On Lebesgue Integral Quadrature

TL;DR

This paper introduces Lebesgue quadrature, a novel numerical integration method that focuses on function values and their probabilities, offering advantages for irregular and stochastic processes compared to traditional Gaussian quadrature.

Contribution

The paper develops a new Lebesgue quadrature method that finds optimal function values and weights, providing a different perspective from Gaussian quadrature and better handling stochastic processes.

Findings

Lebesgue quadrature is formulated as a generalized eigenvalue problem.

It separates outcome values from their probabilities, aiding analysis of non-Gaussian processes.

Software implementation of the method is provided by the authors.

Abstract

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value-nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of the outcome (weight). For this reason, it is especially well-suited for the study of non-Gaussian processes. The software implementing the theory is available from the authors.