Collaborative Research: Computational Geometric Uncertainty Propagation for Hamiltonian Systems on a Lie Group

这一合作研究项目致力于发展精确而有效的计算不确定性传播技术，用于在李群构形空间上发展的非线性随机哈密顿系统。动态系统中的不确定性来自多个来源，如未建模的动态、参数不确定性和初始条件的不确定性。由于它们不能从任何计算实验或物理测量中完全消除，所以在科学和工程问题中，仔细描述不确定性的演变是必不可少的。这个项目涉及到计算几何力学、几何数值积分、非对易调和分析和广义多项式混沌技术的应用，将产生无网格、无坐标的方法来解决哈密顿系统中长期数值稳定的不确定性传播，同时明确地解决系统潜在的随机和几何属性。大多数数学模型都存在不确定性来源，这些不确定性可能源于对物理过程的理解、对参数的精确了解或关于系统当前状态的不完全信息，了解这些模型不确定性如何影响由数学模型产生的预测是很重要的。具体地说，如果计算机预测没有显示出预测的可靠性和可信度，则可能会产生灾难性的误导。该项目旨在解决一项基本任务，即开发准确的数学和数值方法来描述复杂系统中不确定性的影响，这是一项特别及时和紧迫的需要，因为越来越多地依赖复杂系统的数学模型来为具有长期和深远影响的公共政策决策提供信息。将准备一本研究生教科书，并行讨论关于李群的几何力学的连续和离散时间方法，其目的是便于计算科学的专业课程使用，并将在加州大学圣迭戈分校的CSME研究生课程中进行现场测试。这本教科书包括附带的代码，这些代码将促进由该项目资助的计算基础设施在涉及非线性空间上的不确定性传播的其他应用中的重复使用。

This collaborative research project is concerned with the development of accurate and efficient computational uncertainty propagation techniques for nonlinear stochastic Hamiltonian systems that evolve on Lie group configuration spaces. Uncertainties in a dynamic system arise from multiple sources such as unmodeled dynamics, parametric uncertainty, and uncertainty in initial conditions. As they cannot be completely eliminated from any computational experiment or physical measurement, a careful characterization of the evolution of uncertainties is essential in scientific and engineering problems. This project involves the application of computational geometric mechanics, geometric numerical integration, noncommutative harmonic analysis, and generalized polynomial chaos techniques, and will yield mesh-free, coordinate-free methods for the numerically stable long-time propagation of uncertainty in a Hamiltonian system, while explicitly addressing the underlying stochastic and geometric properties of the system.Most mathematical models have sources of uncertainty that may arise from physical processes that are poorly understood, a lack of precise knowledge of the parameters, or incomplete information about the current state of the system, and it is important to understand how these model uncertainties affect the predictions that arise from the mathematical model. In particular, a computer prediction without some indication of the reliability and confidence in the prediction can be disastrously misleading. This project aims to address the essential task of developing accurate mathematical and numerical methods for characterizing the effects of uncertainty in complex systems, which is a particularly timely and pressing need, since mathematical models of complex systems are increasingly relied upon to inform public policy decisions with long lasting and far reaching consequences. A graduate textbook will be prepared that discusses in parallel the continuous and discrete time approach to geometric mechanics on Lie groups that aims to be accessible to professional programs in computational science, and which will be field tested in the CSME graduate program at UCSD. This textbook includes accompanying code that will facilitate the reuse of the computational infrastructure funded by this project in other applications involving uncertainty propagation on nonlinear spaces.