Adaptive design for the estimation of high-variation regions using non-stationary Gaussian process models
Résumé
In the context of expensive deterministic simulations, Gaussian process
(GP) models have become a standard device for sequential design of experiments.
In this context, the objective function f is seen as a black box and assumed to
be one realization of a GP with given mean and covariance. Sampling criteria can
then be worked out to optimally choose the next evaluation points under the GP
assumption. State of the art criteria notably include MSE and IMSE. Further criteria
dedicated to localizing optima, and also to the estimation of target regions have been
proposed. Here we propose two approaches for learning f in cases where it has very
heterogeneous variations across the input space. First, we define a new family of
criteria inspired by MSE and IMSE but with a focus on conditional gradients under
the GP model. Second, we use the classic MSE and IMSE criteria with a specific GP
model accounting for prior knowledge about the heterogeneous variations of f. All
combinations of the considered criteria and GP models are compared on a test case
from the French nuclear safety institute about the mechanical ageing of power plants.
It is found on this application that in the case of stationary GP modelling, one of
the proposed derivative-based criteria outperforms MSE and IMSE both in terms of
local and global approximation error. However, the overall best results in terms of
global approximation are obtained with IMSE and our non-stationary kernel.