Optimal integration of Lipschitz functions with a Gaussian weight
The aim of this work is to study numerical integration of scalar functions over the infinite interval \((-\infty,+\infty)\) with a Gaussian weight. Such integrals, and their multivariate analogs, have numerous applications. The author assumes that the functions satisfy a Lipschitz condition with known constant \(L\). For this class of functions he studies the (worst case) complexity of integration with the Gaussian weight \(\exp(- x^2/(2\sigma))/\sqrt{2\pi\sigma}\), for a positive variance \(\sigma\). First, he studies the problem of finding optimal information. That is, for a given number \(n\), he finds optimal sample points \(x_1,\dots, x_n\) that yield the minimal error \(r(n)\) of the optimal algorithm. The first major result is to show that (asymptotically) the optimal information is given by the quantiles of the Gaussian weight with double variance, i.e., \[ {1\over\sqrt{2\pi(2\sigma)}} \int^{x_i}_0 \exp(- t^2/(2(2\sigma)))dt\sim{i\over n+1}. \] The second major result of this paper is to show that the worst case error \(r(n)\) of the optimal information is \[ r(n)= \sqrt{{\pi\sigma\over 2}}\cdot{L\over n} (1- O(1))\quad\text{as }n\to+\infty. \] Numerical examples are given.
scientific article; zbMATH DE number 1206671
scientific article