Preliminary list of abstracts

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Higher order scrambled digital nets for smooth integrands
Keywords: randomized quasi-Monte Carlo; Numerical integration; digital nets and sequences
Thu, 14:00--14:25
  • Dick, Josef (School of Mathematics and Statistics, The University of New South Wales, Australia)

In this talk we present a random sampling technique to approximate integrals $\int_{[0,1]^s} f(\boldsymbol{x}) \,\mathrm{d} \boldsymbol{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth. We introduce a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order $N^{-\alpha-1/2+\varepsilon}$ provided the integrand has square integrable partial mixed derivatives up to order $\alpha > 1$ in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness.