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

M. Gnewuch,H. Wozniakowski

Published 2007 in Journal of Complexity

ABSTRACT

Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be 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. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. 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 @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(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.

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