@article {SpaceQuantizationMethodIntegration,
title = {A space quantization method for numerical integration},
journal = {J. Comput. Appl. Math.},
volume = {89},
number = {1},
year = {1998},
pages = {1{\textendash}38},
publisher = {Elsevier Science Publishers B. V.},
address = {Amsterdam, The Netherlands},
abstract = {We propose a new method (SQM) for numerical integration of $C^\alpha$ functions ($\alpha \in (0,2]$) defined on a convex subset $C$ of $\mathbb{R}^d$ with respect to a continuous distribution $\mu$. It relies on a space quantization of $C$ by a $n$-tuple $x:= (x_1,\cdots, x_n) \in C^n$. $\int f d\mu$ is approximated by a weighted sum of the $f (x_i)${\textquoteright}s. The integration error bound depends on the distortion $E_n^{z,\mu}(x)$ of the Vorono{\"\i} tessellation of $x$. This notion comes from Information Theoretists. Its main properties (existence of a minimizing $n$-tuple in $C^n$, asymptotics of $\min\limits_{C^n} E_n^{\alpha,\mu}$ as $n \to + \infty$) are presented for a wide class of measures $\mu$. A simple stochastic optimization procedure is proposed to compute, in any dimension $d$, $x^*$ and the characteristics of its Vorono{\"\i} tessellation. Some new results on the Competitive Learning Vector Quantization algorithm (when $\alpha = 2$) are obtained as a by-product. Some tests, simulations and provisional remarks are proposed as a conclusion.},
keywords = {competitive algorithms, error estimation, learning algorithms, numerical integration, numerical methods, optimization method, vector quantization, Voronoi diagram},
issn = {0377-0427},
author = {Gilles Pag{\`e}s}
}