Elsevier

Journal of Complexity

Volume 23, Issue 2, April 2007, Pages 262-295
Journal of Complexity

Generalized tractability for multivariate problems Part I: Linear tensor product problems and linear information

https://doi.org/10.1016/j.jco.2006.06.006Get rights and content
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Abstract

Many papers study polynomial tractability for multivariate problems. Let n(ɛ,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of ɛ for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(ɛ,d) is bounded by a polynomial in ɛ-1 and d and this holds for all (ɛ-1,d)[1,)×N. In this paper we study generalized tractability by verifying when n(ɛ,d) can be bounded by a power of T(ɛ-1,d) for all (ɛ-1,d)Ω, where Ω can be a proper subset of [1,)×N. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set Ω=[1,)×{1,2,,d*}[1,ɛ0-1)×N for some d*1 and ɛ0(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.

Keywords

Tractability
Multivariate problem
Tensor product problem
Information-based complexity
Linear information

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