A Spatially Adaptive Sparse Grid Combination Technique for Numerical Quadrature

Abstract

High-dimensional problems have gained interest in many disciplines such as Machine Learning, Data Analytics, and Uncertainty Quantification. These problems often require an adaptation of a model to the problem as standard methods do not provide an efficient description. Spatial adaptivity is one of these approaches that we investigate in this work. We introduce the Spatially Adaptive Combination Technique using a Split-Extend scheme—a spatially adaptive variant of the Sparse Grid Combination Technique—that recursively refines block adaptive full grids to get an efficient representation of local phenomena in functions. We discuss the method in the context of numerical quadrature and demonstrate that it is suited to refine efficiently for various test functions where common approaches fail. Trapezoidal quadrature rules as well as Gauss-Legendre quadrature are investigated to show its applicability to a wide range of quadrature formulas. Error estimates are used to automate the adaptation process which results in a parameter-free version of our refinement strategy.