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

摘要

Many papers study polynomial tractability for multivariate problems. Let n(ε, d) be the minimal number of information evaluations needed to reduce the initial error by a factor of ε 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(ε, d) is bounded by a polynomial in ε~(-1) and d and this holds for all (ε~(-1), d) ∈ [1, ∞) x N. In this paper we study generalized tractability by verifying when n(ε, d) can be bounded by a power of T(ε~(-1), d) for all (ε~(-1), d) ∈ Ω, where Ω can be a proper subset of [1, ∞) x 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, ∞) x {1,2, ..., d~*} ∪ [1, ε_0~(-1)) x N for some d~* ≥ 1 and ε_0 ∈ (0, 1). We study linear tensor product problems for which we can compute arbitrary linear functional 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.