Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions

TL;DR

This paper proves super-polynomial convergence rates for multivariate integration of infinitely differentiable functions, showing that quasi-Monte Carlo methods with digital nets achieve these rates under certain conditions.

Contribution

It establishes new super-polynomial convergence bounds for multivariate integration in infinitely smooth function spaces, extending tractability results.

Findings

Error bounds of order exp(-c(log n)^2) for general weights

Improved bounds of order exp(-c(log n)^p) for fast-decaying weights

Quasi-Monte Carlo methods with digital nets attain these convergence rates

Abstract

We investigate multivariate integration for a space of infinitely times differentiable functions $\mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty [0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} < \infty \}$, where $\| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_s) \in \mathbb{N}_0^s} \|f^{(\boldsymbol{\alpha})}\|_{L^1}/\prod_{j=1}^s u_j^{\alpha_j}$, $f^{(\boldsymbol{\alpha})} := \frac{\partial^{|\boldsymbol{\alpha}|}}{\partial x_1^{\alpha_1} \cdots \partial x_s^{\alpha_s}}f$ and $\boldsymbol{u} = \{u_j\}_{j \geq 1}$ is a sequence of positive decreasing weights. Let $e(n,s)$ be the minimal worst-case error of all algorithms that use $n$ function values in the $s$-variate case. We prove that for any $\boldsymbol{u}$ and $s$ considered $e(n,s) \leq C(s) \exp(-c(s)(\log{n})^2)$ holds for all $n$, where $C(s)$ and $c(s)$ are constants which may depend on $s$. Further we show that if the weights $\boldsymbol{u}$ decay sufficiently fast then there exist some $1 < p < 2$ and absolute constants $C$ and $c$ such that $e(n,s) \leq C \exp(-c(\log{n})^p)$ holds for all $s$ and $n$. These bounds are attained by quasi-Monte Carlo integration using digital nets. These convergence and tractability results come from those for the Walsh space into which $\mathcal{F}_{s, \boldsymbol{u}}$ is embedded.