Möbius-Transformed Trapezoidal Rule

Abstract

We study numerical integration by combining the trapezoidal rule with a Möbius 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 Möbius 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 Möbius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.

arXiv comment24 pages, 2 figures

References in corpus (9)

• Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha -1/2})$ error independently of the dimension
• Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness
• Lattice rules in non-periodic subspaces of Sobolev spaces
• Scaled lattice rules for integration on ${\mathbb{R}}^d$ achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces
• Infinite-dimensional integration and $L^2$-approximation on Hermite spaces
• How sharp are error bounds? --lower bounds on quadrature worst-case errors for analytic functions
• Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness
• Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces
• On alternative quantization for doubly weighted approximation and integration over unbounded domains