On tractability of weighted integration over bounded and unbounded regions in

We prove that for the space of functions with mixed first derivatives bounded in norm, the weighted integration problem over bounded or unbounded regions is equivalent to the corresponding classical integration problem over the unit cube, provided that the integration domain and weight have product forms. This correspondence yields tractability of the general weighted integration problem.

     

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

Strong tractability of integration using scrambled Niederreiter points

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The -exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

     

TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted l-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series.The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic contin- uous functions spaces.Tractability is the minimal number of function samples required to solve the problem in polynomial in ε~(-1)and d.and the strong tractability is the pres- ence of only a polynomial dependence in ε.This problem has been recently studied for quasi-Monte Carlo quadrature rules.quadrature rules with non-negative coefficients. and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables.The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14]on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative.The arguments are not constructive.     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

The power of standard information for multivariate approximation in the randomized setting

We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted -norm. We consider functions with an arbitrarily large number of variables, , and we focus on the randomized setting with algorithms using standard information consisting of function values at randomly chosen points.

We prove that standard information in the randomized setting is as powerful as linear information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the th minimal errors, i.e., of the minimal errors among all algorithms using function evaluations. Previously, it was only known that standard information in the randomized setting is no more powerful than the linear information in the worst case setting.

We also study (strong) tractability of multivariate approximation in the randomized setting. That is, we study when the minimal number of function evaluations needed to reduce the initial error by a factor is polynomial in  (strong tractability), and polynomial in and (tractability). We prove that these notions in the randomized setting for standard information are equivalent to the same notions in the worst case setting for linear information. This result is useful since for a number of important applications only standard information can be used and verifying (strong) tractability for standard information is in general difficult, whereas (strong) tractability in the worst case setting for linear information is known for many spaces and is relatively easy to check.

We illustrate the tractability results for weighted Korobov spaces. In particular, we present necessary and sufficient conditions for strong tractability and tractability. For product weights independent of , we prove that strong tractability is equivalent to tractability.

We stress that all proofs are constructive. That is, we provide randomized algorithms that enjoy the maximal speed of convergence. We also exhibit randomized algorithms which achieve strong tractability and tractability error bounds.

     

Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules

In this paper we prove the existence of digitally shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be the best possible up to a given degree of smoothness of the integrand. Indeed we even show the existence of polynomial lattice rules which automatically adjust themselves to the smoothness of the integrand up to a certain given degree.Further we show that strong tractability under certain conditions on the weights can be obtained and that polynomial lattice rules exist for which the worst-case error can be bounded independently of the dimension. These results hold independent of the smoothness.     

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an $\epsilon$-approximation for the $d$-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on $d$ and logarithmically on ${\epsilon }^{-1}$. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on $d$ and logarithmic dependence on ${\epsilon }^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $max(s,t)\ge 1$. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential $(s,t)$-weak tractability with $max(s,t)<1$ is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).     

Tractability properties of the weighted star discrepancy

Tractability properties of various notions of discrepancy have been intensively studied in the last decade. In this paper we consider the so-called weighted star discrepancy which was introduced by Sloan and Wo?niakowski. We show that under a very mild condition on the weights one can obtain tractability with s$s$-exponent zero (s$s$ is the dimension of the point set). In the case of product weights we give a condition such that the weighted star discrepancy is even strongly tractable. Furthermore, we give a lower bound for the weighted star discrepancy for a large class of weights. This bound shows that for such weights one cannot obtain strong tractability.     

Simple characterizations of exponential tractability for linear multivariate problems

We study linear multivariate problems defined as the approximation of compact linear multivariate operators over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. In particular, most of these conditions do not require the ordering of these singular values, which in many cases is difficult to achieve.     

On scrambled Halton sequences

Halton's low discrepancy sequence is still very popular in spite of its shortcomings with respect to the correlation between points of two-dimensional projections for large dimensions. As a remedy, several types of scrambling and/or randomization for this sequence have been proposed. We examine empirically some of these by calculating their L_∞- and L₂-discrepancies (D^* resp. T^*), and by performing integration tests.Most investigated sequence types give practically equivalent results for D^*, T^*, and the integration error, with two exceptions: random shift sequences are in some cases less efficient, and the shuffled Halton sequence is no more efficient than a pseudo-random one. However, the correlation mentioned above can only be broken with digit-scrambling methods, even though the average correlation of many randomized sequences tends to zero.     

Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

We consider the approximation of -dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals is . Since this is a worst case result, compared to the expected convergence rate of Monte Carlo, it shows the superiority of quasi-Monte Carlo for this type of integral. This is much faster than the worst case convergence, , of quasi-Monte Carlo.

     

Testing multivariate uniformity and its applications

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in , we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, , or the chi-squared distribution, . A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

     

Strong tractability of multivariate integration of arbitrary order

We show the existence of digitally-shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be best possible up to a given degree of smoothness of the integrand. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

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 ()-th component of the generator vector and the shift are obtained by successive -dimensional searches, with the previous 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 -point rules with prime and all dimensions 1 to requires a total cost of operations. This may be reduced to operations at the expense of storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.

     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L^p discrepancy of points in the d-dimensional unit cube for a sequence γ=(γ₁,γ₂,…) of weights. In this paper we prove a nice formula for the weighted L^p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L^p discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1)^d for which the weighted L^p discrepancy is uniformly bounded in d and goes to zero polynomially in N⁻¹.Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces.