\documentstyle[11pt]{article}
\begin {document}
 \title{Multilevel Monte Carlo Methods}

 \author {S.\ Heinrich\\
Fachbereich Informatik\\
Universit\"at Kaiserslautern\\
D-67653 Kaiserslautern, Germany\\
e-mail: heinrich@informatik.uni-kl.de
}
\date{}
\maketitle
\begin{abstract}
The standard task of Monte Carlo methods is to estimate an unknown
scalar - usually an integral or a functional of the solution of an
integral equation. It is well-known and well-understood that for high
dimensional problems Monte Carlo methods can outperform deterministic ones.
Less clear is the situation, when a whole function instead of a scalar is
to be approximated. Two prominent examples are integrals depending
on a parameter and the approximation of the full solution function of an
integral equation. For both situations various methods were developed
which - as a rule - use a fixed grid and combine the standard estimators
for point values.

In this talk we survey some recent development which goes beyond that
approach. Within a multilevel setting such as multigrids or wavelet
expansions new variance reduction techniques are applied. They
are  based on approximation properties of these multilevel structures
and a balancing of deterministic and stochastic approximation on the
different levels. Substantial reduction of arithmetic cost is reached
this way. Moreover, a complexity theoretic analysis on model classes shows
these methods to be optimal and superior to any deterministic method.

\end{abstract}
\end{document}