Counting Nodes in Smolyak Grids

TL;DR

This paper introduces a unified generating function approach to explicitly count nodes in Smolyak grids, including new formulas for grids based on Chebyshev nodes, unifying and extending previous results.

Contribution

It presents a novel, unified method using generating functions to derive explicit formulas for counting nodes in Smolyak grids, including new formulas for Chebyshev-based grids.

Findings

Derived a new formula for Chebyshev-based Smolyak grid cardinality

Unified existing counting formulas using generating functions

Extended previous results by Bungartz-Griebel and others

Abstract

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, M\"uller-Gronbach, Novak-Ritter and Ullrich.