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

这个合作研究项目关注的是发展精确和有效的计算不确定性传播技术的非线性随机哈密顿系统，李群配置空间上的发展。 动态系统中的不确定性来自多个来源，例如未建模动态、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，因此对不确定性演变的仔细表征在科学和工程问题中至关重要。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，用于数值稳定的长时间传播的不确定性在哈密顿系统，同时明确地处理系统的基本随机和几何特性。大多数数学模型都有不确定性的来源，这些不确定性来自于对物理过程知之甚少、缺乏对参数的精确了解或关于系统当前状态的不完整信息，因此了解这些模型不确定性如何影响数学模型的预测是很重要的。特别是，如果计算机预测没有表明预测的可靠性和可信度，可能会产生灾难性的误导。该项目旨在解决开发精确的数学和数值方法来描述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迫切的需求，因为复杂系统的数学模型越来越多地依赖于为具有长期持久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，旨在为计算科学的专业课程提供便利，并将在UCSD的CSME研究生课程中进行实地测试。这本教科书包括随附的代码，将促进重用的计算基础设施，在其他应用程序中，涉及非线性空间的不确定性传播本项目资助。

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.