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

TL;DR

The paper develops a comprehensive framework for high-dimensional numerical integration of infinitely differentiable non-periodic functions via quasi-Monte Carlo methods. By embedding the target function space $\mathcal{F}_{s,\boldsymbol{u}}$ into Walsh spaces and employing digital nets, the authors establish super-polynomial convergence bounds $e(n,s) \le C(s)\exp(-c(s)(\log n)^2)$, and under fast weight decay, dimension-independent rates $e(n,s) \le C\exp(-c(\log n)^p)$ for $1<p<2$. A duality-based analysis provides matching lower bounds and clarifies the role of weight decay in tractability, with tractable, dimension-robust rates achieved when $a_j$ grows sufficiently fast and $\log(u_j^{-1})$ decays appropriately. Overall, the work offers a rigorous path to efficient high-dimensional integration for smooth non-periodic functions using digital nets and Walsh-space techniques, with clear criteria linking weight decay to convergence and tractability.

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 $\| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbolα = (α_1, \dots, α_s) \in \mathbb{N}_0^s} \|f^{(\boldsymbolα)}\|_{L^1}/\prod_{j=1}^s u_j^{α_j}$, $f^{(\boldsymbolα)} := \frac{\partial^{|\boldsymbolα|}}{\partial x_1^{α_1} \cdots \partial x_s^{α_s}}f$ 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.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 4.1
• Theorem 4.2
• Lemma 5.1
• proof