Randomized Approximation of Sobolev Embeddings

Abstract

Summary. We study approximation of functions belonging to Sobolev spaces W The optimal order of convergence for the case that W r p (Q) is embedded even into C(Q) is well-known. 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.

Randomized approximation of Sobolev embeddings, III

“…Now we combine the proof of Proposition 2 of [10] with that of Theorem 4.2 above. Let η and I η be as defined there; see (117).…”

“…In this paper we study the randomized approximation of Sobolev embeddings of W r p (Q ) into W s q (Q ), continuing the investigations from [10], where the case of s = 0 and Q being a cube was considered, and from [11], concerned with the case s ≥ 0, Q a bounded Lipschitz domain. Now we deal with the case s < 0, again in general Lipschitz domains Q .…”

Randomized approximation of Sobolev embeddings, II

“…Here we extend the analysis of [4] to the case of Sobolev spaces of non-negative smoothness order as target spaces, and to bounded Lipschitz domains. The paper is a continuation of part I [4] (target space L q (Q )), and is followed by part III [5], where the case of a target space with negative smoothness order is studied.…”

“…In this case the rate for randomized approximation is the same as that for the deterministic setting. Recently the case of non-embedding was studied in [4], where it was observed that randomization can bring about a significant speedup over deterministic algorithms. In all these papers the target space was L q (Q ) and the domain Q was a cube.…”

“…The paper is a continuation of part I [4] (target space L q (Q )), and is followed by part III [5], where the case of a target space with negative smoothness order is studied.…”