CAREER: Variational Problems on Arbitrary Sets

技术的快速发展和移动设备的广泛使用导致了爆炸性的数据量，其复杂性使得在合理的时间和资源范围内提取任何有用的信息变得非常困难。这些数据在大小、维度和结构上都很复杂。快速处理如此海量数据的技术仍处于初级阶段。这个项目的主要目标之一是利用一些最新的数学进展来开发和实现用光滑函数拟合数据的快速算法。这可能会有许多实际应用，例如在自动系统的设计中，这些系统可以引导飞机和无人驾驶飞行器通过复杂的地形，避开任意形状的障碍物。这项拟议的研究有望推进避障算法--一种能够在各种约束下规划最优路径的快速算法。该项目将为学生和研究人员开辟新的研究方向，并将增进我们对调和分析、微分几何、偏微分方程组和非线性规划等数学理论的理解。这个为期5年的综合研究和教学项目专注于解决欧氏空间中任意子集上具有边界和/或障碍约束的变分问题。PI提出的方法是非正统的：PI将寻找将能量泛函最小化到普适常数倍数的函数，而不是寻找能量泛函的精确最小化。与传统的求解变分问题的方法不同，PI将直接构造解。最终的目标是发展一个完整的理论，围绕这样一个信念，即任何可以用偏微分方程理论解决的变分问题也可以用可拓理论来处理。PI将关注的特殊问题是障碍问题和高原问题，障碍问题要求极小化对象位于某一障碍之上，高原问题寻求具有指定边界的最小面积曲面。用偏微分方程方法求解变分问题时，通常要求集合的边界具有某种光滑性。在实践中，这是一个严重的限制。该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或正则性做出任何假设。

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.