# Optimal Monte Carlo Methods for $L^2$-Approximation

## TL;DR

This paper develops Monte Carlo algorithms for $L^2$-approximation in Hilbert spaces that achieve optimal convergence rates using limited function samples, matching the performance of any linear information-based method.

## Contribution

It introduces Monte Carlo methods that attain optimal error rates for $L^2$-approximation with minimal function evaluations, matching the best linear information algorithms.

## Key findings

- Monte Carlo methods achieve optimal convergence rates.
- Error behavior matches that of any linear information sampling algorithm.
- Methods are effective for multivariate functions in Hilbert spaces.

## Abstract

We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling $n$ pieces of arbitrary linear information, including function values.