Sequential Designs Based on Bayesian Uncertainty Quantification in Sparse Representation Surrogate Modeling

ABSTRACT

A numerical method, called overcomplete basis surrogate method (OBSM), was recently proposed, which employs overcomplete basis functions to achieve sparse representations. While the method can handle nonstationary response without the need of inverting large covariance matrices, it lacks the capability to quantify uncertainty in predictions. We address this issue by proposing a Bayesian approach that first imposes a normal prior on the large space of linear coefficients, then applies the Markov chain Monte Carlo (MCMC) algorithm to generate posterior samples for predictions. From these samples, Bayesian credible intervals can then be obtained to assess prediction uncertainty. A key application for the proposed method is the efficient construction of sequential designs. Several sequential design procedures with different infill criteria are proposed based on the generated posterior samples. Numerical studies show that the proposed schemes are capable of solving problems of positive point identification, optimization, and surrogate fitting.

KEYWORDS:

• Credible interval
• Overcomplete bases surrogates method
• Posterior sample
• Stochastic search variable selection

Acknowledgments

The authors are grateful to the associate editor, two referees, and Simon Mak for valuable comments. Chen’s research is supported by Ministry of Science and Technology (MOST) of Taiwan 104-2918-I-006-005 and the Mathematics Division of the National Center for Theoretical Sciences in Taiwan. Wang’s research is partially supported by the National Science Council (Taiwan) and the Taida Institute of Mathematical Sciences. Wu’s research is supported by NSF DMS-1308424 and DOE DE-SC0010548.