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

日期：2007年11月21日Proposal：DMS-0747659PI：Leok，Melvin Institution：普渡大学题目：Career：计算几何力学：基础、计算和应用概述对称性以及对不变对象和等变对象的研究，是各种数学领域的深层统一原理。特别是，几何力学的特点是将对称性和微分几何技术应用于拉格朗日力学和哈密顿力学，而几何积分则涉及构造具有几何不变性和等变性的数值方法。计算几何力学融合了这两个领域，并使用几何和力学的自洽离散化来系统地构造几何结构保持的数值格式。这项拟议的研究将结合狄拉克力学和几何、非对易调和分析和不确定性量化的理论和计算工具，极大地扩展计算几何力学和几何控制的适用性，使其适用于本质上在非线性空间上演化的工程问题，如李群和齐次空间。这将提供对狄拉克约束、非完整约束和相互关联系统的正则离散化的见解。此外，几何控制背景下的不确定性研究将提高由此产生的数值和计算工具的稳健性和可靠性。这项研究将通过明确考虑到我们对周围环境的了解所固有的不确定性，提高我们以稳健和高效的方式控制自动驾驶车辆相互关联系统的能力。我们的结果将适用于分布式传感器网络的控制，该网络由一组相互连接的卫星、无人机和水下机器人组成。这样的传感器网络是遥感领域令人兴奋的新发展，它有可能极大地提高我们获得的关于海洋、环境和气候的信息的效率、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一个由更基本的组件组成的相互关联的系统，我们的数学框架将使我们能够更容易地了解复杂系统的组成部分的行为以及它们相互关联的方式。

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.