Construction algorithms for higher order polynomial lattice rules

Abstract

a b s t r a c tHigher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of ''good'' quality has recently been established, but their construction was not addressed.We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time -under certain conditions on the weights -(strong) polynomial tractability. Combining this approach with a sievetype algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.

Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration

“…This is known to be optimal up to log terms (see [21]). In [1,2] there is also a component-by-component (CBC) algorithm to obtain point sets which achieve the same order.…”

“…Notice that w q p is a function but wd q p is an operator. We can rewrite the Walsh function as follows: wal k = w (N1,...,Ns) (1,..., 1) . We see that w j,m and ∂ i,n commute in the next lemma.…”

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

“…Let · 1,I and · 2,I as well as · 1,II and · 2,II be two pairs of seminorms on H, both satisfying (A2) and (A3). In the following we will compare the resulting norms according to (4), and we denote the corresponding reproducing kernels according to Lemma 2.1 by k γ,I and k γ,II , respectively. Proof.…”

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

“…It has been established that this "greedy" algorithm yields lattice rules which achieve the optimal rate of convergence close to order n −1 in the underlying weighted function spaces, with the implied constant independent of s, see [65,19]. This was followed by many further works, most notably the use of fast Fourier transform (FFT) to speed up the computation [85,86], the construction of "extensible lattice sequences" [17,32,57], the use of "tent transform" to achieve close to order n −2 convergence [56,29], the carrying over of the lattice technology to digital nets and sequences [30,26,87], and the revolutionary invention of "higher order digital nets" which allow a convergence rate of order n −α , α > 1, for sufficiently smooth integrands [20,21,22,3,4,5,45]. For surveys of these recent QMC developments see [31,67,27,84,97].…”