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

日期：年月日2007年11月21日提案：DMS-0747659 PI：Leok，Melvin机构：普渡大学标题：职业生涯：计算几何力学：基础，计算和应用摘要对称性，以及不变和等变对象的研究，是一个深层次的和统一的原则基础上的各种数学领域。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，几何积分则关注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，并使用几何和力学的自洽离散化系统地构建几何结构保持数值方案。拟议中的研究将结合联合收割机理论和计算工具所产生的狄拉克力学和几何，非交换谐波分析，不确定性量化，大大扩展适用性的计算几何力学和几何控制的工程问题，演变内在的非线性空间，如李群和齐次空间。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，几何控制背景下的不确定性研究将提高数值和计算工具的鲁棒性和可靠性，通过明确考虑我们对周围环境的知识中固有的不确定性，这项研究将提高我们以鲁棒和有效的方式控制自动驾驶汽车互联系统的能力。我们的研究结果将适用于分布式传感器网络的控制，由一组相互连接的卫星，无人机和水下航行器。这种传感器网络是遥感领域令人兴奋的新发展，有可能大大提高我们获得有关海洋，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都可以表示为一个由更多基本组件组成的互连系统，我们的数学框架将使我们能够更容易地从组件的行为以及它们互连的方式来理解复杂系统。

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.