多重积分的超多项式收敛性与可处理性研究：对无限可微函数空间
Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions
AI总结：
研究了无限次可微函数空间的多重积分收敛性与可处理性，证明了拟蒙特卡洛积分在特定条件下具有超多项式收敛性。
摘要：
我们研究了无限次可微函数空间$\mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty [0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} < \infty \}$的多重积分，其中$\| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbolα = (α_1, \dots, α_s) \in \mathbb{N}_0^s} \|f^{(\boldsymbolα)}\|_{L^1}/\prod_{j=1}^s u_j^{α_j}$，$f^{(\boldsymbolα)} := \frac{\partial^{|\boldsymbolα|}}{\partial x_1^{α_1} \cdots \partial x_s^{α_s}}f$，$\boldsymbol{u} = \{u_j\}_{j \geq 1}$是正递减的权重序列。令$e(n,s)$表示所有使用$n$个函数值的算法的最小最坏误差。我们证明了对于任何$\boldsymbol{u}$和$s$考虑的$e(n,s) \leq C(s) \exp(-c(s)(\log{n})^2)$对所有$n$成立，其中$C(s)$和$c(s)$可能依赖于$s$。进一步我们显示如果权重$\boldsymbol{u}$足够快速地衰减，则存在某些$1 < p < 2$和绝对常数$C$和$c$，使得$e(n,s) \leq C \exp(-c(\log{n})^p)$对所有$s$和$n$成立。这些界限由使用数字网络的拟蒙特卡洛积分获得。这些收敛和可处理性结果来自将$\mathcal{F}_{s, \boldsymbol{u}}$嵌入到Walsh空间中的结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
