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

Abstract: Many papers study polynomial tractability for multivariate problems. Let 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 -variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that is bounded by a polynomial in and and this holds for all . In this paper we study generalized tractability by verifying when can be bounded by a power of for all , where can be a proper subset of . Here 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 for some and . We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on 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. [Copyright &y& Elsevier]