设。