CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications

职业：计算几何力学：基础、计算和应用

• 批准号: 1010687
• 负责人: Melvin Leok
• 金额: $ 42.51万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-29 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1010687&HistoricalAwards=false
• 关键词: CAREER; Computational; Geometric; Mechanics; Foundations

Date: November 21, 2007Proposal: DMS- 0747659PI: Leok, Melvin Institution: Purdue UniversityTitle: CAREER: Computational Geometric Mechanics: Foundations, Computation, and ApplicationsABSTRACTSymmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. The proposed research will combine theoretical and computational tools arising from Dirac mechanics and geometry, noncommutative harmonic analysis, and uncertainty quantification to dramatically extend the applicability of computational geometric mechanics and geometric control to engineering problems that evolve intrinsically on nonlinear spaces, such as Lie groups and homogeneous spaces. This will provide insights into the canonical discretization of Dirac constraints, nonholonomic constraints, and interconnected systems. In addition, the study of uncertainty in the context of geometric control will improve the robustness and reliability of the resulting numerical and computational tools.This research will improve our ability to control interconnected systems of autonomous vehicles in a robust and efficient fashion, by explicitly taking into account the uncertainty inherent in our knowledge of the surrounding environment. Our results will be applicable to the control of distributed sensor networks, consisting of an interconnected set of satellites, unmanned aerial vehicles and underwater vehicles. Such sensor networks are an exciting new development in the field of remote sensing that has the potential to dramatically increase the efficiency, coverage, and reliability of the information we obtain about our oceans, environment, and climate. More broadly, most complex engineering systems can be expressed as an interconnected system of more elementary components, and our mathematical framework will allow us to more readily understand complex systems in terms of the behavior of its component parts and the manner in which they are interconnected.

日期：2007 年 11 月 21 日提案：DMS- 0747659PI：Leok，Melvin 机构：普渡大学标题：职业：计算几何力学：基础、计算和应用摘要对称性以及对不变和等变对象的研究，是各种背后的深刻而统一的原理的数学领域。特别是，几何力学的特点是将对称和微分几何技术应用于拉格朗日和哈密顿力学，而几何积分则涉及具有几何不变和等变性质的数值方法的构造。计算几何力学融合了这些领域，并利用几何和力学的自洽离散化来系统地构建几何结构保持数值格式。拟议的研究将结合狄拉克力学和几何、非交换调和分析和不确定性量化产生的理论和计算工具，以极大地将计算几何力学和几何控制的适用性扩展到非线性空间上本质上演化的工程问题，例如李群和同质空间。这将为狄拉克约束、非完整约束和互连系统的规范离散化提供见解。此外，对几何控制背景下的不确定性的研究将提高所得数值和计算工具的鲁棒性和可靠性。这项研究将通过明确地采用考虑到我们对周围环境的了解所固有的不确定性。我们的结果将适用于分布式传感器网络的控制，该网络由一组互连的卫星、无人机和水下航行器组成。这种传感器网络是遥感领域令人兴奋的新发展，有可能显着提高我们获取有关海洋、环境和气候信息的效率、覆盖范围和可靠性。 更广泛地说，大多数复杂的工程系统可以表示为更基本组件的互连系统，并且我们的数学框架将使我们能够更容易地理解复杂系统的组成部分的行为以及它们互连的方式。