Due to advancements in supercomputing and algorithms for finite element analysis, today's computer simulation models often contain complex calculations that can result in a wealth of knowledge. Gaussian processes (GPs) are highly desirable models for computer experiments for their predictive accuracy and uncertainty quantification. This dissertation addresses GP modeling when data abounds as well as GP adaptive design when simulator expense severely limits the amount of collected data. For data-rich problems, I introduce a localized sparse covariance GP that preserves the flexibility and predictive accuracy of a GP's predictive surface while saving computational time. This locally induced Gaussian process (LIGP) incorporates latent design points, inducing points, with a local Gaussian process built from a subset of the data. Various methods are introduced for the design of the inducing points. LIGP is then extended to adapt to stochastic data with replicates, estimating noise while relying upon the unique design locations for computation. I also address the goal of identifying a contour when data collection resources are limited through entropy-based adaptive design. Unlike existing methods, the entropy-based contour locator (ECL) adaptive design promotes exploration in the design space, performing well in higher dimensions and when the contour corresponds to a high/low quantile. ECL adaptive design can join with importance sampling for the purpose of reducing uncertainty in reliability estimation. / Doctor of Philosophy / Due to advancements in supercomputing and physics-based algorithms, today's computer simulation models often contain complex calculations that can produce larger amounts of data than through physical experiments. Computer experiments conducted with simulation models are sought-after ways to gather knowledge about physical problems but come with design and modeling challenges. In this dissertation, I address both data size extremes - building prediction models with large data sets and designing computer experiments when scarce resources limit the amount of data. For the former, I introduce a strategy of constructing a series of models including small subsets of observed data along with a set of unobserved data locations (inducing points). This methodology also contains the ability to perform calculations with only unique data locations when replicates exist in the data. The locally induced model produces accurate predictions while saving computing time. Various methods are introduced to decide the locations of these inducing points. The focus then shifts to designing an experiment for the purpose of accurate prediction around a particular output quantity of interest (contour). A experimental design approach is detailed that selects new sample locations one-at-a-time through a function to maximize the amount of information gain in the contour region for the overall model. This work is combined with an existing method to estimate the true volume of the contour.

In this thesis, I present a decentralized sparse Gaussian process regression (DSGPR) model with event-triggered, adaptive inducing points. This DSGPR model brings the advantages of sparse Gaussian process regression to a decentralized implementation. Being decentralized and sparse provides advantages that are ideal for multi-agent systems (MASs) performing environmental modeling. In this case, MASs need to model large amounts of information while having potential intermittent communication connections. Additionally, the model needs to correctly perform uncertainty propagation between autonomous agents and ensure high accuracy on the prediction. For the model to meet these requirements, a bounded and efficient real-time sparse Gaussian process regression (SGPR) model is needed. I improve real-time SGPR models in these regards by introducing an adaptation of the mean shift and fixed-width clustering algorithms called radial clustering. Radial clustering enables real-time SGPR models to have an adaptive number of inducing points through an efficient inducing point selection process. I show how this clustering approach scales better than other seminal Gaussian process regression (GPR) and SGPR models for real-time purposes while attaining similar prediction accuracy and uncertainty reduction performance. Furthermore, this thesis addresses common issues inherent in decentralized frameworks such as high computation costs, inter-agent message bandwidth restrictions, and data fusion integrity. These challenges are addressed in part through performing maximum consensus between local agent models which enables the MAS to gain the advantages of decentral- ization while keeping data fusion integrity. The inter-agent communication restrictions are addressed through the contribution of two message passing heuristics called the covariance reduction heuristic and the Bhattacharyya distance heuristic. These heuristics enable user to reduce message passing frequency and message size through the Bhattacharyya distance and properties of spatial kernels. The entire DSGPR framework is evaluated on multiple simulated random vector fields. The results show that this framework effectively estimates vector fields using multiple autonomous agents. This vector field is assumed to be a wind field; however, this framework may be applied to the estimation of other scalar or vector fields (e.g., fluids, magnetic fields, electricity, etc.).