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

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 , ɛ 0 - 1 ) × N for some d * ⩾ 1 and ɛ 0 ∈ ( 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.