Optimal Monte Carlo Methods for $L^2$-Approximation
arXiv:1705.0456721 citationsh-index: 14
AI Analysis
This provides a theoretical optimality result for Monte Carlo methods in function approximation, relevant to numerical analysis and computational mathematics.
The authors develop Monte Carlo methods for L2-approximation in Hilbert spaces that achieve the same convergence rate and preasymptotic behavior as any algorithm using n pieces of linear information, including function values.
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.