CAREER: Variational Problems on Arbitrary Sets

技术的快速发展和移动的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量内提取任何有用的信息变得非常具有挑战性。数据在大小、维度和结构上都很复杂。快速处理如此大量数据的技术仍处于起步阶段。这个项目的主要目标之一是利用一些最近的数学进步，开发和实现快速算法拟合数据的光滑函数。这可能有许多实际应用，例如在自动化系统的设计中，引导飞机和无人驾驶飞行器通过复杂的地形，避免任意形状的障碍物。这项研究有望推进避障算法--一种可以在各种约束条件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，并将提高我们对调和分析，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专注于解决欧几里得空间中任意子集上的边界和/或障碍约束的变分问题。PI提出的方法是非正统的：而不是寻求能量泛函的精确最小化，PI将寻找使能量泛函最小化到普适常数倍数的函数。相对于传统的方法来解决变分问题，这涉及到一个偏微分方程的解决方案，PI将直接构造的解决方案。最终目标是发展一个完整的理论，围绕着这样一个信念，即任何可以用PDE理论解决的变分问题也可以用扩张理论来处理。PI将关注的特殊问题是障碍问题，它要求极小化器位于某个障碍之上;以及高原问题，它寻求具有指定边界的最小面积的表面。用偏微分方程方法求解变分问题时，往往要求集合的边界具有某种光滑性。实际上，这是一个严重的限制。所提出的方法的新奇是，没有必要作出任何假设的几何形状或规则的PI规定的边界条件或约束。

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.