Lattice Rules Research Articles

Convergence, sampling and total order estimator effects on parameter orthogonality in global sensitivity analysis.

Dynamical system models typically involve numerous input parameters whose "effects" and orthogonality need to be quantified through sensitivity analysis, to identify inputs contributing the greatest uncertainty. Whilst prior art has compared total-order estimators' role in recovering "true" effects, assessing their ability to recover robust parameter orthogonality for use in identifiability metrics has not been investigated. In this paper, we perform: (i) an assessment using a different class of numerical models representing the cardiovascular system, (ii) a wider evaluation of sampling methodologies and their interactions with estimators, (iii) an investigation of the consequences of permuting estimators and sampling methodologies on input parameter orthogonality, (iv) a study of sample convergence through resampling, and (v) an assessment of whether positive outcomes are sustained when model input dimensionality increases. Our results indicate that Jansen or Janon estimators display efficient convergence with minimum uncertainty when coupled with Sobol and the lattice rule sampling methods, making them prime choices for calculating parameter orthogonality and influence. This study reveals that global sensitivity analysis is convergence driven. Unconverged indices are subject to error and therefore the true influence or orthogonality of the input parameters are not recovered. This investigation importantly clarifies the interactions of the estimator and the sampling methodology by reducing the associated ambiguities, defining novel practices for modelling in the life sciences.

Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

Abstract Let $f:[0,1]^{d}\to{\mathbb{R}}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^{d}$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^{d}$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $b\geqslant 2$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $b\geqslant 2$ and any dimension $d\geqslant 1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\geqslant 3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\geqslant 2$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.

Theoretical overview on heavy flavor physics at LHCb

Precise prediction of the heavy hadrons decay amplitudes is of great importance in the determination of the parameters of KM matrix and searching for new physics signals beyond the standard model. Due to the high accuracy of the B meson decay constants, the structure dependent QED corrections should be taken into account for B→μ+μ− decays due to the power enhancement, but this effect is numerically negligible for B→τ+τ−. The radiative leptonic decays and double radiative decays are the best channels to determine the B meson distribution amplitudes, but one should systematically deal with the power corrections to reduce the uncertainty. The heavy-to-light form factors is one of the most important nonperturbative quantities, and an improved method to improve the theoretical accuracy is to fit the form factors using combined data from Lattice and light-cone sum rules which works well at large recoil region. The power suppressed weak annihilation contribution is phenomenologically important in the nonleptonic decays, and the previously neglected hard-collinear gluon exchange diagrams have the potential to cancel the endpoint singularity arising from the hard gluon exchange. Therefore, more endeavors need to be paid to this topic.

Randomly Shifted Lattice Rules with Importance Sampling and Applications

In financial and statistical computations, calculating expectations often requires evaluating integrals with respect to a Gaussian measure. Monte Carlo methods are widely used for this purpose due to their dimension-independent convergence rate. Quasi-Monte Carlo is the deterministic analogue of Monte Carlo and has the potential to substantially enhance the convergence rate. Importance sampling is a widely used variance reduction technique. However, research into the specific impact of importance sampling on the integrand, as well as the conditions for convergence, is relatively scarce. In this study, we combine the randomly shifted lattice rule with importance sampling. We prove that, for unbounded functions, randomly shifted lattice rules combined with a suitably chosen importance density can achieve convergence as quickly as O(N−1+ϵ), given N samples for arbitrary ϵ values under certain conditions. We also prove that the conditions of convergence for Laplace importance sampling are stricter than those for optimal drift importance sampling. Furthermore, using a generalized linear mixed model and Randleman–Bartter model, we provide the conditions under which functions utilizing Laplace importance sampling achieve convergence rates of nearly O(N−1+ϵ) for arbitrary ϵ values.

The fast reduced QMC matrix–vector product

We study the approximation of integrals of the form ∫Df(x⊤A)dμ(x), where the measure μ is of product form and A is a matrix, by quasi-Monte Carlo (QMC) rules N−1∑k=0N−1f(xk⊤A). We are interested in cases where the main computational cost in the approximation arises from calculating the products xk⊤A. We design QMC rules for which the computation of xk⊤A, k=0,1,…,N−1, can be done in a fast way, and for which the approximation error of the QMC rule is similar to the standard QMC error. We do not require that the matrix A has any particular structure.Problems of this form arise in some important applications in statistics and uncertainty quantification. For instance, this approach can be used when approximating the expected value of some function with a multivariate normal random variable with some given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients.The speed-up of the computation time of our approach is sometimes better and sometimes worse than the fast QMC matrix–vector product from [Josef Dick, Frances Y. Kuo, Quoc T. Le Gia, and Christoph Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015), no. 3, A1436–A1450]. As in that paper, our approach applies to lattice point sets and polynomial lattice point sets, but also applies to digital nets (we are currently not aware of any approach which allows one to apply the fast QMC matrix–vector paper from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets).The method in this paper does not make use of the fast Fourier transform, instead we use repeated values in the quadrature points to derive a significant reduction in the computation time. Such a situation naturally arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can additionally be used to also reduce the computation time of the underlying QMC rule. One advantage of the present approach is that it can be combined with random (digital) shifts, whereas this does not apply to the fast QMC matrix–vector product from the earlier paper of Dick, Kuo, Le Gia, and Schwab.

Numerical scattering amplitudes with pySecDec

We present a major update of the program pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The new version enables the evaluation of multi-loop integrals as well as amplitudes in a highly distributed and flexible way, optionally on GPUs. The program has been optimised and runs up to an order of magnitude faster than the previous release. A new integration procedure that utilises construction-free median Quasi-Monte Carlo rules is implemented. The median lattice rules can outperform our previous component-by-component rules by a factor of 5 and remove the limitation on the maximum number of sampling points. The expansion by regions procedures have been extended to support Feynman integrals with numerators, and functions for automatically determining when and how analytic regulators should be introduced are now available. The new features and performance are illustrated with several examples. Program summaryProgram Title: pySecDecCPC Library link to program files:https://doi.org/10.17632/dnrkf5jxzh.4Developer's repository link:https://github.com/gudrunhe/secdec, https://secdec.readthedocs.io (Online documentation)Licensing provisions: GNU Public License v3Programming language: Python, Form, C++, CudaExternal routines/libraries:GSL [1], NumPy [2], SymPy [3], Nauty [4], Cuba [5], Form [6], GiNaC and CLN [7], Normaliz [8], GMP [9].Journal reference of previous version: Comput. Phys. Commun. 273 (2022) 108267 [1].Does the new version supersede the previous version?: Yes.Nature of problem: Scattering amplitudes at higher orders in perturbation theory are typically represented as a linear combination of coefficients — containing the kinematic invariants and the space-time dimension — multiplied with loop integrals which contain singularities and whose analytic representation might be unknown.Solution method: Extraction of singularities in the dimensional regularization parameter as well as in analytic regulators for potential spurious singularities is done using sector decomposition. The combined evaluation of the integrals with their coefficients is performed in an efficient way.Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time.

Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration

We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the d-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget n of a maximum allowed number of function evaluations, we uniformly pick a prime p in the range n/2<p≤n. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form O(n−α−1/2+δ), with δ>0, for a Korobov space with smoothness α>1/2 and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors offline ahead of time at cost O(dn4/ln⁡n) when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.

The design of distributed filtering based on lattice rule

For the problem of multiple micro-target tracking and positioning, the high-dynamic and communication coupling are two cases in actual motion. The dynamic evolution and measurement among them are depicted by a nonlinear system. This paper proposes a new distributed lattice Kalman filter (DLKF) to cope with the two cases simultaneously. The prediction is based on the Cranley-Patterson shift and Korobov lattice rule to generate the low-difference sample points. The update stage is based on the weighted average consistency to fusion its innovation and the information of its neighbor. The stability of this distributed filter is proved by constructing the upper and lower bound of the Lyapunov functional. The finite steps consensus iteration strategies for the update fusion are also investigated. To evaluate the efficiency of the DLKF, it is compared with unscented Kalman filtering and its distributed form. The simulation results show that DLKF uses significantly fewer sampling points than other quasi-Monte Carlo (QMC) filters while maintaining the estimation accuracy. The computational complexity is significantly lower, which is the key factor for application to multiple micro-targets.

On a Reduced Component-by-Component Digit-by-Digit Construction of Lattice Point Sets

Abstract In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which is useful for situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. By considering a reduced digit-by-digit construction, we allow an integration algorithm to be less precise with respect to the number of bits in those components of the problem that are considered less important. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied in the paper Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness by Ebert, Kritzer, Nuyens, and Osisiogu, published in the Journal of Complexity in 2021. This improvement is illustrated by numerical results.

Optimizing Air Pollution Modeling with a Highly-Convergent Quasi-Monte Carlo Method: A Case Study on the UNI-DEM Framework

In this study, we present the development of an advanced air pollution modeling approach, which incorporates cutting-edge stochastic techniques for large-scale simulations of long-range air pollutant transportation. The Unified Danish Eulerian Model (UNI-DEM) serves as a crucial mathematical framework with numerous applications in studies concerning the detrimental effects of heightened air pollution levels. We employ the UNI-DEM model in our research to obtain trustworthy insights into critical questions pertaining to environmental preservation. Our proposed methodology is a highly convergent quasi-Monte Carlo technique that relies on a unique symmetrization lattice rule. By fusing the concepts of special functions and optimal generating vectors, we create a novel algorithm grounded in the component-by-component construction method, which has been recently introduced. This amalgamation yields particularly impressive outcomes for lower-dimensional cases, substantially enhancing the performance of the most advanced existing methods for calculating the Sobol sensitivity indices of the UNI-DEM model. This improvement is vital, as these indices form an essential component of the digital ecosystem for environmental analysis.

Study of the b → dℓℓ transitions in the Standard Model and test of New Physics sensitivities

After incorporating all the available experimental data and the most up-to-date Lattice and light cone sum rule (LCSR) inputs on the form factors, we analyze the exclusive b → uℓνℓ and b → dℓℓ decays simultaneously. We have extracted the shapes of all the associated form factors using which we have provided predictions in the standard model for the branching ratios, direct CP asymmetries and isospin asymmetry for B → πℓℓ and various angular observables for B → ρℓℓ transitions. Also, we have tested the sensitivities of these observables towards physics beyond the standard model (BSM). For the B → ρℓℓ decays, we have defined tagged and untagged observables and predicted them in the SM and BSM. In the context of BSM, we have found some compelling information. The respective predictions in a few benchmark scenarios are given, which can be tested in the experiments at the LHCb and the Belle.

Numerical Regularization for 4-loop Self-Energy Feynman Diagrams

In recent work we computed 4-loop integrals for self-energy diagrams with 11 massive internal lines. Presently we perform numerical integration and regularization for diagrams with 8 to 11 lines, while considering massive and massless cases. For dimensional regularization, a sequence of integrals is computed depending on a parameter (ε) that is incorporated via the space-time dimension, and approaches zero. We consider diagrams where the leading term in the expansion is in 1/ε 2 or in 1/ε or finite. The numerical integration methods include non-adaptive – double exponential, and Quasi-Monte Carlo methods with composite lattice rules implemented in CUDA C for acceleration on GPUs. The leading term coefficients in the integral expansion are obtained via linear or nonlinear extrapolation as ε tends to zero.

A note on the CBC-DBD construction of lattice rules with general positive weights

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule for an s-dimensional integral is specified by its generating vectorz∈Zs and its number of points N. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions of good generating vectors for dimensions s≥3. Therefore one resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors for integration of functions in weighted Korobov classes equipped with product weights. Here, we generalize this result to arbitrary positive weights, answering an open question from the paper of Ebert et al. We include a section on how the algorithm can be implemented in the case of POD weights, implying that the CBC-DBD construction is competitive with the classical CBC construction.

Scaled lattice rules for integration on ℝ^{𝕕} achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces

We introduce a new method to approximate integrals∫Rdf(x)dx\int _{\mathbb {R}^d} f(\boldsymbol {x}) \,\mathrm {d}\boldsymbol {x}which simply scales lattice rules from the unit cube[0,1]d[0,1]^dto properly sized boxes onRd\mathbb {R}^d, hereby achieving higher-order convergence that matches the smoothness of the integrand functionffin a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand functionffand does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it “periodic” on a box without changing its integral value over the box is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters,γj\gamma _j, satisfy the summability condition∑j=1∞γj1/α&gt;∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }&gt;\infty, whereα\alphais a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces.In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition∑j=1∞γj1/α&gt;∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }&gt;\inftyholds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

Construction-Free Median Quasi-Monte Carlo Rules for Function Spaces with Unspecified Smoothness and General Weights

We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of $r$ independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these $r$ estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as $r$ increases. Numerical experiments illustrate and support our theoretical findings.

Extrapolated Polynomial Lattice Rule Integration in Computational Uncertainty Quantification

Extrapolated Polynomial Lattice Rule Integration in Computational Uncertainty Quantification

Monte Carlo tree search for generating vectors of lattice rules

Lattice rules are widely studied in the context of quasi-Monte Carlo methods as a means to achieve a small integration error. The rules themselves are determined completely by so called generating vectors, so there is an interest in methods for constructing vectors that perform well. This article introduces a new component-wise construction of a generating vector using the principles of Monte Carlo tree search, with the goal of avoiding local optima. Error bounds are proven for the vectors obtained from this method, which are analogous to existing results for the popular component by component construction.

A fitness sharing based ant clustering method for multimodal optimization of the aircraft longitudinal automatic carrier landing system

With the flight envelope becoming larger and larger, the Automatic Carrier Landing System (ACLS) is becoming a complex large-scale system, and the corresponding control parameter design has been the most important key link to ensure the safety and flight quality of aircraft carrier landing missions. In this paper, a Fitness Sharing based Ant Clustering (FSAC) method is presented for use in multimodal optimization of the longitudinal ACLS and to reveal the hidden properties of the solution space. In doing so, first, the lattice rule based space sampling strategy is used to create the individual ant sequences. Then a fading memory fitness sharing function is applied to modify the clustering strategy and regroup the ants for multimodal optimization of the search space. An adaptive learning strategy is developed to dynamically adjust the search scope of the ant colony. Moreover, an online health monitor is used to replace the weak ants with new ones in a timely manner in order to keep robustness of the FSAC. Finally, the multimodal feasible solutions which are good candidates for the ACLS design are presented and the characteristics of the solution space are also analyzed. The result shows that many large solution sets are located at the bottom of the solution space, whereas on the upper side of the solution space, the number of feasible solutions decreases sharply. An F/A-18 model is used as a test bed and the simulations are carried out in various wind conditions to demonstrate the effectiveness and feasibility of the proposed method.

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify on one hand the regularity of the solution with respect to the random domain mapping required for many multilevel quadrature methods, such as the multilevel quasi-Monte Carlo quadrature using Halton points, the multilevel sparse anisotropic Gauss–Legendre and Clenshaw–Curtis quadratures and multilevel interlaced polynomial lattice rules. On the other hand, we derive the coupling formulation and show by numerical results that the approach is feasible.