Analysis and Construction of Optimal Lattice Based Cubature.

This project develops new theory and analyzes new algorithms which can be used to calculate expected values and other moments of quantities of interest obtained from complicated mathematical models by use of efficient high dimensional numerical integration methods, or "cubature" in short, based on "lattice point sets". Such integration problems with respect to very high dimensional and infinite dimensional measures are one of the most challenging problems in computational mathematics and in particular in the theoretical development of Uncertainty Quantification techniques.

This project will extend the analysis of lattice based methods to higher order convergence in non-periodic Sobolev spaces and aims at achieving optimal randomized error bounds. For such question of optimality it is essential to have proven theoretical upper and lower bounds on the error that match or have as small a gap as possible.