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MiS Preprint

Generalized Tractability for Multivariate Problems

Michael Gnewuch and Henryk Woźniakowski


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.


Related publications

2007 Repository Open Access
Michael Gnewuch and Henryk Wozniakowski

Generalized tractability for multivariate problems. Pt. 1 : Linear tensor product problems and linear information

In: Journal of complexity, 23 (2007) 2, pp. 262-295