15:45
The
notion quantization originates from information theory, where it refers to the
approximation of a continuous signal on a discrete set. Our research on
quantization is mainly motivated by applications in quadrature problems. In
that context, one aims at finding for a given probability measure $\mu$ on a
metric space a discrete approximation that is supported on a finite number of
points, say $N$, and is close to $\mu$ in a Wasserstein metric.
In general it is a hard problem to find close to optimal quantizations, if
$N$ is large and/or $\mu$ is given implicitly, e.g. being the marginal
distribution of a stochastic differential equation. In this talk we analyse the
efficiency of empirical measures in the constructive quantization problem. That
means the random approximating measure is the uniform distribution on $N$
independent $\mu$-distributed elements.
We show that this approach is order order optimal in many cases. Further, we
give fine asymptotic estimates for the quantization error that involve moments
of the density of the absolutely continuous part of $\mu$, so called high
resolution formulas. The talk ends with an outlook on possible applications and
open problems.
The
talk is based on joint work with Michael Scheutzow (TU Berlin) and Reik
Schottstedt (U Marburg).