Uncertainty Quantification for Numerical Methods Based on Sparse Representation: A Bayesian Approach

• Activity Number: 54
• Type: Invited
• Date/Time: Sunday, July 29, 2012 : 4:00 PM to 5:50 PM
• Sponsor: Statistical and Applied Mathematical Sciences Institute
• Author(s): Jeff Wu*+
• Companies: Georgia Institute of Technology

Abstract

Because of the known numerical stability problem of the kriging method, alternatives based on efficient and stable numerical approximations have become popular for modeling computer experiments. A challenge for numerical based method is its lack of a stochastic structure on which uncertainty quantification is based. In this work we focus on a class of numerical method which employs a sparse representation from an over-complete dictionary of basis functions. The only unknown parameters in the representation are the linear coefficients attached to each basis function. But the number of coefficients is large. To deal with linearity in high dimension, we impose a Bayesian normal prior on the large space of coefficients. Then we use the stochastic search variable selection (SSVS) method to identify and estimate the significant coefficients. The search is very efficient because the number of significant coefficient is much smaller (i.e., effect sparsity). MCMC is used to do the Bayesian computations. Posterior distributions are automatically obtained, which can provide the Bayesian credible intervals for the predicted values. The approach is illustrated successfully on two real data sets. (Based on joint work with R. B. Chen and W. C. Wang.)