Sparse Grid Interpolation

TLDR: This thesis presents the classical sparse grid where the problem is discretized and solved on a certain sequence of conventional grids with uniform mesh sizes in each coordinate direction and describes the sparse grid combination technique to demonstrate that it is competitive to the classical bare grid approaches with respect to quality and run time.

Abstract: For the approximation of multidimensional functions, using classical numerical discretization schemes such as full grids suffers the curse of dimensionality which is still a roadblock for the numerical treatment of high-dimensional problems. The number of basis functions or nodes (grid points) have to be stored and processed depend exponentially on the number of dimensions, where efficient computation are challenging in the implementation. Recently, the technique of sparse grids has been introduced to significantly reduce the cost to approximate high-dimensional functions under certain regularity conditions. In this thesis, we present the classical sparse grid where the problem is discretized and solved on a certain sequence of conventional grids with uniform mesh sizes in each coordinate direction. Furthermore, the different types of sparse grids,i.e. Clenshaw Curtis sparse grid, have been taken into consideration to compare the accuracy and complexity of these algorithms. We then describe the sparse grid combination technique to demonstrate that it is competitive to the classical sparse grid approaches with respect to quality and run time and give proof that the interpolation by using combination approach is the classical sparse grid. We give details on the basic features of sparse grids and we consider several test problems up to dimensions. The results of numerical experiments report on the quality of approximation generated by the sparse grids, and, finally, employ the sparse grid interpolation for a real-world case to reduce a computationally expensive simulation model. We aim to obtain an efficient surrogate approximation based on a small number of simulations.