Lattice based integration algorithms: Kronecker sequences and rank-1 lattices

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness B̊p,θ, which also comprise the concept of Sobolev spaces of dominating mixed smoothness H̊ s p as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from TN (Z d) ∩ [0, 1)d, where TN is a real invertible generator matrix of size d. For such rules the worst-case error can be bounded in terms of the Zaremba index of the lattice XN = TN (Z d). We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to logN -factors for arbitrary smoothness s. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.