NUMA is a research section within the Department of Computer Science of KU Leuven, with 12 permanent staff members and approximately 40 PhD and postdoctoral researchers.
NUMA works on the crossing of numerical analysis and computational mathematics and develops numerical algorithms and software for large-scale problems in science and engineering.
This position is on a project in cooperation with the research group of Frances Kuo of the School of Mathematics and Statistics of the University of New South Wales, Sydney, Australia.
This position is part of an FWO project in cooperation with UNSW Sydney which develops new algorithms for approximating high dimensional integrals.
Very high dimensional integrals are one of the most challenging problems in computational mathematics and in particular in the theoretical development of Uncertainty Quantification techniques (a very hot topic these days).
To approximate high dimensional integrals we make use of sampling point sets based on lattices. The simplest form are "lattice rules" and they have been extensively studied, mostly in the worst case setting, by the project supervisors (both in Leuven and in Sydney).
This project extends the analysis of lattice based methods to higher order convergence in non-periodic spaces aiming for optimal randomized error bounds.
Very recently the supervisors were able to achieve the optimal order for the root mean square error using a randomized algorithm based on lattice rules.
The new algorithm was a very surprising and exciting result. Previously, methods for usage with digital nets (a different type of sampling points) were known, but a method for lattice rules kept being elusive for a very long time.
There are two PhD positions on this project. This job posting is concerned with developing and analyzing such randomized algorithms for non-periodic integrands in two particular non-periodic function spaces of smoothness 1 and smoothness 2.
The two PhD students will also work together on a more general framework for the randomization of lattice based point sets.