Simulation on Rank-1 Lattices

Rank-1 lattices are available in any dimension for any number of lattice points and because their generation is so efficient, they often are used in quasi-Monte Carlo methods. Applying the Fourier transform to functions sampled on rank-1 lattice points turns out to be simple and efficient if the number of lattice points is a power of two. Considering the Voronoi diagram of a rank-1 lattice as a partition of the simulation domain and its dual, the Delauney tessellation, as a mesh for display and interpolation, rank-1 lattices are an interesting alternative to tensor product lattices. Instead of classical criteria, we investigate lattices selected by maximized minimum distance, because then the Delauney tessellation becomes as equilateral as possible. Similar arguments apply for the selection of the wave vectors. We explore the use of rank-1 lattices for the examples of stochastic field synthesis and a simple fluid solver with periodic boundary conditions.