The rapid technological growth, whilst desirable, comes with increased complexity in real-world problems that engineers need to solve. Computational methods have been commonly employed in the design and analyses of such complex problems. Reducing computational costs of their solution methods becomes key in efficiently performing such tasks. Kriging has been a familiar solution to engineers to inexpensively represent function evaluations of complex physical systems. To be effective, ensuring good computational efficiency and predictive ability is imperative before using kriging as a surrogate in engineering applications. This requires prudent selections of model structure (including kernels) and samples. Kernel selections—which are important to establish the input-output relation of the...[ Read more ]

The rapid technological growth, whilst desirable, comes with increased complexity in real-world problems that engineers need to solve. Computational methods have been commonly employed in the design and analyses of such complex problems. Reducing computational costs of their solution methods becomes key in efficiently performing such tasks. Kriging has been a familiar solution to engineers to inexpensively represent function evaluations of complex physical systems. To be effective, ensuring good computational efficiency and predictive ability is imperative before using kriging as a surrogate in engineering applications. This requires prudent selections of model structure (including kernels) and samples. Kernel selections—which are important to establish the input-output relation of the system—often require domain expertise, which in most cases are not transferable to other applications. Sample selection has to be done such that the sample points can adequately span the design space, and yet not too many that the computational cost becomes prohibitive. The distribution of samples also affects the well-posedness of kriging model structure. In addition, kriging construction and usage suffer from the curse of dimensionality with high-dimensional problems, thereby limiting its widespread applications. To address this issue, researchers have sought to integrate various dimensionality reduction strategies into the model structure of kriging. While this approach can make kriging scalable to higher-dimensional problems, the model accuracy is often sacrificed.

In this thesis, I introduce new strategies to address the three challenges mentioned above. Towards improving the predictive ability and computational efficiency of kriging-based models in high-dimensional problems, I develop the kriging with joint mutual information maximization (KJMIM). Its structure-preserving property proves superior than other dimensionality reduction techniques in ensuring high model accuracy. Multiple kernels, instead of a single one, are used to add more flexibility in fitting complex profiles using kriging-based models. To this end, I develop mixed kernel learning (MiKL) and multidimensional composite kernel learning (MCKL) that can automatically avoid nonperforming kernels and select the optimum kernel combinations and hyperparameters. A systematic study on the relationship between multimodality of the function and kernel selection in high-dimensional problems is also presented in this thesis. Lastly, I develop an active learning strategy with multiple stopping criteria to improve the robustness of sample selection. In particular, the likelihood information and expected improvement are exploited to select the most informative points to be added to the sample set. In this thesis, I also demonstrate that the combination of the proposed active learning strategy and effective kernel selections can notably improve the predictive capability of kriging on complex problems. The works presented in this thesis contribute to enhancing the efficacy and practicality of kriging-based models in real-world engineering applications. The exposition on methodology, results, and observation is intended to provide more intuition and insight into the effects of kriging structure and parameters on its performance.