Date of Award
Doctor of Philosophy (PhD)
Characterizing and incorporating uncertainties when simulating physical phenomena is essential for improving model-based predictions. These uncertainties may stem from a lack of knowledge regarding the underlying physical processes or from imprecise measurements of quantities that describe properties of the physical system. Uncertainty quantification (UQ) is a tool that seeks to characterize the impact of these uncertainties on solutions of computational models, resulting in improved predictive models. In practice, these uncertainties are either treated as random parameters to inform the statistics of the solution of interest (forward UQ), or their statistics are inferred from noisy observations of the solutions (inverse UQ).
For systems exhibiting high-dimensional uncertainty, performing either forward or inverse UQ presents a significant computational challenge, as these methods require a large number forward solves of the high-fidelity model, that is, the model that accurately captures the physics of the problem. For large-scale problems, this may result in the need for a possibly infeasible number of simulations. Prominent methods have been developed to reduce the burdens related to these challenges, including multilevel Monte Carlo (for forward UQ) and low-rank approximations to the posterior covariance (for inverse UQ). However, these methods may still require many forward solves of the high-fidelity model.
To reduce the cost of performing UQ on high-dimensional systems, we apply multi-fidelity strategies to both the forward problem, in order to estimate moments of the quantity of interest, and inverse problem, to approximate the posterior covariance. In particular, we formulate multi-fidelity methods that exploit the low-rank structure of the solution of interest and utilize models of lower fidelity (which are computationally cheaper to simulate) than the intended high-fidelity model, in a nonintrusive manner. Doing so results in surrogate models that may have accuracies closer to that of the high-fidelity model, yet have computational costs comparable to that of the low-fidelity models. Theoretical error analysis, cost comparisons, and numerical examples are provided to to show the promise of these novel methods.
Fairbanks, Hillary Ruth, "Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems" (2018). Applied Mathematics Graduate Theses & Dissertations. 114.