The fast reduced QMC matrix–vector product

Abstract

We study the approximation of integrals of the form ∫Df(x⊤A)dμ(x), where the measure μ is of product form and A is a matrix, by quasi-Monte Carlo (QMC) rules N−1∑k=0N−1f(xk⊤A). We are interested in cases where the main computational cost in the approximation arises from calculating the products xk⊤A. We design QMC rules for which the computation of xk⊤A, k=0,1,…,N−1, can be done in a fast way, and for which the approximation error of the QMC rule is similar to the standard QMC error. We do not require that the matrix A has any particular structure.Problems of this form arise in some important applications in statistics and uncertainty quantification. For instance, this approach can be used when approximating the expected value of some function with a multivariate normal random variable with some given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients.The speed-up of the computation time of our approach is sometimes better and sometimes worse than the fast QMC matrix–vector product from [Josef Dick, Frances Y. Kuo, Quoc T. Le Gia, and Christoph Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015), no. 3, A1436–A1450]. As in that paper, our approach applies to lattice point sets and polynomial lattice point sets, but also applies to digital nets (we are currently not aware of any approach which allows one to apply the fast QMC matrix–vector paper from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets).The method in this paper does not make use of the fast Fourier transform, instead we use repeated values in the quadrature points to derive a significant reduction in the computation time. Such a situation naturally arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can additionally be used to also reduce the computation time of the underlying QMC rule. One advantage of the present approach is that it can be combined with random (digital) shifts, whereas this does not apply to the fast QMC matrix–vector product from the earlier paper of Dick, Kuo, Le Gia, and Schwab.