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.
There is always a gap between the mathematical description of a practical engineering system, and the physical reality. Being able to quantify the reliability and confidence one can have in a mathematical model is extremely important, and one needs to take into account the sources of uncertainty in the model. These uncertainties might arise from unmodeled physics, variability in the manufacturing process, or incomplete information about the environment. In engineered systems, this is addressed through the use of sensors which allow us to measure the state of the system, but due to cost or performance limitations, the sensor readings are often sparse, and it is necessary to incorporate sensor information over time in order to accurately estimate the state of a system. The project addressed the critical problem 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. In particular, we developed numerical methods for quantifying uncertainty in engineered systems with complex configuration spaces, which take into account the range of motion constraints associated with things like joints in a human prosthetic device or a robotic arm. The methods we have developed are extremely accurate and much more efficient, which could potentially allow these methods to be used in small embeddable devices. One of the postdoctoral scholars who worked on the project is currently working in industry on adapting these methods for use in mobile devices. The project also supported a graduate textbook that is written with broad accessibility to the engineering community in mind, so as to broadly disseminate these sophisticated mathematical techniques and thereby promote the use of these methods for the practical real world engineering problems for which they were developed.
