Generalized Tractability for Multivariate Problems
Michael Gnewuch and Henryk Woźniakowski
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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
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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.