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

该合作研究项目致力于为在李群配置空间上演化的非线性随机哈密顿系统开发准确有效的计算不确定性传播技术。 动态系统中的不确定性来自多种来源，例如未建模的动力学、参数不确定性和初始条件的不确定性。 由于它们无法从任何计算实验或物理测量中完全消除，因此仔细表征不确定性的演变对于科学和工程问题至关重要。该项目涉及计算几何力学、几何数值积分、非交换调和分析和广义多项式混沌技术的应用，并将产生无网格、无坐标的方法，用于哈密顿系统中不确定性的数值稳定长期传播，同时明确解决系统的潜在随机和几何性质。大多数数学模型的不确定性来源可能来自对物理过程的了解甚少、缺乏对参数的精确了解或有关系统当前状态的不完整信息，并且它是了解这些模型不确定性如何影响数学模型的预测非常重要。特别是，如果计算机预测没有表明预测的可靠性和置信度，则可能会产生灾难性的误导。该项目旨在解决开发准确的数学和数值方法来表征复杂系统中不确定性影响的基本任务，这是一个特别及时和紧迫的需求，因为复杂系统的数学模型越来越依赖于为具有长期和深远影响的公共政策决策提供信息。将编写一本研究生教科书，并行讨论李群几何力学的连续和离散时间方法，旨在供计算科学专业课程使用，并将在加州大学圣地亚哥分校的 CSME 研究生课程中进行现场测试。本教科书包含随附的代码，这些代码将有助于在涉及非线性空间不确定性传播的其他应用中重用该项目资助的计算基础设施。

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.