Mathematical models are commonly used in science and industry to simulate complex physical processes. These models are implemented by computer codes which are often complex. For this reason, the codes are also expensive in terms of computation time, and this limits the number of simulations in an experiment. The codes are also deterministic, which means that output from a code has no measurement error. <br /><br /> One modelling approach in dealing with deterministic output from computer experiments is to assume that the output is composed of a drift component and systematic errors, which are stationary Gaussian stochastic processes. A Bayesian approach is desirable as it takes into account all sources of model uncertainty. Apart from prior specification, one of the main challenges in a complete Bayesian model is integration. We take a Bayesian approach with a Jeffreys prior on the model parameters. To integrate over the posterior, we use two approximation techniques on the log scaled posterior of the correlation parameters. First we approximate the Jeffreys on the untransformed parameters, this enables us to specify a uniform prior on the transformed parameters. This makes Markov Chain Monte Carlo (MCMC) simulations run faster. For the second approach, we approximate the posterior with a Normal density. <br /><br /> A large part of the thesis is focused on the problem of integration. Integration is often a goal in computer experiments and as previously mentioned, necessary for inference in Bayesian analysis. Sampling strategies are more challenging in computer experiments particularly when dealing with computationally expensive functions. We focus on the problem of integration by using a sampling approach which we refer to as "GaSP integration". This approach assumes that the integrand over some domain is a Gaussian random variable. It follows that the integral itself is a Gaussian random variable and the Best Linear Unbiased Predictor (BLUP) can be used as an estimator of the integral. We show that the integration estimates from GaSP integration have lower absolute errors. We also develop the Adaptive Sub-region Sampling Integration Algorithm (ASSIA) to improve GaSP integration estimates. The algorithm recursively partitions the integration domain into sub-regions in which GaSP integration can be applied more effectively. As a result of the adaptive partitioning of the integration domain, the adaptive algorithm varies sampling to suit the variation of the integrand. This "strategic sampling" can be used to explore the structure of functions in computer experiments.

Mathematical models are commonly used in science and industry to simulate complex physical processes. These models are implemented by computer codes which are often complex. For this reason, the codes are also expensive in terms of computation time, and this limits the number of simulations in an experiment. The codes are also deterministic, which means that output from a code has no measurement error. <br /><br /> One modelling approach in dealing with deterministic output from computer experiments is to assume that the output is composed of a drift component and systematic errors, which are stationary Gaussian stochastic processes. A Bayesian approach is desirable as it takes into account all sources of model uncertainty. Apart from prior specification, one of the main challenges in a complete Bayesian model is integration. We take a Bayesian approach with a Jeffreys prior on the model parameters. To integrate over the posterior, we use two approximation techniques on the log scaled posterior of the correlation parameters. First we approximate the Jeffreys on the untransformed parameters, this enables us to specify a uniform prior on the transformed parameters. This makes Markov Chain Monte Carlo (MCMC) simulations run faster. For the second approach, we approximate the posterior with a Normal density. <br /><br /> A large part of the thesis is focused on the problem of integration. Integration is often a goal in computer experiments and as previously mentioned, necessary for inference in Bayesian analysis. Sampling strategies are more challenging in computer experiments particularly when dealing with computationally expensive functions. We focus on the problem of integration by using a sampling approach which we refer to as "GaSP integration". This approach assumes that the integrand over some domain is a Gaussian random variable. It follows that the integral itself is a Gaussian random variable and the Best Linear Unbiased Predictor (BLUP) can be used as an estimator of the integral. We show that the integration estimates from GaSP integration have lower absolute errors. We also develop the Adaptive Sub-region Sampling Integration Algorithm (ASSIA) to improve GaSP integration estimates. The algorithm recursively partitions the integration domain into sub-regions in which GaSP integration can be applied more effectively. As a result of the adaptive partitioning of the integration domain, the adaptive algorithm varies sampling to suit the variation of the integrand. This "strategic sampling" can be used to explore the structure of functions in computer experiments.

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This thesis proposes new analysis tools for simulation models in the presence of data. To achieve a representation close to reality, simulation models are typically endowed with a set of inputs, termed parameters, that represent several controllable, stochastic or unknown components of the system. Because these models often utilize computationally expensive procedures, even modern supercomputers require a nontrivial amount of time, money, and energy to run for complex systems. Existing statistical frameworks avoid repeated evaluations of deterministic models through an emulator, constructed by conducting an experiment on the code. In high dimensional scenarios, the traditional framework for emulator-based analysis can fail due to the computational burden of inference. This thesis proposes a new class of experiments where inference from half a million observations is possible in seconds versus the days required for the traditional technique. In a case study presented in this thesis, the parameter of interest is a function as opposed to a scalar or a set of scalars, meaning the problem exists in the high dimensional regime. This work develops a new modeling strategy to nonparametrically study the functional parameter using Bayesian inference. Stochastic simulations are also investigated in the thesis. I describe the development of emulators through a framework termed quantile kriging, which allows for non-parametric representations of the stochastic behavior of the output whereas previous work has focused on normally distributed outputs. Furthermore, this work studied asymptotic properties of this methodology that yielded practical insights. Under certain regulatory conditions, there is the following result: By using an experiment that has the appropriate ratio of replications to sets of different inputs, we can achieve an optimal rate of convergence. Additionally, this method provided the basic tool for the study of defect patterns and a case study is explored.

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With remarkable advances in computing power, computer experiments continue to expand the boundaries and drive down the cost of various scientific discoveries. New challenges keep arising from designing, analyzing, modeling, calibrating, optimizing, and predicting in computer experiments. This dissertation consists of six chapters, exploring statistical methodologies in sequential learning, model calibration, and uncertainty quantification for heteroskedastic computer experiments and large-scale computer experiments. For heteroskedastic computer experiments, an optimal lookahead based sequential learning strategy is presented, balancing replication and exploration to facilitate separating signal from input-dependent noise. Motivated by challenges in both large data size and model fidelity arising from ever larger modern computer experiments, highly accurate and computationally efficient divide-and-conquer calibration methods based on on-site experimental design and surrogate modeling for large-scale computer models are developed in this dissertation. The proposed methodology is applied to calibrate a real computer experiment from the gas and oil industry. This on-site surrogate calibration method is further extended to multiple output calibration problems. / Doctor of Philosophy / With remarkable advances in computing power, complex physical systems today can be simulated comparatively cheaply and to high accuracy through computer experiments. Computer experiments continue to expand the boundaries and drive down the cost of various scientific investigations, including biological, business, engineering, industrial, management, health-related, physical, and social sciences. This dissertation consists of six chapters, exploring statistical methodologies in sequential learning, model calibration, and uncertainty quantification for heteroskedastic computer experiments and large-scale computer experiments. For computer experiments with changing signal-to-noise ratio, an optimal lookahead based sequential learning strategy is presented, balancing replication and exploration to facilitate separating signal from complex noise structure. In order to effectively extract key information from massive amount of simulation and make better prediction for the real world, highly accurate and computationally efficient divide-and-conquer calibration methods for large-scale computer models are developed in this dissertation, addressing challenges in both large data size and model fidelity arising from ever larger modern computer experiments. The proposed methodology is applied to calibrate a real computer experiment from the gas and oil industry. This large-scale calibration method is further extended to solve multiple output calibration problems.

This thesis consists of three chapters on the statistical adjustment, calibration, and uncertainty quantification of complex computer models with applications in engineering. The first chapter systematically develops an engineering-driven statistical adjustment and calibration framework, the second chapter deals with the calibration of potassium current model in a cardiac cell, and the third chapter develops an emulator-based approach for propagating input parameter uncertainty in a solid end milling process. Engineering model development involves several simplifying assumptions for the purpose of mathematical tractability which are often not realistic in practice. This leads to discrepancies in the model predictions. A commonly used statistical approach to overcome this problem is to build a statistical model for the discrepancies between the engineering model and observed data. In contrast, an engineering approach would be to find the causes of discrepancy and fix the engineering model using first principles. However, the engineering approach is time consuming, whereas the statistical approach is fast. The drawback of the statistical approach is that it treats the engineering model as a black box and therefore, the statistically adjusted models lack physical interpretability. In the first chapter, we propose a new framework for model calibration and statistical adjustment. It tries to open up the black box using simple main effects analysis and graphical plots and introduces statistical models inside the engineering model. This approach leads to simpler adjustment models that are physically more interpretable. The approach is illustrated using a model for predicting the cutting forces in a laser-assisted mechanical micromachining process and a model for predicting the temperature of outlet air in a fluidized-bed process. The second chapter studies the calibration of a computer model of potassium currents in a cardiac cell. The computer model is expensive to evaluate and contains twenty-four unknown parameters, which makes the calibration challenging for the traditional methods using kriging. Another difficulty with this problem is the presence of large cell-to-cell variation, which is modeled through random effects. We propose physics-driven strategies for the approximation of the computer model and an efficient method for the identification and estimation of parameters in this high-dimensional nonlinear mixed-effects statistical model. Traditional sampling-based approaches to uncertainty quantification can be slow if the computer model is computationally expensive. In such cases, an easy-to-evaluate emulator can be used to replace the computer model to improve the computational efficiency. However, the traditional technique using kriging is found to perform poorly for the solid end milling process. In chapter three, we develop a new emulator, in which a base function is used to capture the general trend of the output. We propose optimal experimental design strategies for fitting the emulator. We call our proposed emulator local base emulator. Using the solid end milling example, we show that the local base emulator is an efficient and accurate technique for uncertainty quantification and has advantages over the other traditional tools.