Hierarchical Nonlinear Approximation for Experimental Design and Statistical Data Fitting

This paper proposes a hierarchical nonlinear approximation scheme for scalar-valued multivariate functions, where the main objective is to obtain an accurate approximation with using only very few function evaluations. To this end, our iterative method combines at any refinement step the selection of suitable evaluation points with kriging, a standard method for statistical data analysis. Particular improvements over previous non-hierarchical methods are mainly concerning the construction of new evaluation points at run time. In this construction process, referred to as experimental design, a flexible two-stage method is employed, where adaptive domain refinement is combined with sequential experimental design. The hierarchical method is applied to statistical data analysis, where the data is generated by a very complex and computationally expensive computer model, called a simulator. In this application, a fast and accurate statistical approximation, called an emulator, is required as a cheap surrogate of the expensive simulator. The construction of the emulator relies on computer experiments using a very small set of carefully selected input configurations for the simulator runs. The hierarchical method proposed in this paper is, for various analysed models from reservoir forecasting, more efficient than existing standard methods. This is supported by numerical results, which show that our hierarchical method is, at comparable computational costs, up to ten times more accurate than traditional non-hierarchical methods, as utilized in commercial software relying on the response surface methodology (RSM).