Multidimensional Hierarchical Interpolation Method on Sparse Grids for the Absorption Problem

Xuesong Chen,Lili Zhang,Heng Mai

doi:10.1109/access.2019.2956809

Abstract

The numerical integration of multidimensional functions using some variables of the sparse grid method for the absorption problem is presented in this paper. The multivariate quadrature expressions are constructed by combining tensor of suited one dimensional formula. We develop a multidimensional adaptive quadrature algorithm for the implementation of sparse grid based on a hierarchical basis. Furthermore, we obtain a new error bound at each sparse grid point. The numerical examples are shown to demonstrate the efficiency of our algorithm for the absorption problem and confirm the theoretical estimates.

Highlights

Multivariate integrals arise in many scientific and engineering application fields such as statistical mechanics, the valuation of financial derivatives etc
The main contributions of this paper are: 1) We develop a novel algorithm for implementing sparse grid method based on the hierarchical polynomials for high dimensional numerical integration, the characteristics method addressed in this paper is causality free and has wonderful parallelism
The sparse grid approach is one of the popular methods recently used to reduce the computational cost associated with spatial discretization in solving high dimensional integration problems

Summary

INTRODUCTION

Multivariate integrals arise in many scientific and engineering application fields such as statistical mechanics, the valuation of financial derivatives etc. Sparse grid techniques were firstly introduced by Smolyak [6] to overcome the curse of dimension to a certain extent In this method, multivariate quadrature formulas are constructed using combinations of tensor products of suited. The main contributions of this paper are: 1) We develop a novel algorithm for implementing sparse grid method based on the hierarchical polynomials for high dimensional numerical integration, the characteristics method addressed in this paper is causality free and has wonderful parallelism. It is shown that the computation results using sparse grid method on the absorption problem are much more accurate than the results using Quasi-Monte Carlo integration approach in [3].

HIERARCHICAL BASES AND SPARSE GRID

MULTIDIMENSIONAL HIERARCHICAL INTERPOLATION

ALGORITHM

NUMERICAL EXAMPLES

CONCLUSIONS AND DISCUSSIONS