Generalized Tractability for Multivariate Problems
Michael Gnewuch and Henryk Wozniakowski
Contact the author: Please use for correspondence this email.
Submission date: 09. Jan. 2006
published in: Journal of complexity, 23 (2007) 2, p. 262-295
DOI number (of the published article): 10.1016/j.jco.2006.06.006
with the following different title: Generalized tractability for multivariate problems. Pt. 1 : Linear tensor product problems and linear information
Download full preprint: PDF (375 kB), PS ziped (262 kB)
There are many papers studying polynomial tractability for multivariate problems. Polynomial tractability means that the minimal number of information evaluations needed to reduce the initial error by a factor of for a multivariate problem defined on a space of d-variate functions may be bounded by a polynomial in and d and this holds for all .
We propose to study generalized tractability by verifying when can be bounded by a power of for all , where can be a proper subset of . 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 for some and . 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.