Computing Integrals Involved the Gaussian Function with a Small Standard Deviation

TL;DR

This paper introduces specialized numerical integration methods for efficiently computing integrals involving a Gaussian function with a small standard deviation, addressing accuracy issues caused by rapid derivative changes.

Contribution

The paper presents novel quadrature schemes based on graded meshes and Chebyshev interpolation tailored for small-variance Gaussian integrals, achieving polynomial and exponential accuracy.

Findings

Quadrature schemes achieve high accuracy for small standard deviations.

Numerical results confirm the effectiveness of the proposed methods.

Methods outperform traditional approaches in precision for challenging integrals.

Abstract

We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to the integral normally lead to poor accuracy due to the rapid change in high order derivatives of its integrand when the standard deviation is small. The proposed quadrature schemes are based on graded meshes designed according to the standard deviation so that the quadrature errors on the resulting subintervals are approximately equal. The integral in each subinterval is then computed by considering the Gaussian function as a weight function and interpolating the smooth factor of the integrand at the Chebyshev points of the first kind. For a finite order differentiable factor, we design a quadrature scheme having accuracy of a polynomial order and for an infinitely differentiable factor of the integrand, we design a quadrature scheme having accuracy of an exponential order. Numerical results are presented to confirm the accuracy of these proposed quadrature schemes.