Date of Award
Summer 7-17-2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics, Statistics and Computer Science
First Advisor
Elaine Spiller
Second Advisor
Anthony Parolari
Third Advisor
Cheng-Han Yu
Abstract
Computer models, or simulators, are mathematical representations of real world phenomena that, to run at new input settings, are often computationally intensive and prohibitively slow. Surrogate models, or emulators, provide a method of rapidly predicting simulator outputs with uncertainty at untested configurations by treating the computer model output as a single realization of a stochastic process, specifically a Gaussian Process (GP).In this dissertation, we consider three projects: Correlated Linked GP Emulation, Spatially Correlated Parallel Partial Emulation, and Optimizing the Zero-censored Gaussian Process with Spatially Correlated Sampling. First, the Correlated Linked Emulator, focuses on developing a model for the covariance of the linked GP. The linked GP emulator is a surrogate designed for composite functions. As of yet, the linked GP has not been framed as a true Gaussian process, only predicting at a single untested input as a Normal random variable. The covariance model we present allows the linked GP to be expressed as a Gaussian process, making correlated sampling possible for multiple untested inputs. Second, the Spatially Correlated Parallel Partial Emulator, is an extension of the parallel partial emulator (PPE). The PPE models simulators with vector-valued outputs in a highly efficient manner using a shared correlation structure with a distinct mean trend and scalar variance for each output component. Spatially correlated samples are one efficient mechanism to quantify uncertainty inherent in using a GP surrogate. Spatially correlated samples are not available from PPE, but can be now obtained using the process we propose in this chapter. Details of the construction and utilization of the spatially correlated PPE are presented and applied to applications with three pde-based simulators. Lastly, Optimizing the Zero-censored Gaussian Process with Spatially Correlated Sampling discusses the novel zero-censored Gaussian process, which was designed to enable Gaussian process predictions for simulators with constrained outputs, such as those with non-negative outputs. We introduce two methods for sampling it along an entire spatial dimension, rather than its previously offered point-wise sampling. Promising results from this study suggest that further optimization of these methods could greatly improve the uncertainty predictions in geophysical problems with constrained outputs.