Limiting Ranges of Function Values of Sparse Grid Surrogates

Abstract

Sparse grid interpolants of high-dimensional functions do not maintain the range of function values. This is a core problem when one is dealing with probability density functions, for example. We present a novel approach to limit range of function values of sparse grid surrogates. It is based on computing minimal sets of sparse grid indices that extend the original sparse grid with properly chosen coefficients such that the function value range of the resulting surrogate function is limited to a certain interval. We provide the prerequisites for the existence of minimal extension sets and formally derive the intersection search algorithm that computes them efficiently. The main advantage of this approach is that the surrogate remains a linear combination of basis functions and, therefore, any problem specific post-processing operation such as evaluation, quadrature, differentiation, regression, density estimation, etc. can remain unchanged. Our sparse grid approach is applicable to arbitrarily refined sparse grids.