Session III.5 - Information-Based Complexity
Monday, June 19, 15:30 ~ 16:00
On quadratures with optimal weights for spaces with bounded mixed derivatives
Michael Griebel
INS, University of Bonn, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We discuss quadrature rules in $d$ dimensions with optimal weights for given Quasi Monte Carlo point sets. Here we focus on spaces $H^r_{mix}$ with bounded $r$-th mixed derivatives. It turns out that we obtain a convergence rate of the order $O(N^{-r}\log(N)^{\alpha(d,r)})$ for some $\alpha$ which depends on the dimension $d$ and the smoothness $r$. The main order term $N^{-r}$ is then substantially improved in contrast to conventional plain QMC, which achieves just a main order term of $N^{-1}$.
Moreover. we consider optimal weights for spaces $H^s_{mix}$ with bounded $s$-th mixed derivatives and use these weights in a quadrature rule for functions from $H^r_{mix}$. There are now two situations: $ s \gt r $ and $s \lt r$. We observe that we obtain an error of the order $O(N^{-\min(r,s)} \log(N)^{\beta(d,r,s)})$ for some $\beta$ which depends on the dimension $d$ and the smoothness $r$ and $s$.
This way, if we do not know a-priorily the smoothness of our integrand class, it makes sense to choose optimal weights for some large $r$. Then, we always gain uniformly the best main error rate possible.
Joint work with Uta Seidler (INS, University of Bonn, Germany).