Computational Higher Order Quasi-Monte Carlo Integration

Abstract

The efficient construction of higher-order interlaced polynomial lattice rules introduced recently in [Dick et al. SIAM Journal of Numerical Analysis, 52(6):2676–2702, 2014] is considered and the computational performance of these higher-order QMC rules is investigated on a suite of parametric, high-dimensional test integrand functions. After reviewing the principles of their construction by the “fast component-by-component” (CBC) algorithm due to Nuyens and Cools as well as recent theoretical results on their convergence rates from [Dick, J., Kuo, F.Y., Le Gia, Q.T., Nuyens, D., Schwab, C.: Higher order QMC Petrov–Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52(6) (2014), pp. 2676–2702], we indicate algorithmic aspects and implementation details of their efficient construction. Instances of higher order QMC quadrature rules are applied to several high-dimensional test integrands which belong to weighted function spaces with weights of product and of SPOD type. Practical considerations that lead to improved quantitative convergence behavior for various classes of test integrands are reported. The use of (analytic or numerical) estimates on the Walsh coefficients of the integrand provide quantitative improvements of the convergence behavior. The sharpness of theoretical, asymptotic bounds on memory usage and operation counts, with respect to the number of QMC points N and to the dimension s of the integration domain is verified experimentally to hold starting with dimension as low as $s=10$ and with $N=128$. The efficiency of the proposed algorithms for computation of the generating vectors is investigated for the considered classes of functions in dimensions $s=10,...,1000$. A pruning procedure for components of the generating vector is proposed and computationally investigated. The use of pruning is shown to yield quantitative improvements in the QMC error, but also to not affect the asymptotic convergence rate, consistent with recent theoretical findings from [Dick, J., Kritzer, P.: On a projection-corrected component-by-component construction. Journal of Complexity (2015) DOI 10.1016/j.jco.2015.08.001].