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.
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 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.
Recently, quasi-Monte Carlo methods have been successfully used for approximating multiple integrals in hundreds of dimensions in mathematical finance, and were significantly more efficient than Monte Carlo methods. To understand the apparent success of quasi-Monte Carlo methods for multiple integration, one popular approach is to study worst-case error bounds in weighted function spaces in which the importance of the variables is moderated by some sequences of weights. Ideally, a family of quasi-Monte Carlo methods in some weighted function space should be strongly tractable. Strong tractability means that the minimal number of quadrature points n needed to reduce the initial error by a factor of ε is bounded by a polynomial in ε⁻¹ independently of the dimension d. Several recent publications show the existence of lattice rules that satisfy the strong tractability error bounds in weighted Korobov spaces of periodic integrands and weighted Sobolev spaces of non-periodic integrands. However, those results were non-constructive and thus give no clues as to how to actually construct these lattice rules. In this thesis, we focus on the construction of quasi-Monte Carlo methods that are strongly tractable. We develop and justify algorithms for the construction of lattice rules that achieve strong tractability error bounds in weighted Korobov and Sobolev spaces. The parameters characterizing these lattice rules are found ‘component-by-component’: the (d + 1)-th components are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The cost of these algorithms vary from O(nd²) to O(n³d²) operations. With currently available technology, they allow construction of rules easily with values of n up to several million and dimensions d up to several hundred.
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.
Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the cube but is otherwise independent of the product measure.