Computational Higher Order Quasi-Monte Carlo Integration

Abstract

The efficient construction of higher-order interlaced polynomial lattice rules introduced recently in [4] is considered. After briefly reviewing the principles of their construction by the "fast component-by-component" (CBC) algorithm due to [1,10] as well as recent theoretical results on their convergence rates, we indicate algorithmic details of their construction. Instances of such rules are applied to highdimensional 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) bounds on the Walsh coefficients of the integrand are found to improve the convergence behavior. The sharpness of theoretical bounds on memory usage and operation counts, with respect to the number of points N and dimension s of the integration domain is verified experimentally. The efficiency of the proposed algorithms for computation of the generating vectors is confirmed for the considered classes of functions in dimensions s = 10, ..., 1000.

Multilevel higher-order quasi-Monte Carlo Bayesian estimation

“…The parametric sparsity of the countably parametric model problem considered in the numerical experiments was moderate (p 0 = 1/2 in (5.12)); for classes of uncertain input data u with higher sparsity, i.e. smaller values of p 0 , the gains of the presently proposed, HoQMC-based algorithms over MLMC are predicted to be correspondingly higher, as a consequence of the present theoretical results, and supported by extensive numerical experiments in [12]. 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11…”

Richardson extrapolation allows truncation of higher-order digital nets and sequences

“…Let us move on to the high-dimensional setting. Following Dick et al (2019) and Gantner & Schwab (2016), we consider the following two test functions: When 1 < c 1 < 2, the second derivative of the function x → x c1 is not absolutely continuous but is in L q ([0, 1)) for any q < 1/(2 − c 1 ), which means that f 3 ∈ W s,2,q,r for q < 1/(2 − c 1 ). On the other hand, f 4 is analytic and belongs to W s,α,q,r for any α ≥ 2 and q ≥ 1.…”

“…On the other hand, f 4 is analytic and belongs to W s,α,q,r for any α ≥ 2 and q ≥ 1. As stated by Gantner & Schwab (2016), f 4 is designed to mimic the behavior of parametric solution families of partial differential equations. We employ this test function to see potential applicability to such problems.…”

“…Since then, on the one hand, further theoretical investigations on them have been made (see, e.g., Baldeaux et al 2011, Goda et al 2017, 2018, Hinrichs et al 2016). On the other hand, how to efficiently search for good quadrature node sets in a weighted function space setting as considered by Sloan & Woźniakowski (1998) has also attracted some interest (see, e.g., Baldeaux et al 2012, Gantner & Schwab 2016, Goda 2015, Goda & Dick 2015. In particular, so-called interlaced polynomial lattice rules originated by Goda & Dick (2015) and Goda (2015), which are based on the digit interlacing composition due to Dick (2007Dick ( , 2008, have been applied in the context of partial differential equations with random coefficients (see, e.g., Dick et al 2014, Kuo & Nuyens 2016.…”

Application of Quasi-Monte Carlo Methods to PDEs with Random Coefficients – An Overview and Tutorial

“…Essentially, due to the linearity of a with respect to each yj, if we differentiate once then we obtain ψj , and if we differentiate a second time with respect to any variable we get 0. (16) into (15) and separating out the m = 0 term, we obtain…”