A randomised trapezoidal quadrature (2011.15086v2)

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.

• The paper introduces a randomised trapezoidal quadrature rule that improves convergence order from O(N^(-σ)) to O(N^(-σ-1/2)) for low-regularity functions.
• It leverages uniform random sampling within each partition to successfully handle functions in fractional Sobolev spaces, extending the traditional method.
• Numerical experiments confirm the enhanced performance, indicating potential efficiency gains in fields like financial mathematics and stochastic simulations.

An Examination of Randomised Trapezoidal Quadrature for Fractional Sobolev Spaces

The paper introduces a new randomised trapezoidal quadrature rule tailored for continuous functions with lower regularity, particularly targeting functions within fractional Sobolev spaces. This enhancement of the classical trapezoidal rule aims to address scenarios where the integrand is less smooth, thus bolstering the order of convergence from $\mathcal{O}(N^{-\sigma})$ to $\mathcal{O}(N^{-\sigma-1/2})$.

Overview of Classical and Randomised Quadrature

Traditionally, the trapezoidal rule requires the integrand to have a minimum of second-order differentiability. In such contexts, the method achieves an order of convergence of 2. By considering integrands that reside within a fractional Sobolev space, $W^{\sigma,p}(0,T)$, the paper utilizes a randomised approach to enhance convergence rates. This fractional space is defined based on the Sobolev-Slobodeckij norm, which suits functions where the regularity parameter $\sigma$ lies between 1 and 2, and $p$ is greater than or equal to 2.

The classical trapezoidal quadrature captures an integral through a key partitioning of the domain, relying on evaluations strictly at equidistant points. Conversely, the proposed randomised trapezoidal quadrature selects points within each partition interval via uniformly distributed random variables. The central claim is that incorporating stochasticity in this manner yields a superior convergence order, specifically $\mathcal{O}(N^{-\sigma-\frac{1}{2}})$.

Performance Analysis

Error estimates for both classical and randomised quadrature rules are rigorously derived. For classical quadrature, the error is bounded by $$CT^{1-\frac{1}{p}}h^\sigma \|g\|_{W^{\sigma,p}(0,T)}$$, highlighting its dependency on the fractional Sobolev norm of the integrand. The randomised scheme not only retains the expectation properties of classical quadrature but further establishes its error relative to convergence rate improvements of a half-order.

These theoretical results are supported through numerical experiments with functions of varying smoothness and a practical scenario involving integration with respect to a discretely sampled Brownian motion. The findings illustrate the practical benefits of the randomised scheme, where gains in convergence are observed particularly as the regularity parameter $\sigma$ approaches its upper limit.

Implications and Future Work

The proposed randomised trapezoidal quadrature expands the toolkit for numerical integration by offering robust performance for less regular functions, a common occurrence in practical computational problems, including those in financial mathematics and other applied fields involving stochastic differential equations. The paper's approach facilitates better integration accuracy without necessitating finer discretization, leading to potential computational efficiencies.

Future research may explore further optimization of the random partitioning strategy, perhaps incorporating adaptive mechanisms that cater to the local regularity of the integrand. Another important direction is the potential extension of this quadrature method to multidimensional cases and its adaptation for other forms of irregular integrals beyond those constructed over Sobolev spaces.

In essence, the paper provides a significant contribution to numerical analysis by systematically demonstrating how randomness can be strategically leveraged to offset the constraints imposed by integrand regularity, yielding improvements in computational efficiency and integrative accuracy.