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.
