Chebyshev lattices for high-dimensional integration

Quasi-Monte Carlo (QMC) methods are widely used as a strategy of simulation and can overcome “the curse of dimensionality”, which refers to the extremely rapid rise in difficulty of integration problems as the number of variables increase. Chebyshev lattice rule is a new framework for QMC methods which arise from Chebyshev approximations of multivariate functions. Chebyshev lattice is based on a cosine mapping of a classical lattice, e.g. rank-one lattice rule, but good classical lattice doesn’t result in good Chebyshev lattice. Recently, a complete framework for approximating non-periodic functions in a multivariate Chebyshev basis using FFT-based algorithms on lattice based sampling points has been formulated by scholars. However, we are working on completing the theory of error analysis and pushing the limits of Chebyshev lattice rules.