Average case tractability of multivariate approximation with Gevrey type kernels

TL;DR

This work analyzes multivariate approximation in the average-case setting with a zero-mean Gaussian measure whose covariance kernel is the periodic Gevrey kernel $K_{d,\alpha,\beta,p}$. By characterizing the covariance operator $C_{\nu_d}$ via eigenvalues $\lambda_{d,k}$ that mirror $\exp(-2\beta|\mathbf{k}|_{p}^{\alpha})$, the authors derive the average-case error $e^{avg}(n,d)=(\sum_{k>n} \lambda_{d,k})^{1/2}$ and establish sharp necessary and sufficient conditions for ALG- and EXP-tractability. The main contributions include precise criteria for ALG-$(s,t)$-WT and EXP-$(s,t)$-WT, the identification of the curse of dimensionality via $\alpha\le p$, and the finding that in the average case the EXP-exponent is $p^{*,avg}(d)=\alpha/d$ with no UEXP, alongside detailed comparisons to worst-case results. These results illuminate how Gevrey smoothness and kernel decay govern high-dimensional tractability and inform algorithm design for large-scale approximation problems on $\mathbb{T}^d$.

Abstract

We consider multivariate approximation problems in the average case setting with a zero mean Gaussian measure whose covariance kernel is a periodic Gevrey kernel. We investigate various notions of algebraic tractability and exponential tractability, and obtain necessary and sufficient conditions in terms of the parameters of the problem.

Theorems & Definitions (9)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof