Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness

TL;DR

This paper analyzes the convergence rates of Monte Carlo methods for approximating functions in Sobolev spaces with mixed smoothness on the torus, revealing potential speedups over worst-case scenarios and resolving open questions.

Contribution

It determines the convergence rates for Monte Carlo approximation in specific Sobolev spaces, including cases previously unresolved by Fang and Duan.

Findings

Monte Carlo methods can achieve up to 50% faster convergence rates.

The paper resolves open cases in the approximation theory of Sobolev spaces.

Provides theoretical bounds for approximation errors in mixed smoothness spaces.

Abstract

We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness defined on the torus $\mathbb{T}^d$ into the space $L_{\infty}(\mathbb{T}^d)$ via methods that use arbitrary linear information. These cases are interesting because we can gain a speedup of up to $1/2$ in the main rate compared to the worst case approximation. In doing so we determine the rate for some cases that have been left open by Fang and Duan.