Gaussian Process Regression on Nested Spaces

As computer codes simulate complex physical phenomena, they involve a very large number of variables. To gain time, industrial experts build metamodels on a restricted set of variables, the most influential ones, while the others are fixed. The set of variables is then enlarged progressively to improve knowledge on the studied output. Several designs of experiment are generated, which belong to subspaces included in each other and of increasing dimension. The goal of this paper is to create a metamodel adapted to this inefficient design process, that exploits the structure of all previous runs. An approach based on Gaussian process regression and called seqGPR (sequential Gaussian process regression) is introduced. At each new step of the study (when new variables are released), the output is supposed to be the realization of the sum of two independent Gaussian processes. The first one models the output at the previous step. The second one is a correction term which must be null on the subspace studied at the previous step, that is to say null on a continuum of points. First, some candidate Gaussian processes for the correction terms are suggested. Then, an EM (expectation-maximization) algorithm is implemented to estimate the parameters of the processes. Finally, the metamodel seqGPR is compared to a standard kriging metamodel on three test cases and gives better results.