Average case complexity of linear multivariate problems

Abstract:

We study the average case complexity of a linear multivariate problem (LMP) defined on functions of d variables. We consider two classes of information. The first ${\Lambda ^{std}}$ consists of function values and the second ${\Lambda ^{all}}$ of all continuous linear functionals. Tractability of LMP means that the average case complexity is $O({(1/\varepsilon )^p})$ with p independent of d. We prove that tractability of an LMP in ${\Lambda ^{std}}$ is equivalent to tractability in ${\Lambda ^{all}}$, although the proof is not constructive. We provide a simple condition to check tractability in ${\Lambda ^{all}}$. We also address the optimal design problem for an LMP by using a relation to the worst case setting. We find the order of the average case complexity and optimal sample points for multivariate function approximation. The theoretical results are illustrated for the folded Wiener sheet measure.