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

日期：2007年11月21日提案：DMS- 0747659PI: Leok， Melvin机构：普渡大学职称：职业：计算几何力学：基础，计算和应用摘要对称性，以及对不变和等变物体的研究，是一个深刻而统一的原理，它隐含在许多数学领域中。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，几何积分涉及构造具有几何不变和等变性质的数值方法。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来系统地构建几何结构保持的数值方案。提出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的理论和计算工具，极大地扩展计算几何力学和几何控制在非线性空间（如李群和齐次空间）上发展的工程问题的适用性。这将提供对狄拉克约束、非完整约束和互联系统的规范离散化的见解。此外，在几何控制背景下对不确定性的研究将提高所得到的数值和计算工具的鲁棒性和可靠性。通过明确考虑到我们对周围环境的了解中固有的不确定性，这项研究将提高我们以稳健有效的方式控制自动驾驶汽车互联系统的能力。我们的结果将适用于分布式传感器网络的控制，包括一组相互连接的卫星，无人驾驶飞行器和水下航行器。这种传感器网络是遥感领域一个令人兴奋的新发展，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被表示为一个由更基本的组件组成的相互连接的系统，我们的数学框架将使我们更容易地从其组成部分的行为和它们相互连接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.