Information Theory and Quadrature Rules

TL;DR

This paper explores the connection between information theory and quadrature rules, proposing new rules based on Kolmogorov complexity and compressed sensing, applicable to sparse functions and higher dimensions.

Contribution

It introduces a novel framework linking information theory with quadrature rule design, enabling the creation of rules without smoothness assumptions and extending to higher dimensions.

Findings

Standard rules relate to low Kolmogorov complexity strings

Good rules exist for sparse functions satisfying an error--information duality

Error bounds depend on compressed sensing concepts

Abstract

Quadrature rules estimate the value of an integral when the function is given by a table of values. Every binary string defines a quadrature rule by choosing which endpoint of each interval represents the interval. The standard rules, such as Simpson's Rule, correspond to strings of low Kolmogorov complexity, making it possible to define new quadrature rules with no smoothness assumptions, as well as in higher dimensions. Error results depend on concepts from compressed sensing. Good quadrature rules exist for "sparse" functions, which also satisfy an error--information duality principle.