Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions
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| Format: | Preprint |
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2015
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| _version_ | 1866908682233577472 |
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| author | Suzuki, Kosuke |
| author_facet | Suzuki, Kosuke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1505_02003 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions Suzuki, Kosuke Numerical Analysis 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. |
| title | Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/1505.02003 |