In this thesis, two distinct problems in data-driven computational science are considered. The main problem of interest is the multiobjective optimization problem, where the tradeoff surface (called the Pareto front) between multiple conflicting objectives must be approximated in order to identify designs that balance real-world tradeoffs. In order to solve multiobjective optimization problems that are derived from computationally expensive blackbox functions, such as engineering design optimization problems, several methodologies are combined, including surrogate modeling, trust region methods, and adaptive weighting. The result is a numerical software package that finds approximately Pareto optimal solutions that are evenly distributed across the Pareto front, using minimal cost function evaluations. The second problem of interest is the closely related problem of multivariate interpolation, where an unknown response surface representing an underlying phenomenon is approximated by finding a function that exactly matches available data. To solve the interpolation problem, a novel algorithm is proposed for computing only a sparse subset of the elements in the Delaunay triangulation, as needed to compute the Delaunay interpolant. For high-dimensional data, this reduces the time and space complexity of Delaunay interpolation from exponential time to polynomial time in practice. For each of the above problems, both serial and parallel implementations are described. Additionally, both solutions are demonstrated on real-world problems in computer system performance modeling. / Doctor of Philosophy / Science and engineering are full of multiobjective tradeoff problems. For example, a portfolio manager may seek to build a financial portfolio with low risk, high return rates, and minimal transaction fees; an aircraft engineer may seek a design that maximizes lift, minimizes drag force, and minimizes aircraft weight; a chemist may seek a catalyst with low viscosity, low production costs, and high effective yield; or a computational scientist may seek to fit a numerical model that minimizes the fit error while also minimizing a regularization term that leverages domain knowledge. Often, these criteria are conflicting, meaning that improved performance by one criterion must be at the expense of decreased performance in another criterion. The solution to a multiobjective optimization problem allows decision makers to balance the inherent tradeoff between conflicting objectives. A related problem is the multivariate interpolation problem, where the goal is to predict the outcome of an event based on a database of past observations, while exactly matching all observations in that database. Multivariate interpolation problems are equally as prevalent and impactful as multiobjective optimization problems. For example, a pharmaceutical company may seek a prediction for the costs and effects of a proposed drug; an aerospace engineer may seek a prediction for the lift and drag of a new aircraft design; or a search engine may seek a prediction for the classification of an unlabeled image. Delaunay interpolation offers a unique solution to this problem, backed by decades of rigorous theory and analytical error bounds, but does not scale to high-dimensional "big data" problems. In this thesis, novel algorithms and software are proposed for solving both of these extremely difficult problems.

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Derivative free optimization algorithms are implementations of trust region based derivative-free methods using multivariate polynomial interpolation. These are designed to minimize smooth functions whose derivatives are not available or costly to compute. The trust region based multilevel optimization algorithms for solving large scale unconstrained optimization problems resulting by discretization of partial differential equations (PDEs), make use of different discretization levels to reduce the computational cost. In this thesis, a derivative free multilevel optimization algorithm is derived and its convergence behavior is analyzed. The effectiveness of the algorithms is demonstrated on a shape optimization problem.

Many problems in science and engineering are solved through experimental investigations. Because experiments can be costly and time consuming, it is important to efficiently design the experiment so that maximum information about the problem can be obtained. It is also important to devise efficient statistical methods to analyze the experimental data so that none of the information is lost. This thesis makes contributions on several aspects in the field of design and analysis of experiments. It consists of two parts. The first part focuses on physical experiments, and the second part on computer experiments. The first part on physical experiments contains three works. The first work develops Bayesian experimental designs for robustness studies, which can be applied in industries for quality improvement. The existing methods rely on modifying effect hierarchy principle to give more importance to control-by-noise interactions, which can violate the true effect order of a system because the order should not depend on the objective of an experiment. The proposed Bayesian approach uses a prior distribution to capture the effect hierarchy property and then uses an optimal design criterion to satisfy the robustness objectives. The second work extends the above Bayesian approach to blocked experimental designs. The third work proposes a new modeling and design strategy for mixture-of-mixtures experiments and applies it in the optimization of Pringles potato crisps. The proposed model substantially reduces the number of parameters in the existing multiple-Scheffé model and thus, helps the engineers to design much smaller experiments. The second part on computer experiments introduces two new methods for analyzing the data. The first is an interpolation method called regression-based inverse distance weighting (RIDW) method, which is shown to overcome some of the computational and numerical problems associated with kriging, particularly in dealing with large data and/or high dimensional problems. In the second work, we introduce a general nonparametric regression method, called kernel sum regression. More importantly, we make an interesting discovery by showing that a particular form of this regression method becomes an interpolation method, which can be used to analyze computer experiments with deterministic outputs.

Humidity measurement has been increasingly important in many industries and process control applications. This thesis research focus mainly on humidity sensor calibration and characterization. The humidity sensor instrumentation is briefly described. The testing infrastructure was designed for sensor data acquisition, in order to compensate the humidity sensor’s temperature coefficient, temperature chambers using Peltier elements are used to achieve easy-controllable stable temperatures. The sensor characterization falls into a multivariate interpolation problem. Neuron networks is tried for non-linear data fitting, but in the circumstance of limited training data, an innovative algorithm was developed to utilize shape preserving polynomials in multiple planes in this kind of multivariate interpolation problems.