Complexity of approximation with relative error criterion in worst, average, and probabilistic settings

Abstract

The complexity of approximating a continuous linear functional defined on a separable Banach space equipped with a Gaussian measure is studied. The quality of the approximation is measured by a relative error criterion. The complexity is studied in the worst case, average case, and probabilistic settings. In the worst and average case settings, the complexity is infinite. In the probabilistic setting, the complexity is finite under a mild assumption. Tight lower and upper complexity bounds are established and an almost optimal algorithm is constructed. We briefly indicate how some of the results generalize for linear operators. In particular, in the worst case setting the complexity remains infinite, whereas in the average case setting the complexity becomes finite if the dimension of the range of a linear operator is at least two.