Lattice Rules Research Articles

On the approximation of smooth functions using generalized digital nets

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α > 1 in each variable. The approximation error is studied in the worst case setting in the L 2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.

Quasi-Monte Carlo methods with applications in finance

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space

We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the “generalised weighted star discrepancy”. This version of the discrepancy corresponds to the classic L ∞ weighted star discrepancy via a mapping to the unit cube. The weights here are general weights rather than the product weights considered in earlier works on integrals over R d . Known methods based on an averaging argument are used to show the existence of these lattice rules, while the component-by-component technique is used to construct the generating vector of these shifted lattice rules. We prove that the bound on the weighted star discrepancy considered here is of order O ( n − 1 + δ ) for any δ > 0 and with the constant involved independent of the dimension. This convergence rate is better than the O ( n − 1 / 2 ) achieved so far for both Monte Carlo and quasi-Monte Carlo methods.

Automated generation of lattice QCD Feynman rules

The derivation of the Feynman rules for lattice perturbation theory from actions and operators is complicated, especially for highly improved actions such as HISQ. This task is, however, both important and particularly suitable for automation. We describe a suite of software to generate and evaluate Feynman rules for a wide range of lattice field theories with gluons and (relativistic and/or heavy) quarks. Our programs are capable of dealing with actions as complicated as (m)NRQCD and HISQ. Automated differentiation methods are used to calculate also the derivatives of Feynman diagrams. Program summary Program title: HiPPY, HPsrc Catalogue identifier: AEDX_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEDX_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPLv2 (see Additional comments below) No. of lines in distributed program, including test data, etc.: 513 426 No. of bytes in distributed program, including test data, etc.: 4 893 707 Distribution format: tar.gz Programming language: Python, Fortran95 Computer: HiPPy: Single-processor workstations. HPsrc: Single-processor workstations and MPI-enabled multi-processor systems Operating system: HiPPy: Any for which Python v2.5.x is available. HPsrc: Any for which a standards-compliant Fortran95 compiler is available Has the code been vectorised or parallelised?: Yes RAM: Problem specific, typically less than 1 GB for either code Classification: 4.4, 11.5 Nature of problem: Derivation and use of perturbative Feynman rules for complicated lattice QCD actions. Solution method: An automated expansion method implemented in Python (HiPPy) and code to use expansions to generate Feynman rules in Fortran95 (HPsrc). Restrictions: No general restrictions. Specific restrictions are discussed in the text. Additional comments: The HiPPy and HPsrc codes are released under the second version of the GNU General Public Licence (GPL v2). Therefore anyone is free to use or modify the code for their own calculations. As part of the licensing, we ask that any publications including results from the use of this code or of modifications of it cite Refs. [1,2] as well as this paper. Finally, we also ask that details of these publications, as well as of any bugs or required or useful improvements of this core code, would be communicated to us. Running time: Very problem specific, depending on the complexity of the Feynman rules and the number of integration points. Typically between a few minutes and several weeks. The installation tests provided with the program code take only a few seconds to run.

Recent topics in numerical integration

AbstractIn this article, we describe some recent topics in the approximation of univariate and multivariate integrals. We especially pay attention to progress in the area of oscillatory integrals and in quasi‐Monte Carlo methods using lattice rules. Many pointers to the relevant literature are provided. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

RENORMALIZATION OF ATOMISTIC GROWTH MODELS

We review a general procedure for the multiscale analysis of atomistic lattice models of fluctuating interfaces driven by the deposition of new material. Beginning with a lattice Langevin formulation of site fluctuations, stochastic differential equations are derived by regularizing the lattice transition rules. Subsequent coarse graining is accomplished by applying the renormalization group, which yields trajectories from initial conditions determined by the regularized atomistic models. These trajectories correspond to hierarchies of continuum equations that describe the original lattice models over expanding length and time scales as the extent of coarse graining increases. This provides a systematic method for the derivation of continuum equations from the transition rules of lattice models appropriate for any length and time scales, and thereby establishes a quantitative link between atomistic transition rules and the collective behavior of the system. The results obtained with this method are confirmed by all available kinetic Monte Carlo simulations and, in some cases, have provided new interpretations of previous experimental observations. In this review, we first discuss the elements of our multiscale method in general terms, and then illustrate their implementation for specific growth models.

Determination of the rank of an integration lattice

The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations. In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q() directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice ? . We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deciency of modulo p versions of B.

Construction of lattice rules with a trigonometric d-property on the basis of extreme lattices

Lattice rules with the trigonometric d-property that are optimal with respect to the number of points are constructed for the approximation of integrals over an n-dimensional unit cube. An extreme lattice for a hyperoctahedron at n = 4 is used to construct lattice rules with the trigonometric d-property and the number of points $$ 0.80822 \ldots \cdot \Delta ^4 (1 + o(1)), \Delta \to \infty $$ (d = 2Δ − 1 ≥ 3 is an arbitrary odd number). With few exceptions, the resulting lattice rules have the highest previously known effectiveness factor.

Intermediate rank lattice rules and applications to finance

This paper discusses the intermediate rank lattice rule for the general case when the number of quadrature points is n t m , where m is a composite integer, t is the rank of the rule, n is an integer such that ( n , m ) = 1 . By using the technique of averaging the quantity P α , we obtained a general expression for the average of P α over a subset of Z s , derived an upper bound and the asymptotic rate for intermediate rank lattice rule. The results recover the cases of the conventional good lattice rule and the maximal rank rule. Computer search results show that P 2 's by the intermediate rank lattice rules are smaller than those by good lattice rule, while searching intermediate rank lattice points is much faster than that of good lattice points for very close numbers of quadrature points. Numerical tests for application to an option pricing problem show that the intermediate rank lattice rules are not worse than the conventional good lattice rule on average. All the lattice rules show superiority over the Sobol' sequence, which beats the pseudo-random point sets.

Periodization strategy may fail in high dimensions

We discuss periodization of smooth functions f of d variables for approximation of multivariate integrals. The benefit of periodization is that we may use lattice rules, which have recently seen significant progress. In particular, we know how to construct effectively a generator of the rank-1 lattice rule with n points whose worst case error enjoys a nearly optimal bound C d,p n −p . Here C d,p is independent of d or depends at most polynomially on d, and p can be arbitrarily close to the smoothness of functions belonging to a weighted Sobolev space with an appropriate condition on the weights. If F denotes the periodization for f then the error of the lattice rule for a periodized function F is bounded by C d,p n −p ∣∣F∣∣ with the norm of F given in the same Sobolev space. For small or moderate d, the norm of F is not much larger than the norm of f. This means that for small or moderate d, periodization is successful and allows us to use optimal properties of lattice rules also for non-periodic functions. The situation is quite different if d is large since the norm of F can be exponentially larger than the norm of f. This can already be seen for f = 1. Hence, the upper bound of the worst case error of the lattice rule for periodized functions is quite bad for large d. We conjecture not only that this upper bound is bad, but also that all lattice rules fail for large d. That is, if we fix the number of points n and let d go to infinity then the worst case error of any lattice rule is bounded from below by a positive constant independent of n. We present a number of cases suggesting that this conjecture is indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.

An overview of fast component‐by‐component constructions of lattice rules and lattice sequences

AbstractSince the initial work by I. H. Sloan and his collaborators on the component‐by‐component construction of good lattice rules for the approximation of multivariate integrals, a lot of variations on this theme have been published. These include various function spaces, prime and composite number of points, intermediate‐rank rules, polynomial lattice rules and extensible rules. We sketch the different variations and discuss the properties needed to have a fast component‐by‐component construction. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Strong tractability of multivariate integration of arbitrary order

AbstractWe 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)

Constructions of general polynomial lattices for multivariate integration

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed moduloarbitrarypolynomials.

Atypical grain boundaries in ZnO layers deposited on sapphire by rf magnetron sputtering on (0001) sapphire

The crystalline quality of ZnO is investigated, in thin films prepared by RF magnetron sputtering on c-plane sapphire. The ZnO layers exhibit a columnar growth and the average column diameter depends on the deposition temperature. Along the [101¯0] zone axis of sapphire, the diffraction pattern exhibits two zone axis patterns; the central rows belong to the two zones, with the 0002n spots clearly underlined by the overlap. On each part, the two inner rows belong to the [112¯0] zone and the next two to the [101¯0] zone. Therefore, the two main epitaxial relationships in mismatched growth of wurtzite structures coexist in these layers, and adjacent columns are rotated, with respect to one other, by 90∘ around the [0001] direction. In cross section observations, the interface between two grains is abrupt, but it is not easy to determine its atomic structure. Observations in planar view show that although the long range rotation between grains is in agreement with the theoretical epitaxial relationships, the local angles oscillate between 27∘ and 32∘. This discrepancy is explained by the formation of grain boundaries which are found to follow the coincidence site lattice rules which make them settle into minimal energy configurations.

The construction of extensible polynomial lattice rules with small weighted star discrepancy

In this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance. The construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest. We also show that the construction can be done component by component.

Lattice rule algorithms for multivariate approximation in the average case setting

We study multivariate approximation for continuous functions in the average case setting. The space of d variate continuous functions is equipped with the zero mean Gaussian measure whose covariance function is the reproducing kernel of a weighted Korobov space with the smoothness parameter α > 1 and weights γ d , j for j = 1 , 2 , … , d . The weight γ d , j moderates the behavior of functions with respect to the j th variable, and small γ d , j means that functions depend weakly on the j th variable. We study lattice rule algorithms which approximate the Fourier coefficients of a function based on function values at lattice sample points. The generating vector for these lattice points is constructed by the component-by-component algorithm, and it is tailored for the approximation problem. Our main interest is when d is large, and we study tractability and strong tractability of multivariate approximation. That is, we want to reduce the initial average case error by a factor ε by using a polynomial number of function values in ε - 1 and d in the case of tractability, and only polynomial in ε - 1 in the case of strong tractability. Necessary and sufficient conditions on tractability and strong tractability are obtained by applying known general tractability results for the class of arbitrary linear functionals and for the class of function values. Strong tractability holds for the two classes in the average case setting iff sup d ⩾ 1 ∑ j = 1 d γ d , j s < ∞ for some positive s < 1 , and tractability holds iff sup d ⩾ 1 ∑ j = 1 d γ d , j t / log ( d + 1 ) < ∞ for some positive t < 1 . . The previous results for the class of function values have been non-constructive. We provide a construction in this paper and prove tractability and strong tractability error bounds for lattice rule algorithms. This paper can be viewed as a continuation of our previous paper where we studied multivariate approximation for weighted Korobov spaces in the worst case setting. Many technical results from that paper are also useful for the average case setting. The exponents of ε - 1 and d corresponding to our error bounds are not sharp. However, for α close to 1 and for slow decaying weights, we obtain almost the minimal exponent of ε - 1 . We also compare the results from the worst case and the average case settings in weighted Korobov spaces.

Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels

We consider Fredholm integral equations of the second kind of the form f ( x ) = g ( x ) + ∫ k ( x - y ) f ( y ) d y , where g and k are given functions from weighted Korobov spaces. These spaces are characterized by a smoothness parameter α > 1 and weights γ 1 ≥ γ 2 ≥ ⋯ . The weight γ j moderates the behavior of the functions with respect to the jth variable. We approximate f by the Nyström method using n rank-1 lattice points. The combination of convolution and lattice group structure means that the resulting linear system can be solved in O ( n log n ) operations. We analyze the worst case error measured in sup norm for functions g in the unit ball and a class of functions k in weighted Korobov spaces. We show that the generating vector of the lattice rule can be constructed component-by-component to achieve the optimal rate of convergence O ( n - α / 2 + δ ) , δ > 0 , with the implied constant independent of the dimension d under an appropriate condition on the weights. This construction makes use of an error criterion similar to the worst case integration error in weighted Korobov spaces, and the computational cost is only O ( n log nd ) operations. We also study the notion of QMC-Nyström tractability: tractability means that the smallest n needed to reduce the worst case error (or normalized error) to ɛ is bounded polynomially in ɛ - 1 and d; strong tractability means that the bound is independent of d. We prove that strong QMC-Nyström tractability in the absolute sense holds iff ∑ j = 1 ∞ γ j < ∞ , and QMC-Nyström tractability holds in the absolute sense iff lim sup d → ∞ ∑ j = 1 d γ j / log ( d + 1 ) < ∞ .

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.

The construction of good extensible Korobov rules

In this paper, we introduce construction algorithms for Korobov rules for numerical integration which work well for a given set of dimensions simultaneously. The existence of such rules was recently shown by Niederreiter. Here we provide a feasible construction algorithm and an upper bound on the worst-case error in certain reproducing kernel Hilbert spaces for such quadrature rules. The proof is based on a sieve principle recently used by the authors to construct extensible lattice rules. We only treat classical lattice rules. The same ideas apply for polynomial lattice rules.

Constructing Robust Good Lattice Rules for Computational Finance

The valuation of many financial derivatives leads to high‐dimensional integrals. The constructions of robust or universal good lattice rules for financial applications are both important and challenging. An important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low‐dimensional functions, i.e., functions that depend on only a small number of variables (usually just two variables). For numerical integration of such functions the quality of the low‐order (i.e., low‐dimensional) projections of the node set is crucial. In this paper we propose methods to construct good lattice points with “optimal” low‐order projections. The quality of a point set is measured by a new measure called elementary order‐$\ell$ discrepancy, which measures the quality of all order‐$\ell$ projections and is more informative than usual measures. Two constructions, namely the Korobov and the component‐by‐component constructions, are studied such that the low‐order projections are optimized. Numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order‐2 projections that are better than those of the Sobol’ points and random points and with higher‐order projections that are no worse (while the Sobol’ points lost the advantage over random points in order‐2 projections on the average). The new lattice rules have the potential to improve upon the accuracy for favorable functions, while doing no harm for unfavorable ones. Their applications for pricing path‐dependent options and American options (based on the least‐square Monte Carlo method) are studied and their high efficiency is demonstrated. A nice surprise revealed is the robustness property of such lattice rules: the good projection property and the suitability for a large range of problems. The potential possibility and limitations of good lattice points in achieving good quality of moderate‐ and high‐order projections is investigated. The reason why classical lattice rules may not be efficient for high‐dimensional finance problems is also discussed.