In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and O(k) complexity estimation is provided for a problem with k generators.

The optimal quantization theory is applied for approximating law-invariant comonotonic coherent risk measures. Simple Lp-norm estimates for the risk measures provide the rate of convergence of that approximation as the number of quantization points goes to infinity.

We elucidate the asymptotics of the Ls -quantization error induced by a sequence of Lr -optimal n-quantizers of aprobability distribution P on $\mathbb{R}^d$ when s > r. In particular we show that under natural assumptions, the optimal rate is preserved aslong as s < r+d (and for everys in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubatureformulae in numerical integration on $\mathbb{R}^d$ and on the Wiener space.