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MiS Preprint
2/2006
Generalized Tractability for Multivariate Problems
Michael Gnewuch and Henryk Woźniakowski
Abstract
There are many papers studying polynomial tractability for multivariate problems. Polynomial tractability means that the minimal number $n(\varepsilon,d)$ of information evaluations needed to reduce the initial error by a factor of $\varepsilon$ for a multivariate problem defined on a space of $d$-variate functions may be bounded by a polynomial in $\varepsilon^{-1}$ and $d$ and this holds for all $(\varepsilon^{-1},d) \in [1,\infty)\times \mathbb{N}$.
We propose to study {\it generalized} tractability by verifying when $n(\varepsilon,d)$ can be bounded by a power of $T(\varepsilon^{-1},d)$ for all $(\varepsilon^{-1},d)\in \Omega$, where $\Omega$ can be a proper subset of $[1,\infty) \times \mathbb{N}$. Here $T$ is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity.
In this paper we study the set $\Omega=[1,\infty) \times \{1,2,\dots,d^*\,\} \cup [1,\varepsilon_0^{-1})\times \mathbb{N}$ for some $d^* \geq 1$ and $\varepsilon_0\in(0,1)$. We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on $T$ such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does hold.