Randomized lattice based algorithms for high-dimensional integration

Very high dimensional integrals are one of the most challenging problems in computational mathematics and in the theoretical development of uncertainty quantification techniques. To approximate high dimensional integrals we will make use of sampling point sets based on lattices. We will extend the analysis of lattice based methods to higher order convergence in non-periodic spaces aiming for optimal randomized error bounds. We will develop and analyze randomized algorithms for non-periodic integrands in two particular non-periodic function spaces of smoothness 1 and smoothness 2.