Heteroscedastic metamodeling has gained popularity as an effective tool for analyzing and optimizing complex stochastic systems. A heteroscedastic metamodel provides an accurate approximation of the input-output relationship implied by a stochastic simulation experiment whose output is subject to input-dependent noise variance. Several challenges remain unsolved in this field. First, in-depth investigations into the consistency of heteroscedastic metamodeling techniques, particularly from the sequential prediction perspective, are lacking. Second, sequential heteroscedastic metamodel-based level-set estimation (LSE) methods are scarce. Third, the increasingly high computational cost required by heteroscedastic Gaussian process-based LSE methods in the sequential sampling setting is a concern. Additionally, when constructing a valid uniform bound for a heteroscedastic metamodel, the impact of noise variance estimation is not adequately addressed. This dissertation aims to tackle these challenges and provide promising solutions. First, we investigate the information consistency of a widely used heteroscedastic metamodeling technique, stochastic kriging (SK). Second, we propose SK-based LSE methods leveraging novel uniform bounds for input-point classification. Moreover, we incorporate the Nystrom approximation and a principled budget allocation scheme to improve the computational efficiency of SK-based LSE methods. Lastly, we investigate empirical uniform bounds that take into account the impact of noise variance estimation, ensuring an adequate coverage capability. / Doctor of Philosophy / In real-world engineering problems, understanding and optimizing complex systems can be challenging and prohibitively expensive. Computer simulation is a valuable tool for analyzing and predicting system behaviors, allowing engineers to explore different scenarios without relying on costly physical prototypes. However, the increasing complexity of simulation models leads to a higher computational burden. Metamodeling techniques have emerged to address this issue by accurately approximating the system performance response surface based on limited simulation experiment data to enable real-time decision-making. Heteroscedastic metamodeling goes further by considering varying noise levels inherent in simulation outputs, resulting in more robust and accurate predictions. Among various techniques, stochastic kriging (SK) stands out by striking a good balance between computational efficiency and statistical accuracy. Despite extensive research on SK, challenges persist in its application and methodology. These include little understanding of SK's consistency properties, an absence of sequential SK-based algorithms for level-set estimation (LSE) under heteroscedasticity, and the increasingly low computational efficiency of SK-based LSE methods in implementation. Furthermore, a precise construction of uniform bounds for the SK predictor is also missing. This dissertation aims at addressing these aforementioned challenges. First, the information consistency of SK from a prediction perspective is investigated. Then, sequential SK-based procedures for LSE in stochastic simulation, incorporating novel uniform bounds for accurate input-point classification, are proposed. Furthermore, a popular approximation technique is incorporated to enhance the computational efficiency of the SK-based LSE methods. Lastly, empirical uniform bounds are investigated considering the impact of noise variance estimation.

<p>This dissertation presents methods and associated performance bounds for spectral image processing tasks such as reconstruction and target detection, which are useful in a variety of applications such as astronomical imaging, biomedical imaging and remote sensing. The key idea behind our spectral image processing methods is the fact that important information in a spectral image can often be captured by low-dimensional manifolds embedded in high-dimensional spectral data. Based on this key idea, our work focuses on the reconstruction of spectral images from <italic>photon-limited</italic>, and distorted observations. </p><p>This dissertation presents a partition-based, maximum penalized likelihood method that recovers spectral images from noisy observations and enjoys several useful properties; namely, it (a) adapts to spatial and spectral smoothness of the underlying spectral image, (b) is computationally efficient, (c) is near-minimax optimal over an <italic>anisotropic</italic> Holder-Besov function class, and (d) can be extended to inverse problem frameworks.</p><p>There are many applications where accurate localization of desired targets in a spectral image is more crucial than a complete reconstruction. Our work draws its inspiration from classical detection theory and compressed sensing to develop computationally efficient methods to detect targets from few projection measurements of each spectrum in the spectral image. Assuming the availability of a spectral dictionary of possible targets, the methods discussed in this work detect targets that either come from the spectral dictionary or otherwise. The theoretical performance bounds offer insight on the performance of our detectors as a function of the number of measurements, signal-to-noise ratio, background contamination and properties of the spectral dictionary. </p><p>A related problem is that of level set estimation where the goal is to detect the regions in an image where the underlying intensity function exceeds a threshold. This dissertation studies the problem of accurately extracting the level set of a function from indirect projection measurements without reconstructing the underlying function. Our partition-based set estimation method extracts the level set of proxy observations constructed from such projection measurements. The theoretical analysis presented in this work illustrates how the projection matrix, proxy construction and signal strength of the underlying function affect the estimation performance.</p> / Dissertation