论文标题
线性加权张量产品问题的指数障碍性在最差的线性功能的最差案例设置中
Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
论文作者
论文摘要
我们研究了在某些加权张量产品希尔伯特空间上定义的紧凑型线性算子的近似值。信息复杂性定义为最小数量的任意线性函数，以获取$ \ varepsilon $ -Approximation所需的$ d $ variate问题。它是根据权重和单变量奇异值完全确定的。指数障碍性意味着信息复杂性受某个函数的界限，该功能在$ d $上取决于$ d $，并在$ \ varepsilon^{ - 1} $上进行对数。希克内尔（Hickernell），克里特策（Kritzer）和沃尼亚科夫斯基（Woitniakowski）最近研究了相应的非加权问题，对指数障碍性有许多负面结果。本文中研究的产品权重改变了情况。 Depending on the form of polynomial dependence on $d$ and logarithmic dependence on $\varepsilon^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $\max(s,t)\ge1$.对于所有这些指数障碍性的概念，我们在权重和单变量的奇异值上建立了必要和充分的条件，实际上确实可以实现相应的指数障碍概念。指数$（s，t）$ - $ \ max（s，t）<1 $的较弱的障碍性均留下来进行以后的研究。
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals which is needed to obtain an $\varepsilon$-approximation for the $d$-variate problem. It is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function which depends polynomially on $d$ and logarithmically on $\varepsilon^{-1}$. The corresponding un-weighted problem was studied recently by Hickernell, Kritzer and Woźniakowski with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on $d$ and logarithmic dependence on $\varepsilon^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $\max(s,t)\ge1$. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential $(s,t)$-weak tractability with $\max(s,t)<1$ is left for future study.
