Tractability and Strong Tractability of Linear Multivariate Problems

Linear multivariate problems are defined as the approximation of linear operators on functions of d variables. We study the complexity of computing an ϵ-approximation in different settings. We are particularly interested in large d and/or large ϵ−1. Tractability means that the complexity is bounded by c(d) K(d, ϵ), where c(d) is the cost of one information operation and K(d, ϵ) is a polynomial in d and/or in ϵ−1. Strong tractability means that K(d, ϵ) is a polynomial in ϵ−1, independent of d. We provide necessary and sufficient conditions for linear multivariate problems to be tractable or strongly tractable in the worst case, average case, randomized, and probabilistic settings. This is done for the class Λall where an information operation is defined as the computation of any continuous linear functional. We also consider the class Λstd where an information operation is defined as the computation of a function value. We show under mild assumptions that tractability in the class Λall is equivalent to tractability in the class Λstd. The proof is, however, not constructive. Finally, we consider linear multivariate problems over reproducing kernel Hilbert spaces, showing that such problems are strongly tractable even in the worst case setting.