[Paper Review] A note about EC-$(s,t)$-weak tractability of multivariate approximation with analytic Korobov kernels

Heping Wang|arXiv (Cornell University)|Aug 4, 2018

Mathematical Approximation and Integration17 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for exponential convergence $(s,t)$-weak tractability (EC-$(s,t)$-WT) in multivariate approximation using analytic Korobov kernels in both worst and average case settings. It proves that EC-$(s,t)$-WT holds under specific decay conditions on the sequence $\{a_k\}$, with distinct criteria depending on the values of $s$ and $t$, particularly distinguishing cases for $s<1$, $s\geq1$, $t<1$, and $t\geq1$, and provides a complete characterization for both settings using information complexity bounds.

ABSTRACT

This note is devoted to discussing multivariate approximation of continuous functions on $[0,1]^d$ with analytic Korobov kernels in the worst and average case settings. We only consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We study EC-$(s, t)$-weak tractability under the absolute or normalized error criterion, and obtain necessary and sufficient conditions for $00$ in the average case setting.

Motivation & Objective

• To determine necessary and sufficient conditions for EC-$(s,t)$-weak tractability in multivariate approximation problems with analytic Korobov kernels.
• To analyze the behavior of information complexity $n(\varepsilon, S_d)$ as a function of $\varepsilon^{-1}$ and $d$ under absolute and normalized error criteria.
• To establish a complete characterization of EC-$(s,t)$-WT in both worst and average case settings for $s,t>0$ and $0<\min(s,t)<1$, $\max(s,t)\leq1$.
• To clarify the role of the sequence $\{a_k\}$ in determining tractability, particularly when $\lim_{k\to\infty}a_k = \infty$ or $\lim_{k\to\infty}a_k < \infty$.
• To compare tractability in worst and average case settings, showing equivalence under certain conditions for ABS (absolute error) and NOR (normalized error) settings.

Proposed method

• Uses reproducing kernel Hilbert spaces $H(K_{d,{\bf a},{\bf b}})$ with product-form analytic Korobov kernels defined via $K_{d,{\bf a},{\bf b}}({\bf x},{\bf y}) = \sum_{{\bf h} \in \mathbb{Z}^d} \omega^{\sum_{k=1}^d a_k |h_k|^{b_k}} \exp(2\pi i {\bf h} \cdot ({\bf x} - {\bf y}))$.
• Analyzes the information complexity $n(\varepsilon, S_d)$ via bounds on the logarithm of the minimal error and the number of functionals required, using $\ln n(\varepsilon, S_d)$ as a proxy for tractability.
• Applies known results from prior literature (e.g., [13], [14], [2]) to relate EC-$(s,t)$-WT in average case to worst case, especially through the equivalence between ABS and NOR settings under certain conditions.
• Derives necessary and sufficient conditions via asymptotic analysis of $\ln n(\varepsilon, S_d)$ as $\varepsilon^{-1} + d \to \infty$, focusing on the growth rates of $\ln n(\varepsilon, S_d)$ relative to $(\ln \varepsilon^{-1})^s + d^t$.
• Employs Young’s inequality and logarithmic bounds to control the growth of $s_d$ and $\ln n(\varepsilon, S_d)$, particularly in the case $s>1$, $t<1$.
• Uses the identity $\frac{\ln n^{\rm avg,ABS}(\varepsilon,d)}{(\ln \varepsilon^{-1})^s + d} = \frac{[\ln(e^{\rm avg}(0,d)\varepsilon^{-1})]^s + d}{(\ln \varepsilon^{-1})^s + d} \cdot \frac{\ln n^{\rm avg,NOR}((e^{\rm avg}(0,d))^{-1}\varepsilon,d)}{[\ln(e^{\rm avg}(0,d)\varepsilon^{-1})]^s + d}$ to relate average case complexity across settings.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on the sequence $\{a_k\}$ for EC-$(s,t)$-weak tractability to hold in the worst case setting when $0 < \min(s,t) < 1$ and $\max(s,t) \leq 1$?
• RQ2 Under what conditions does EC-$(s,t)$-weak tractability hold in the average case setting for $s,t > 0$?
• RQ3 How do the conditions for EC-$(s,t)$-WT differ between the worst case and average case settings, especially when $s<1$, $t<1$, or $s>1$, $t<1$?
• RQ4 What role does the divergence of $\{a_k\}$, i.e., $\lim_{k\to\infty} a_k = \infty$, play in ensuring EC-$(s,t)$-WT for $s \geq 1$, $t=1$?
• RQ5 Can EC-$(s,t)$-WT with $s>1$, $t<1$ be characterized solely by the decay of $a_k$, and if so, what is the precise condition?

Key findings

• For the worst case setting, EC-$(s,t)$-WT holds if and only if $\lim_{k\to\infty} a_k = \infty$ when $s \geq 1$ and $t=1$, and for $s<1$, $t\leq1$, it holds if and only if $\sum_{k=1}^\infty \omega^{a_k} < \infty$.
• In the average case setting, EC-$(s,t)$-WT holds for $0<s\leq1$, $t>1$ unconditionally, and for $s>1$, $t<1$, it holds if and only if $\sum_{k=1}^\infty \omega^{a_k} < \infty$.
• For $s>1$, $t<1$, EC-$(s,t)$-WT holds in the average case setting if and only if $\sum_{k=1}^\infty \omega^{a_k} < \infty$, which is also the necessary and sufficient condition for EC-$(s,t)$-WT with $s<1$, $t\leq1$.
• When $s=1$, $t<1$, EC-$(s,t)$-WT holds in the average case setting if and only if $\sum_{k=1}^\infty \omega^{a_k} < \infty$, which is equivalent to the condition $\lim_{k\to\infty} a_k = \infty$.
• The equivalence between ABS and NOR settings in average case tractability is established via the identity relating $n^{\rm avg,ABS}$ and $n^{\rm avg,NOR}$, allowing transfer of results between normalized and absolute error criteria.
• For $s>1$, $t<1$, the proof shows $\lim_{\varepsilon^{-1}+d\to\infty} \frac{\ln n^{\rm avg,ABS}(\varepsilon,d)}{(\ln \varepsilon^{-1})^s + d^t} = 0$ under $\sum_{k=1}^\infty \omega^{a_k} < \infty$, confirming the condition is sufficient and necessary.