A randomised trapezoidal quadrature

TL;DR

This paper introduces a randomized trapezoidal quadrature rule that effectively integrates functions with fractional Sobolev regularity, achieving improved convergence rates over classical methods.

Contribution

The paper proposes a novel randomized quadrature rule applicable to less regular functions, with proven error bounds and enhanced convergence order.

Findings

Improved convergence order by half compared to classical trapezoidal rule

Error bounds established for the randomized quadrature

Applicable to functions with fractional Sobolev regularity

Abstract

A randomised trapezoidal quadrature rule is proposed for continuous functions which enjoys less regularity than commonly required. Indeed, we consider functions in some fractional Sobolev space. Various error bounds for this randomised rule are established while an error bound for classical trapezoidal quadrature is obtained for comparison. The randomised trapezoidal quadrature rule is shown to improve the order of convergence by half.