33 research outputs found
The construction of good lattice rules and polynomial lattice rules
A comprehensive overview of lattice rules and polynomial lattice rules is
given for function spaces based on semi-norms. Good lattice rules and
polynomial lattice rules are defined as those obtaining worst-case errors
bounded by the optimal rate of convergence for the function space. The focus is
on algebraic rates of convergence for
and any , where is the decay of a series representation
of the integrand function. The dependence of the implied constant on the
dimension can be controlled by weights which determine the influence of the
different dimensions. Different types of weights are discussed. The
construction of good lattice rules, and polynomial lattice rules, can be done
using the same method for all ; but the case is special
from the construction point of view. For the
component-by-component construction and its fast algorithm for different
weighted function spaces is then discussed
04401 Abstracts Collection -- Algorithms and Complexity for Continuous
From 26.09.04 to 01.10.04, the Dagstuhl Seminar ``Algorithms and Complexity for Continuous Problems\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Quadrature Points via Heat Kernel Repulsion
We discuss the classical problem of how to pick weighted points on a
dimensional manifold so as to obtain a reasonable quadrature rule
This problem, naturally, has a long history; the purpose of our paper is to
propose selecting points and weights so as to minimize the energy functional
\sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) }
\rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d}, is the
geodesic distance and is the dimension of the manifold. This yields point
sets that are theoretically guaranteed, via spectral theoretic properties of
the Laplacian , to have good properties. One nice aspect is that the
energy functional is universal and independent of the underlying manifold; we
show several numerical examples