Fast CBC construction of randomly shifted lattice rules achievingO(n−1+δ)convergence for unbounded integrands over

James A. Nichols

¹

,

Frances Y. Kuo

²

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

“…A suitable (but "non-standard") function space setting for the integral (4.1) has been studied in [21,28] (see also the earlier papers [16,22,37,38]), and it is known that in this case randomly shifted lattice rules can still be constructed to achieve the optimal rate of convergence close to O(n −1 ). The norm in this case is given by…”

“…The norm (4.3) is said to be "unanchored" because the inactive variables are integrated out, as opposed to being "anchored" at some fixed value, say, 0. The unanchored norm (4.3) was first considered in [28] (which allowed also a general interval of integration instead of R); an anchored norm was considered in [21].…”

“…Coordinate-dependent weight functions ψ j were first considered in [28], while [21] used the same weight function for all coordinates, i.e. ψ j = ψ for all j.…”

“…This is due to the presence of the inverse cumulative distribution function −1 s , which is unbounded near the boundary of the unit cube. We will therefore consider a function space setting which uses a sequence of weight functions ψ j to counteract the growth of the mixed first derivatives of F( y) = G(u s h (·, y)) as the components of y go to ±∞, and we will make use of the corresponding error analysis for randomly shifted lattice rules from [21,28].…”

“…Then, with appropriately chosen weight parameters γ u , the QMC convergence rate can be arbitrarily close to order n −1 , with a constant that is independent of the truncation dimension s. In our analysis, as in [19], we choose the weight parameters γ u so as to minimize a certain upper bound on the QMC quadrature error (i.e., the second term in (1.10)), and then show that the constant is indeed independent of the dimension s. Moreover, the resulting weight parameters γ u , again as in [19], are of a special form called POD weights (which stands for "product and order dependent weights"). This POD structure of the weight parameters γ u allows in turn for a fast algorithm to construct lattice rules that are tailored to our problem (for details see [28]). In the penultimate section we summarize the overall error.…”