Metamodeling For Computer Experiments Based On Sparseness Prior

Along with the rapid development of economics and society and the growing popularity of global economic integration, enterprises are urgently required to achieve product design of short cycle, low cost, and high quality, in order to improve product competitiveness in the global market. In the realization process of product, the design phase is the headstream of improving the quality of product, the concept of which has gradually been accepted. At present, the product quality design is often based on the traditional physical experimental design, which is, however, notorious of many inadequacies, such as limited number of design factors, high costs of experimental running, strong disturbance of random noise, etc. In order to get over these difficulties of the design of experiments, computer experiments with great practice and application, is a new developing frontier branch of science. In order to achieve quality control of complex products more effectively, an in-depth analysis and research is made on the issue of deterministic modeling of computer experiments in this thesis, with expectations for providing more effective technical means to product quality design.The thesis focuses mainly on the metamodeling of computer experiments based on sparseness prior and its practical application. The innovation theory and results we have obtained are listed as follows.(1) The number of input variables in computer simulation experiments is up to 15-20, or even more than 100. Fist of all, a metamodeling approach for computer experiments is proposed to achieve variable selection, through minimizing the non-convex Lp norms(0<p<1) of candidate variables, as well as our derived non-convex robust penalty functions capable of measuring sparseness. The estimation of sparse regression coefficient is implemented approximately using the half-quadratic regularization algorithm and reweighted least squares. In the case study of Borehole Model, the proposed metamodeling approaches performs more effectively compared with several variable selection methods in literature.(2) A Bayesian metamodeling approach is proposed based on hierarchical sparseness priors for computer experiments. Specifically, the sparseness prior imposed on the regression coefficient is defined either via variance-varying Gaussian and uniform PDF (probability density function) or variance-varying Gaussian, double exponential, and Gamma PDF. The proposed approach is not only capable of simultaneously and automatically screening out the noise factors and estimating the coefficients corresponding to important design factors at the same time, but also capable of fast Bayesian inference leading to efficient metamodeling for computer experiments. Experimental results on the Borehole and Piston Slap Noise Models show that, our approach has provided a concise surrogate model and made an efficient and effective prediction.(3) Taking into consideration of the disadvantage of Gaussian Kriging metamodeling based on maximum likelihood estimation (MLE), a new and effective Kriging metamodeling approach is proposed through imposing a hierarchical Bayesian prior on the correlation parameter in Kriging. Specifically, first impose the variance-varying Gaussian PDF directly on the correlation parameter, then impose Jeffery's non-informative hyper prior on the varying variances, and finally achieve Bayesian inference via the expectation maximization algorithm and Fisher's scoring algorithm. As for both the artificial simulation and actual computer experiments, e.g., Piston Slap Noise, Exhaust Manifold, the mean square error, median of absolute residual, and absolute deviation map all demonstrate the superiority of our proposed approach over several previous methods.(4) Since regression basis functions in Universal Kriging are seldom known, the simple Ordinary Kriging is more preferred in practice, whose overall trend is just described by a constant. To make Universal Kriging more practical, we propose a sparse Blind Kriging metamodeling approach through incorporating the mentioned hierarchical Bayesian variable selection scheme into the Universal Kriging metamodeling process. Case studies of both the artificial simulation and Piston Slap Noise experiments demonstrate that, sparse Blind Kriging performs much better than several other known methods, such as Ordinary Kriging, Universal Kriging, SCAD-Kriging (Smooth Clipped Absolute Deviation), and so on.