Lattice Rules Research Articles

Lattice rules in non-periodic subspaces of Sobolev spaces

We investigate quasi-Monte Carlo (QMC) integration over the $s$-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $\alpha>1$, where $\boldsymbol{\gamma}$ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter $\alpha>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $\alpha$, denoted by $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $\alpha$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ as sets for $\alpha \geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbol{\gamma},s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{\alpha,\boldsymbol{\gamma},s}^{\mathrm{sob}})$. We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case.

High-speed evaluation of loop integrals using lattice rules

Lattice rules are implemented in CUDA for many-core computations on GPUs. A high-speed evaluation of Feynman loop integrals is presented, based on lattice rules and suitable transformations. The accuracy and efficiency of the method are compared for higher order sinm-transformations. Extensive results are reported for classes of diagrams including 2-loop box and 3-loop self-energy diagrams with massive internal lines. The method is further combined with an extrapolation with respect to the dimensional regularization parameter.

Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients

Abstract Quasi–Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion of some system of functions. Graham et al. (2015, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math., 131, 329–368) developed a lattice-based QMC theory for this problem and established a quadrature error decay rate ≈ 1 with respect to the number of quadrature points. The key assumption there was a suitable summability condition on the aforementioned system of functions. As a consequence, product-order-dependent weights were used to construct the lattice rule. In this paper, a different assumption on the system is considered. This assumption, originally considered by Bachmayr et al. (2017c, Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM Math. Model. Numer. Anal., 51, 321–339) to utilise the locality of support of basis functions in the context of polynomial approximations applied to the same type of the diffusion problem, is shown to work well in the same lattice-based QMC method considered by Graham et al.: the assumption leads us to product weights, which enables the construction of the QMC method with a smaller computational cost than Graham et al. A quadrature error decay rate ≈ 1 is established, and the theory developed here is applied to a wavelet stochastic model. By a characterisation of the Besov smoothness, it is shown that a wide class of path smoothness can be treated with this framework.

Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function $a(x,{y}) = \bar{a}(x) + \sum_{j\geq 1} y_j \psi_j(x)$ in a bounded domain $D\subset \mathbb{R}^d$. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where $\psi_j$ was assumed to have global support in the domain $D$, we assume in the present paper that ${supp}(\psi_j)$ is localized in $D$ and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903--920] which scale linearly with respect to the parameter dimension $s$. The dimension-independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher-order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.

Extraction of |Vcb| from B \u2192 D(\u2217)\u2113\u03bd\u2113 and the Standard Model predictions of R(D(\u2217))

We extract |Vcb| from the available data in the decay B → D(∗)ℓνℓ. B → D∗ℓνℓ, the unfolded binned differential decay rates of four kinematic variables including the q2 bins have been used. In the CLN and BGL parameterizations of the form factors, the combined fit to all the available data along with their correlations yields |Vcb| = (39.77 ± 0.89) × 10−3 and (40.90 ± 0.94) × 10−3 respectively. In these fits, we have used the inputs from lattice and light cone sum rule (LCSR) along with the data. Using our fit results and the HQET relations (with the known corrections included) amongst the form factors, and parameterizing the unknown higher order corrections (in the ratios of HQET form factors) with a conservative estimate of the normalizing parameters, we obtain R(D∗) = 0.259 ± 0.006 (CLN) and R(D∗) = 0.257 ± 0.005 (BGL).

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

AbstractThe notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤb. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤbin a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of Lpdiscrepancy for all 1 ≤ p < ∞.

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens et al. (Adv Comput Math 42(1):55–84, 2016), the authors derived an upper estimate for the nth minimal worst case error for such problems and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-1 lattice rule that obtains a rate of convergence arbitrarily close to \(\mathcal {O}(n^{-\alpha })\), where \(\alpha >1/2\) denotes the smoothness of our function space and n is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets that can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension d is significantly improved.

A Multi-Index Quasi--Monte Carlo Algorithm for Lognormal Diffusion Problems

We present a multi-index quasi--Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm constructs an estimator for the expected value of some functional of the solution. The efficiency of this new method is illustrated on a three-dimensional subsurface flow problem with lognormal diffusion coefficient with underlying Matérn covariance function. This example is particularly challenging because of the small correlation length considered, and thus the large number of uncertainties that must be included. We show numerical evidence that it is possible to achieve a cost inversely proportional to the requested tolerance on the root-mean-square error, for problems with a smoothly varying random field.

Multivariate integration over [formula omitted] with exponential rate of convergence

In this paper we analyze the approximation of multivariate integrals over the Euclidean space for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with different mesh sizes in each direction to sample the function, and then truncate it. In our analysis, the mesh sizes and the truncated domain are chosen by optimally balancing the truncation error and the discretization error.This paper derives results in comparable function space settings, extended to Rs, as which were recently obtained in the unit cube by Dick, Larcher, Pillichshammer and Woźniakowski, see Dick et al. (2011). They showed that both lattice rules and regular grids, with different mesh sizes in each direction, attain exponential rates, hence motivating us to analyze only cubature formula based on regular meshes. We further also amend the analysis of older publications, e.g., Sloan and Osborn (1987) and Sugihara (1987), using lattice rules on Rs by taking the truncation error into account and extending them to take the anisotropy of the function space into account.

Influence of dynamical changes on concept lattice and implication rules

Concept lattices are widely used as a knowledge structure in many real-life applications and the updating of a formal context is inevitable in most cases. How to effectively update the corresponding concept lattice as well as the embodied implication rules is still an open, interesting and important problem. The main objective of this paper is to give a solution to this problem. At first, we prove that in the updating process of the present concept lattice, whether to create a new concept and where to insert the new concept are solely determined by the lately created concepts or modified ones. And then, based on this finding, we propose an algorithm for the updating of concept lattice. Moreover, we carry out time complexity analysis and conduct experiments to show the effectiveness of our algorithm. Finally, we put forward some useful strategies for the updating of implication rules based on minimal generators.

Hiding the weights—CBC black box algorithms with a guaranteed error bound

The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the “weighted” function space setting introduced by Sloan and Woźniakowski. The “weights” that define such spaces are needed as inputs into the CBC algorithm, and so a natural question is, for a given problem how does one choose the weights? This paper introduces two new CBC algorithms which, given bounds on the mixed first derivatives of the integrand, produce a randomly shifted lattice rule with a guaranteed bound on the root-mean-square error. This alleviates the need for the user to specify the weights. We deal with “product weights” and “product and order dependent (POD) weights”. Numerical tables compare the two algorithms under various assumed bounds on the mixed first derivatives, and provide rigorous upper bounds on the root-mean-square integration error.

Performance-based assessment of roadway maintenance and rehabilitation activities from user’s perception in Taiwan

In Taiwan, even though the highway agencies have been making effort to improve the overall serviceability of the road system, more often than not, road users were not satisfied with the performance of the maintenance and rehabilitation (M&R) activities. This paper focuses on the performance of M&R activities in Taiwan from the road users’ perspective. Surveys were conducted to collect the users’ perception (serving as the performance) of the M&R activities. The Decision Making Trial and Evaluation Laboratory (DEMATEL) and Analytic Network Process (ANP) are jointly used to quantify the performance of M&R activities. DEMATEL is employed to understand the influence among the criteria/elements, and ANP is used to determine the relative weights of the elements. It is found that the respondents were most concerned about unsafe conditions due to manhole covers, damaged manhole covers, insufficient street lighting, potholes and bumps on pavements, and uneven pavements. Furthermore, the performance of M&R activities implemented on six major road types in Taiwan is analysed using the Formal Concept Analysis (FCA) method. Based on the concept lattice and association rules derived from FCA, directions for the improvement of M&R activities are proposed.

Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.

Algorithm 958

We introduce a new software tool and library named Lattice Builder, written in C++, that implements a variety of construction algorithms for good rank-1 lattice rules. It supports exhaustive and random searches, as well as component-by-component (CBC) and random CBC constructions, for any number of points, and for various measures of (non)uniformity of the points. The measures currently implemented are all shift-invariant and represent the worst-case integration error for certain classes of integrands. They include, for example, the weighted Pα square discrepancy, the Rα criterion, and figures of merit based on the spectral test, with projection-dependent weights. Each of these measures can be computed as a finite sum. For the Pα and Rα criteria, efficient specializations of the CBC algorithm are provided for projection-dependent, order-dependent, and product weights. For numbers of points that are integer powers of a prime base, the construction of embedded rank-1 lattice rules is supported through any of these algorithms, and through a fast CBC algorithm, with a variety of possibilities for the normalization of the merit values of individual embedded levels and for their combination into a single merit value. The library is extensible, thanks to the decomposition of the algorithms into decoupled components, which makes it easy to implement new types of weights, new search domains, new figures of merit, and so on.

The lattice‐based approaches for mining association rules: a review

The traditional methods for mining association rules (ARs) include two phrases: mining frequent itemsets (FIs)/frequent closed itemsets (FCIs)/frequent maximal itemsets (FMIs) and generating ARs from FIs/FCIs/FMIs. Lattice‐based approaches (LBAs) for mining ARs are new approaches including two phrases: frequent itemset lattice (FIL)/frequent closed itemset lattice (FCIL) building and generating ARs from the lattice. Total mining time of LBAs for mining ARs outperforms the traditional methods for mining ARs. Besides, the most important advantage of LBAs for mining ARs is that the algorithms only build the lattice once and mine ARs with many different confidences or many different minimum supports (the thresholds have to be greater than or equal to the threshold used to build lattices) without mining FIs/FCIs again. In this article, we describe a number of existing LBAs for mining ARs on static databases including lattice building and rule generation. In addition, in today's online system, the data often change in several operations such as insert, delete, and update. Hence, a number of LBAs for mining ARs on dynamic databases are mentioned. Finally, complexity analysis of the LBAs for mining ARs is also thoroughly discussed. WIREs Data Mining Knowl Discov 2016, 6:140–151. doi: 10.1002/widm.1181This article is categorized under: Algorithmic Development > Association Rules

Charmonia decay constants from the QCD lattice and QCD sum rules

Using lattice QCD and QCD sum rules we compute the lowest state charmonia JPC=0−+(ηc), 1−(J/ψ), and 1+−(hc) decay constants. For calculating the decay constant of J/ψ we use both the vector and tensor currents and compare the results. Lattice QCD results are obtained from the unquenched (Nf=2) simulations using twisted mass QCD at four lattice spacings and taking the continuum limit. In the QCD sum rule calculation we apply the moment sum rules. We also comment the phenomenological implications of calculated charmonia decay constants in ηc→γγ decay, and B→Xcc‾K decays.

The Hermite normal form for certain rank-1 circulant and skew-circulant lattice rules

Characterisations are provided of the Hermite normal form of certain integer circulant and skew-circulant matrices. These matrices are associated with rank-1 circulant and skew-circulant lattice rules. Previous computer searches for lattice rules of specified trigonometric degree have indicated that there is some merit in searches of such lattice rules.We also consider the question of whether a Hermite normal form with the characterisation for these rank-1 circulant lattice rules is actually associated to a circulant lattice rule. Though the answer is negative in the general case, several examples where the answer is positive will be given.

Formula omitted] form factors reexamined in the whole kinematically accessible region

We reexamine the B→πℓ+ℓ−(ℓ=e,μ,τ) form factors, f+B→π(q2), f0B→π(q2) and fTB→π(q2), in the entire region of the momentum transfer squared q2, by taking advantage of the complementarity between lattice QCD (LQCD) simulation and light cone sum rule approach (LCSR), and analyticity of the form factors. A LCSR calculation with a chiral current correlator, which could avoid pollution by twist-3 components, is performed at twist-2 next-to-leading order (NLO) accuracy, to determinate the form factor shapes in the small and intermediate q2 region. Further, fitting simultaneously the LCSR results for these form factors and the related LQCD ones (available or based on a SUF(3) symmetry breaking ansatz) to a Bourrely–Caprini–Lellouch (BCL) parametrization, we get a global understanding of their q2 behaviors. Our findings turn out to be consistent with the recent study by Ali, Parkhomenko and Rusov, and the resulting observation for the vector form factor provides support for the existing LQCD as well as LCSR predictions extrapolated to the entire kinematically accessible region.

On a projection-corrected component-by-component construction

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.