This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness 1 . Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of O ( n − 1 / 4 ( log n ) 1 / 2 ) , with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, in contrast to the result of Kazashi and Sloan.
Shifted rank-1 lattice rules, a special class of quasi-Monte Carlo methods, have recently been proposed by the present authors for the integration of functions belonging to certain "weighted" Sobolev spaces. The shifts in these rules were generated in a deterministic manner. In contrast, in this paper we generate these shifts randomly. This allows probabilistic estimates for the error in a given integral. It also reduces the number of operations required to find the generating vectors for the underlying lattice rules component-by-component. The rules thus constructed achieve a worst-case strong tractability error bound in an average or probabilistic sense.
We develop and justify an algorithm for the construction of quasi–Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found “component-by-component”: the ( d + 1 d+1 )-th component of the generator vector and the shift are obtained by successive 1 1 -dimensional searches, with the previous d d components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for n n -point rules with n n prime and all dimensions 1 to d d requires a total cost of O ( n 3 d 2 ) O(n^3d^2) operations. This may be reduced to O ( n 3 d ) O(n^3d) operations at the expense of O ( n 2 ) O(n^2) storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and n n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by
n−1/2 times the
L2-norm of the Fourier transform outside a region around the origin, where
n is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces
Hps with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is
n−s−1/2 for
p≥2, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than
n−1/2 for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.
In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases
p=(p1,...,ps) for which we study the root mean square (RMS) worst-case error. The randomization method is a
p-adic shift which is based on
p-adic arithmetic. The obtained error bounds are optimal in the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension
s. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS
L2-discrepancy. Except for the
p-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.
In the present paper we study quasi-Monte Carlo rules for approximating integrals over the
d-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules. Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number
n of used integration nodes. For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.
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 the 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
Fb with
α consecutive sizes of nodes,
bm−α+1,…,bm, 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)NlogN, which improves the currently known result for the interlaced polynomial lattice rule, which is of order
sαNlogN. 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.