Generalized tractability for multivariate problems Part I. Linear tensor product problems and linear information
Many papers study polynomial tractability for multivariate problems. Let n(epsilon, d) be the minimal number of information evaluations needed to reduce the initial error by a factor of epsilon fora 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(epsilon, d) is bounded by a polynomial in epsilon(-1) and d and this holds for all (epsilon(-1), d) is an element of [1, infinity) x N. In this paper we study generalized tractability by verifying when n(epsilon, d) can be bounded by a power of T(epsilon(-1), d) for all (epsilon(-1), d) is an element of Omega, where Omega can be a proper subset of [1, infinity) 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 Omega=[1, infinity) x {1, 2,..., d*} boolean OR [1, epsilon(-1)(0)) x N for some d* >= 1 and epsilon(0) is an element of (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. (c) 2006 Elsevier Inc. All rights reserved.
tractability multivariate problem tensor product problem linear information information-based complexity