Recovery of Sobolev functions restricted to iid sampling

Abstract

We studyLqL_q-approximation and integration for functions from the Sobolev spaceWps(Ω)W^s_p(\Omega )and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use identically distributed (iid) sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption in learning and uncertainty quantification. The only exception is whenp=q=∞p=q=\infty, where a logarithmic loss cannot be avoided.