Average case complexity of linear multivariate problems I. Theory

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We study the average case complexity of linear multivariate problems, that is, the approximation of continuous linear operators on functions of d variables. The function spaces are equipped with Gaussian measures. We consider two classes of information. The first class Λ std consists of function values, and the second class Λ all consists of all continuous linear functionals. Tractability of a linear multivariate problem means that the average case complexity of computing an ϵ-approximation is O ((1/ ε ) p ) with p independent of d . The smallest such p is called the exponent of the problem. Under mild assumptions, we prove that tractability in Λ all is equivalent to tractability in Λ std and that the difference of the exponents is at most 2. The proof of this result is not constructive. We provide a simple condition to check tractability in Λ all . We also address the issue of how to construct optimal (or nearly optimal) sample points for linear multivariate problems. We use relations between average case and worst case settings. These relations reduce the study of the average case to the worst case for a different class of functions. In this way we show how optimal sample points from the worst case setting can be used in the average case. In Part II we shall apply the theoretical results to obtain optimal or almost optimal sample points, optimal algorithms, and average case complexity functions for linear multivariate problems equipped with the folded Wiener sheet measure. Of particular interest will be the multivariate function approximation problem.