Fast CBC construction of randomly shifted lattice rules achieving O(n−1+δ)O(n−1+δ) convergence for unbounded integrands over RsRs in weighted spaces with POD weights

This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over RsRs arising from practical applications. For an integral of the form ∫Rsf(y)∏j=1sϕ(yj)dy with a univariate probability density ϕϕ, our general strategy is to first map the integral into the unit cube [0,1]s[0,1]s using the inverse of the cumulative distribution function of ϕϕ, and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube rarely falls within the standard QMC setting of Sobolev spaces of functions with mixed first derivatives. Therefore, a non-standard function space setting for integrands over RsRs, previously considered by Kuo, Sloan, Wasilkowski and Waterhouse (2010), is required for the analysis. Motivated by the needs of three applications, the present paper extends the theory of the aforementioned paper in several non-trivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored variant for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD (“product and order dependent”) weights.