Richardson Extrapolation of Polynomial Lattice Rules

Abstract

We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness α ≥ 2 defined over the s-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over F b with α consecutive sizes of nodes, b m−α+1 , . . . , b m , and ii) recursive application of Richardson extrapolation to a chain of α approximate values of the integral obtained by consecutive polynomial lattice rules.We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order (s + α)N log N , which improves the currently known result for interlaced polynomial lattice rule, that is of order sαN log N . We also study the dependence of the worst-case error bound on the dimension.A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known.Numerical experiments for test integrands support our theoretical result.

Tribological Behavoiur of Silicon Carbide/ Rice Husk ash Reinforced Al7075 Hybrid Composites Using Optimization Techniques

“…The standard errors are equal and are at same distance ρ from the center of the region for all the points. ie., [35] x1u 2…”

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.…”

“…Recently, a new alternative approach to interlaced polynomial lattice rules has been developed by Dick et al (2019). Instead of searching for a single interlaced polynomial lattice point set, their approach is to search for α classical polynomial lattice point sets with geometric spacing of N first, and then to apply Richardson extrapolation recursively to α numerical values A N (f ).…”

“…In this paper, as a continuation of Dick et al (2019), we push forward the idea of applying Richardson extrapolation to QMC rules for achieving a high order of convergence for multivariate numerical integration. In particular, we consider QMC rules using truncated higher order digital nets or se-quences as quadrature nodes, where truncation is done in the following way: we apply the following map tr m : [0, 1) → [0, 1) component-wise to each node x h = (x h,1 .…”

Lattice rules in non-periodic subspaces of Sobolev spaces

“…for the non-negative parameters c 1 , c 2 and ω j . These smooth integrands were originally used in [6]. Note that the exact values of the integrals for f 1 and for f 2 with the special cases c 2 = 1 and c 2 = 2 are known: I(f 1 ) = 1 and…”