Lattice Rules Research Articles

On the convergence rate of the component-by-component construction of good lattice rules

We prove error bounds on the worst-case error for integration in certain Korobov and Sobolev spaces using rank-1 lattice rules with generating vectors constructed by the component-by-component algorithm. For a prime number of points n a rate of convergence of the worst-case error for multivariate integration in Korobov spaces of O(n −α/2+δ) , where α>1 is a parameter of the Korobov space and δ is an arbitrary positive real number, has been shown by Kuo. First we improve the constant of this error bound. Further, we prove an error bound which shows that the rate of convergence is optimal up to a power of log n for prime n. These error bounds are then generalised to the case where the number of points is not a prime number. Numerical results comparing the worst-case errors and the error bounds are presented.

On Korobov Lattice Rules in Weighted Spaces

This paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the Korobov form, in which the generating vector for an n-point rule with n prime has the form (1,a,...,ad -1) (mod,n). With the parameter a chosen optimally, we establish new error bounds for Korobov lattice rules in weighted Korobov spaces. In particular, we prove that if the weights decay sufficiently fast, the optimal Korobov lattice rule has an error bound of order O(n^{-\alpha/2 +\delta})$ (for arbitrary $\delta \,{>}\, 0$), with the implied constant depending at worst polynomially on the dimension. Here $\alpha \,{>}\,1$ is the smoothness parameter of the weighted Korobov spaces. We generalize the construction to the case where n is a product of arbitrary distinct prime numbers, with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules. A corresponding result is deduced for weighted Sobolev spaces of nonperiodic functions, using randomly shifted optimal Korobov lattice rules. A comparison of the worst-case errors for Korobov lattice rules and the recent component-by-component constructions is presented. The investigation establishes the usefulness of (shifted) optimal Korobov lattice rules for integration, even in high dimensions, if the weights which characterize the weighted spaces are suitably chosen.

Finite-order weights imply tractability of multivariate integration

Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can be well approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order (where the order is the number of variables in a group), and ignore all groups of variables of higher order. In this paper we consider multivariate integration for the anchored and unanchored (non-periodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst-case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ε depends on s and ε −1. If there is no dependence on s, and only polynomial dependence on ε −1, we have strong tractability, whereas with polynomial dependence on both s and ε −1 we have tractability. We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for all bounded finite-order weights. In both cases, the dependence on ε −1 is quadratic. We can improve the dependence on ε −1 at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ε −1 with a polynomial dependence on s whose degree is proportional to the order of the weights. We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well-known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to error bounds with almost linear dependence on n −1 and polynomial dependence on d. We present explicit worst-case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration.

Five- and six-dimensional lattice rules generated by structured matrices

We describe the results of a computer-based search for five and six-dimensional lattice rules of specified trigonometric degree. In this search only lattice rules that can be generated by a circulant or skew-circulant generator matrix are considered, which makes this approach significantly faster than earlier approaches. The drawback is that we do not necessarily obtain optimal lattice rules. We also present some new families of lattice rules parametrized by the trigonometric degree.

Reducing the construction cost of the component-by-component construction of good lattice rules

The construction of randomly shifted rank- 1 1 lattice rules, where the number of points n n is a prime number, has recently been developed by Sloan, Kuo and Joe for integration of functions in weighted Sobolev spaces and was extended by Kuo and Joe and by Dick to composite numbers. To construct d d -dimensional rules, the shifts were generated randomly and the generating vectors were constructed component-by-component at a cost of O ( n 2 d 2 ) O(n^2d^2) operations. Here we consider the situation where n n is the product of two distinct prime numbers p p and q q . We still generate the shifts randomly but we modify the algorithm so that the cost of constructing the, now two, generating vectors component-by-component is only O ( n ( p + q ) d 2 ) O(n(p+q)d^2) operations. This reduction in cost allows, in practice, construction of rules with millions of points. The rules constructed again achieve a worst-case strong tractability error bound, with a rate of convergence O ( p − 1 + δ q − 1 / 2 ) O(p^{-1+\delta }q^{-1/2}) for δ > 0 \delta >0 .

The Existence of Good Extensible Polynomial Lattice Rules

Extensible (polynomial) lattice rules have been introduced recently and they are convenient tools for quasi-Monte Carlo integration. It is shown in this paper that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomial lattice rules such that for all these infinitely many moduli and all dimensions s the quantity R (s) and the star discrepancy are small. The case of Korobov-type polynomial lattice rules is also considered.

Four-dimensional lattice rules generated by skew-circulant matrices

We introduce the class of skew-circulant lattice rules. These are s-dimensional lattice rules that may be generated by the rows of an s × s skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for δ = d + 1 ≤ 47, we have constructed an infinite sequence of rules Q(4, δ) that has a limit rho index of 27/34 ≈ 0.79. This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.

The second moment of the pion light cone wave function

We present a preliminary result for second moment of the light cone wave function of the pion. This parameter is the subject of a discrepancy between theoretical predictions (coming from lattice and sum rules) and a recent experimental result (that remarkably agrees with purely perturbative predictions). In this work we exploit lattice hypercubic symmetries to remove power divergences and, moreover, implement a full 1-loop matching for all the contributing operators.

Mott-Schottky Analysis of Nanoporous Semiconductor Electrodes in Dielectric State Deposited on SnO2 ( F ) Conducting Substrates

This paper analyzes the dark capacitance of nanostructured electrodes in the dielectric state, with particular emphasis on electrodes deposited over a transparent conducting substrate of It is shown that at those potentials where the nanostructure is in the dielectric state, the capacitance is controlled by the contact The partial or total covering of the substrate by a dielectric medium causes a modification of the Mott-Schottky plot of the bare substrate. We provide a mapping of the various Mott-Schottky curves that will appear depending on the film characteristics. If the dielectric nanoparticles completely block part of the substrate surface, the slope of the Mott-Schottky plot increases (with the same apparent flatband potential) as an effect of area reduction. The covering of a significant fraction of the surface by a thin dielectric layer shifts the apparent flatband negatively. Measurements on several nanostructured electrodes show that the capacitance contribution of the semiconductor network in the dielectric state is very low, indicating that the field lines penetrate little into the network, not much further than the first particle. The different surface covering observed for rutile-anatase and pure anatase colloids is explained by lattice matching rules with the substrate. By comparing different electrodes, the Helmholtz capacitance at the interface was calculated and the apparent flatband potential was corrected for the effect of band unpinning. © 2003 The Electrochemical Society. All rights reserved.

Notes on lattice rules

An elementary introduction to lattices, integration lattices and lattice rules is followed by a description of the role of the dual lattice in assessing the trigonometric degree of a lattice rule. The connection with the classical lattice-packing problem is established: any s-dimensional cubature rule can be associated with an index ρ= δ s / s! N, where δ is the enhanced degree of the rule and N its abscissa count. For lattice rules, this is the packing factor of the associated dual lattice with respect to the unit s-dimensional octahedron. An individual cubature rule may be represented as a point on a plot of ρ against δ. Two of these plots are presented. They convey a clear idea of the relative cost-effectiveness of various individual rules and sequences of rules.

Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces

It is known from the analysis by Sloan and Woźniakowski that under appropriate conditions on the weights, the optimal rate of convergence for multivariate integration in weighted Korobov spaces is O( n − α/2+ δ ) where α>1 is some parameter of the spaces, δ is an arbitrary positive number, and the implied constant in the big O notation depends only on δ, and is independent on the number of variables. Similarly, the optimal rate for weighted Sobolev spaces is O( n −1+ δ ). However, their work did not show how to construct rules achieving these rates of convergence. The existing theory of the component-by-component constructions developed by Sloan, Kuo and Joe for the Sobolev case yields the rules achieving O( n −1/2) error bounds. Here we present theorems which show that those lattice rules constructed by the component-by-component algorithms in fact achieve the optimal rate of convergence under appropriate conditions on the weights.

Methods for enhancing numerical integration

Abstract We give a survey of common strategies for numerical integration (adaptive, Monte-Carlo, Quasi-Monte Carlo), and attempt to delineate their realm of applicability. The inherent accuracy and error bounds for basic integration methods are given via such measures as the degree of precision of cubature rules, the index of a family of lattice rules, and the discrepancy of uniformly distributed point sets. Strategies incorporating these basic methods often use paradigms to reduce the error by, e.g., increasing the number of points in the domain or decreasing the mesh size, locally or uniformly. For these processes the order of convergence of the strategy is determined by the asymptotic behavior of the error, and may be too slow in practice for the type of problem at hand. For certain problem classes we may be able to improve the effectiveness of the method or strategy by such techniques as transformations, absorbing a difficult part of the integrand into a weight function, suitable partitioning of the domain, transformations and extrapolation or convergence acceleration. Situations warranting the use of these techniques (possibly in an “automated” way) are described and illustrated by sample applications.

Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation

Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasi-Monte Carlo set of tools. A theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, are special cases of this method.

Component-By-Component Construction of Good Intermediate-Rank Lattice Rules

It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a number of times in the first few dimensions to yield an intermediate-rank lattice rule. We show that the generating vector (and in weighted Sobolev spaces, the shift also) of an intermediate-rank lattice rule can also be constructed component-by-component to achieve strong tractability error bounds. In certain circumstances, these bounds are better than the corresponding bounds for rank-1 lattice rules.

The Number of Lattice Rules of Specified Upper Class and Rank

The upper class of a lattice rule is a convenient entity for classification and other purposes. The rank of a lattice rule is a basic characteristic, also used for classification. By introducing a rank proportionality factor and obtaining certain recurrence relations, we show how many lattice rules of each rank exist in any prime upper class. The Sylow p-decomposition may be used to obtain corresponding results for any upper class.

Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points

We develop algorithms to construct rank-1 lattice rules in weighted Korobov spaces of periodic functions and shifted rank-1 lattice rules in weighted Sobolev spaces of non-periodic functions. Analyses are given which show that the rules so constructed achieve strong QMC tractability error bounds. Unlike earlier analyses, there is no assumption that n, the number of quadrature points, be a prime number. However, we do assume that there is an upper bound on the number of distinct prime factors of n. The generating vectors and shifts characterizing the rules are constructed ‘component-by-component,’ that is, the ( d+1)th components of the generating vectors and shifts are obtained using one-dimensional searches, with the previous d components kept unchanged.

Status of the QCD sum rule determinations of the light quark masses

We summarize the current status of QCD sum rule determinations of the light quark masses. We show that results obtained using the sum rule and lattice approaches are in good agreement, and that the sum rule errors have been significantly reduced in the last few years. We also comment on the status of pseudoscalar-sum-rule-based quark mass bounds which, in the past, were sometimes said to rule out the low quark mass values produced by the lattice, pointing out that bounds associated with sum rules displaying good OPE convergence are in fact all compatible with the present values obtained from lattice and explicit sum rule determinations.

DECAY CONSTANTS OF HEAVY–LIGHT MESONS IN THE RELATIVISTIC QUARK MODEL

The decay constants of pseudoscalar and vector heavy–light mesons are calculated in the framework of the relativistic quark model with the completely relativistic treatment of the light quark. It is argued that relativistic effects play a significant role. Good agreement of the model predictions with recent lattice and QCD sum rule calculations as well as available experimental data are found.

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 (d+1)-th component of the generator vector and the shift are obtained by successive 1-dimensional searches, with the previous 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-point rules with n prime and all dimensions 1 to d requires a total cost of O(n3d2) operations. This may be reduced to O(n3d) operations at the expense of O(n2) storage. Numerical values of parameters and worst-ease errors are given for dimensions up to 40 and n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.

Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces

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.