Heping Wang

Jia Chen, Heping Wang

We study the problem of approximating functions of $d$ variables in the average case setting for the $L_2$ space $L_{2,d}$ with the standard Gaussian weight equipped with a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure takes the form of a Gaussian kernel with non-increasing positive shape parameters $γ_j^2$ for $j = 1, 2, \dots, d$. The error of approximation is defined in the norm of $L_{2,d}$. We study the average case error of algorithms that use at most $n$ arbitrary continuous linear functionals. The information complexity $n(\varepsilon, d)$ is defined as the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most $\varepsilon$. We study different notions of tractability or exponentially-convergent tractability (EC-tractability) which the information complexity $n(\varepsilon, d)$ describe how behaves as a function of $d$ and $\varepsilon^{-1}$ or as one of $d$ and $(1+\ln\varepsilon^{-1})$. We find necessary and sufficient conditions on various notions of tractability and EC-tractability in terms of shape parameters. In particular, for any positive $s>0$ and $t\in(0,1)$ we obtain that the sufficient and necessary condition on $γ^2_ j$ for which $$\lim_{d+\varepsilon^{-1}\to\infty}\frac{n(\varepsilon,d)}{\varepsilon^{-s}+d^t}=0$$ holds is $$ \lim_{j\to \infty}j^{1-t}γ_j^2\,\ln^+ γ_j^{-2}=0,$$where $\ln^+ x=\max(1,\ln x)$.

Heping Wang

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 $0<\min(s,t)<1$ and $\max(s,t)\le 1$ in the worst case setting and for $s,t>0$ in the average case setting.

Jia Chen, Heping Wang, Jie Zhang

We study $d$-variate problem in the average case setting with respect to a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure is a product of univariate kernels and satisfies some special properties. We study $(s, t)$-weak tractability of this multivariate problem, and obtain a necessary and sufficient condition for $s>0$ and $t\in(0,1)$. Our result can apply to the problems with covariance kernels corresponding to Euler and Wiener integrated processes, Korobov kernels, and analytic Korobov kernels.

Zirong Liu, Heping Wang, Kai Wang

This paper is devoted to discussing the weighted linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential $(s, t)$-weak tractability (EXP-$(s, t)$-WT) with $\max(s,t)<1$ and exponential uniform weak tractability (EXP-UWT) under the absolute or normalized error criterion. We solve the problem by filling the remaining gaps left open on EXP-tractability. That is, we obtain necessary and sufficient conditions for EXP-$(s, t)$-WT with $\max(s, t) < 1$ and for EXP-UWT.

JIa Chen, Heping Wang

In this paper, we obtain the preasymptotic and asymptotic behavior and strong equivalences of the approximation numbers of the embeddings from the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces $W_2^{\infty}(\Bbb T^d)$ of the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.

Feng Dai, Heping Wang

Let $Π_n^d$ denote the space of all spherical polynomials of degree at most $n$ on the unit sphere $\sph$ of $\mathbb{R}^{d+1}$, and let $d(x, y)$ denote the usual geodesic distance $\arccos x\cdot y$ between $x, y\in \sph$. Given a spherical cap $$ B(e,\al)=\{x\in\sph: d(x, e) \leq \al\}, (e\in\sph, \text{$\al\in (0,π)$ is bounded away from $π$}),$$ we define the metric $$ρ(x,y):=\frac 1{\al} \sqrt{(d(x, y))^2+\al(\sqrt{\al-d(x, e)}-\sqrt{\al-d(y,e)})^2}, $$ where $x, y\in B(e,\al)$. It is shown that given any $\be\ge 1$, $1\leq p<\infty$ and any finite subset $\Ld$ of $B(e,\al)$ satisfying the condition $\dmin_{\sub{ξ,η\in \Ld ξ\neq η}} ρ(ξ,η) \ge \f \da n$ with $\da\in (0,1]$, there exists a positive constant $C$, independent of $\al$, $n$, $\Ld$ and $\da$, such that, for any $f\inΠ_{n}^d$, \begin{equation*} \sum_{\og\in \Ld} (\max_{x,y\in B_ρ(\og, \be\da/n)}|f(x)-f(y)|^p) |B_ρ(\og, \da/n)| \le (C \dz)^p \int_{B(e,\al)} |f(x)|^p d\sa(x),\end{equation*} where $d\sa(x)$ denotes the usual Lebesgue measure on $\sph$, $$B_ρ(x, r)=\Bl\{y\in B(e,\al): ρ(y,x)\leq r\Br\}, (r>0),$$ and $$\Bl|B_ρ(x, \f\da n)\Br|=\int_{B_ρ(x, \da/n)} d\sa(y) \sim \al ^{d}\Bl[ (\f{\da}n)^{d+1}+ (\f\da n)^{d} \sqrt{1-\f{d(x, e)}\al}\Br].$$ As a consequence, we establish positive cubature formulas and Marcinkiewicz-Zygmund inequalities on the spherical cap $B(e,\al)$.