Randomized Approximation of Sobolev Embeddings

Summary

We study approximation of functions belonging to Sobolev spaces W ^r _p (Q) by randomized algorithms based on function values. Here 1 ≤ p ≤ ∞, Q = [0, 1] _d , and r, d ∈ N. The error is measured in L _q (Q), with 1 ≤ q < ∞, and we assume r/d > 1/p − 1/q, guaranteeing that W ^r _p (Q) is embedded into L _q (Q). The optimal order of convergence for the case that W ^r _p (Q) is embedded even into C(Q) is wellknown. It is ^{n−r/d+max(1/p−1/q,0)} (n the number of function evaluations). This rate is already reached by deterministic algorithms, and randomization gives no speedup.

In this paper we are concerned with the case that W ^r _p (Q) is not embedded into C(Q) (but, of course, still into L _q (Q)). For this situation approximation based on function values was not studied before. We prove that for randomized algorithms the above rate also holds, while for deterministic algorithms no rate whatsoever is possible. Thus, in the case of low smoothness, Monte Carlo approximation algorithms reach a considerable speedup over deterministic ones (up to n ^−1+ε for any ε > 0).

We also give some applications to integration of functions and to approximation of solutions of elliptic PDE.