M\"obius-Transformed Trapezoidal Rule

TL;DR

This paper introduces a M"obius-transformed trapezoidal rule for numerical integration that achieves optimal convergence rates for functions in weighted Sobolev spaces, without requiring derivative information or sampling from probability measures.

Contribution

It develops a novel transformation-based trapezoidal rule that attains optimal convergence for weighted Sobolev space functions and extends to various approximation and integration methods.

Findings

Achieves optimal convergence rate for weighted Sobolev functions

Does not require derivative evaluations or probability sampling

Extends to multivariate and randomized integration methods

Abstract

We study numerical integration by combining the trapezoidal rule with a M\"obius transformation that maps the unit circle onto the real line. We prove that the resulting transformed trapezoidal rule attains the optimal rate of convergence if the integrand function lives in a weighted Sobolev space with a weight that is only assumed to be a positive Schwartz function decaying monotonically to zero close to infinity. Our algorithm only requires the ability to evaluate the weight at the selected nodes, and it does not require sampling from a probability measure defined by the weight nor information on its derivatives. In particular, we show that the M\"obius transformation, as a change of variables between the real line and the unit circle, sends a function in the weighted Sobolev space to a periodic Sobolev space with the same smoothness. Since there are various results available for integrating and approximating periodic functions, we also describe several extensions of the M\"obius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.