Optimal Quadrature Rules Research Articles

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

Machine learning discovery of optimal quadrature rules for isogeometric analysis

We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor products. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on adaptive gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.

Weighted averaged Gaussian quadrature rules for modified Chebyshev measures

This paper is concerned with the approximation of integrals of a real-valued integrand over the interval [−1,1] by Gauss quadrature. The averaged and optimal averaged quadrature rules ([13,21]) provide a convenient method for approximating the error in the Gauss quadrature. However, they are applicable to all integrands that are continuous on the interval [−1,1] only if their nodes are internal, i.e. if they belong to this interval.We discuss two approaches to determine averaged quadrature rules with nodes in [−1,1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of the first, second, and third kinds that are modified by a linear over linear rational factor, and discuss the internality of averaged, optimal averaged, and truncated optimal averaged quadrature rules. Moreover, we show that the weighting yields internal averaged rules if a weighting parameter is properly chosen, and we provide bounds for this parameter that guarantee internality. Finally, we illustrate that the weighted averaged rules give more accurate estimates of the quadrature error than the truncated optimal averaged rules.

Optimal quadrature rules for numerical solution of the nonlinear Fredholm integral equations

In this paper, an iterative method of successive approximations to the approximate solution of nonlinear Hammerstein-Fredholm integral equations using an optimal quadrature formula for classes of functions of Lipschitz types is provided. Also, the convergence analysis and numerical stability of the proposed method are proved. Finally, some numerical examples verify the theoretical results and show the accuracy of the method.

Note on Corrected Optimal Quadrature Formulas in Sense Nikolski

The optimal 2-points quadrature formulas of open type are derived and the error estimates in terms of a variety of norms involving the second derivative are considered. Also, the corrected formulas of the optimal quadrature rules are obtained.

Asymptotically optimal quadrature rules for uniform splines over the real line

We provide explicit asymptotically optimal quadrature rules for uniform Ck-splines, k = 0,1, over the real line. The nodes of these quadrature rules are given in terms of the zeros of ultraspherical polynomials (Gegenbauer polynomials) and related polynomials. We conjecture that our derived rules are the only possible periodic asymptotically optimal quadrature rules for these spline spaces.

Fast algorithms using orthogonal polynomials

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

Optimization of Compatible Meshfree Quadrature Rule for Nonlocal Problems with Applications to Peri Dynamics Study

The optimization of a compatible meshfree quadrature rule for nonlocal problems with applications to Peridynamics is investigated in this study. Peridynamics is a nonlocal continuum mechanics theory that models material behavior based on interactions between material points within a finite neighbourhood. In Peridynamics, the accurate evaluation of nonlocal integral equations is crucial for obtaining reliable and efficient solutions. Traditional numerical integration methods, such as Gaussian quadrature, are not directly applicable to nonlocal problems due to their local nature. Hence, the development of a compatible meshfree quadrature rule that can effectively handle nonlocal interactions is of great importance. The objective it to optimize a quadrature rule that accurately captures the nonlocal interactions in Peridynamics while maintaining computational efficiency. The optimization process involves the selection of appropriate quadrature points and weights that minimize the quadrature error and maximize the computational efficiency. Various optimization techniques, such as genetic algorithms, particle swarm optimization, or machine learning algorithms, are explored to search for an optimal quadrature rule. The optimized quadrature rule is then applied to several Peridynamics problems, including fracture mechanics, material failure, and dynamic response analysis. The performance of the optimized quadrature rule is evaluated by comparing the results with those obtained using traditional quadrature methods and analytical solutions, when available. The accuracy, stability, and computational efficiency of the optimized quadrature rule are analysed and discussed. The findings provide valuable insights into the development and optimization of compatible meshfree quadrature rules for nonlocal problems, particularly in the context of Peridynamics. The optimized quadrature rule offers improved accuracy in capturing nonlocal interactions and reduces the computational cost compared to traditional methods. The applications of the optimized quadrature rule to various Peridynamics problems demonstrate its effectiveness and reliability.
 The implications of it extend beyond Peridynamics and can be applicable to other nonlocal models in the field of computational mechanics. The optimized quadrature rule has the potential to enhance the accuracy and efficiency of numerical simulations involving nonlocal phenomena, enabling more reliable predictions of material behavior and structural response. The findings of this contribute to the advancement of numerical methods for nonlocal problems and provide a foundation for further research and developments in the field of compatible meshfree quadrature rules for nonlocal problems.

Optimal Sample Weights for Hemispherical Integral Quadratures

AbstractThis paper proposes optimal quadrature rules over the hemisphere for the shading integral. We leverage recent work regarding the theory of quadrature rules over the sphere in order to derive a new theoretical framework for the general case of hemispherical quadrature error analysis. We then apply our framework to the case of the shading integral. We show that our quadrature error theory can be used to derive optimal sample weights (OSW) which account for both the features of the sampling pattern and the bidirectional reflectance distribution function (BRDF). Our method significantly outperforms familiar Quasi Monte Carlo (QMC) and stochastic Monte Carlo techniques. Our results show that the OSW are very effective in compensating for possible irregularities in the sample distribution. This allows, for example, to significantly exceed the regular convergence rate of stochastic Monte Carlo while keeping the exact same sample sets. Another important benefit of our method is that OSW can be applied whatever the sampling points distribution: the sample distribution need not follow a probability density function, which makes our technique much more flexible than QMC or stochastic Monte Carlo solutions. In particular, our theoretical framework allows to easily combine point sets derived from different sampling strategies (e.g. targeted to diffuse and glossy BRDF). In this context, our rendering results show that our approach overcomes MIS (Multiple Importance Sampling) techniques.

Spectral approximation properties of isogeometric analysis with variable continuity

We study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis. Such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh, which allows for large computational savings during the solution of the resulting linear system. The continuity reduction results in extra degrees of freedom that modify the approximation properties of the method. The convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis. We show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra, such as stopping bands and outliers, and present a unified description of these effects in finite element method, isogeometric analysis, and refined isogeometric analysis. Accuracy of the refined isogeometric analysis approximations can be improved by using non-standard quadrature rules. In particular, optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence, as it occurs in standard isogeometric analysis.

An atomistic fingerprint algorithm for learning ab initio molecular force fields.

Molecular fingerprints, i.e., feature vectors describing atomistic neighborhood configurations, is an important abstraction and a key ingredient for data-driven modeling of potential energy surface and interatomic force. In this paper, we present the density-encoded canonically aligned fingerprint algorithm, which is robust and efficient, for fitting per-atom scalar and vector quantities. The fingerprint is essentially a continuous density field formed through the superimposition of smoothing kernels centered on the atoms. Rotational invariance of the fingerprint is achieved by aligning, for each fingerprint instance, the neighboring atoms onto a local canonical coordinate frame computed from a kernel minisum optimization procedure. We show that this approach is superior over principal components analysis-based methods especially when the atomistic neighborhood is sparse and/or contains symmetry. We propose that the "distance" between the density fields be measured using a volume integral of their pointwise difference. This can be efficiently computed using optimal quadrature rules, which only require discrete sampling at a small number of grid points. We also experiment on the choice of weight functions for constructing the density fields and characterize their performance for fitting interatomic potentials. The applicability of the fingerprint is demonstrated through a set of benchmark problems.

Numerical Integration in Multiple Dimensions with Designed Quadrature

We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions and geometric constraints on nodes and weights. We use penalty methods to address the geometric constraints, and subsequently solve a quadratic minimization problem via the Gauss-Newton method. Our analysis provides guidance on requisite sizes of quadrature rules for a given polynomial subspace, and furnishes useful user-end stability bounds on error in the quadrature rule in the case when the polynomial moment conditions are violated by a small amount due to, e.g., finite precision limitations or stagnation of the optimization procedure. We present several numerical examples investigating optimal low-degree quadrature rules, Lebesgue constants, and 100-dimensional quadrature. Our capstone examples compare our quadrature approach to popular alternatives, such as sparse grids and quasi-Monte Carlo methods, for problems in linear elasticity and topology optimization.

Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis

We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.

Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis

We introduce optimal quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. Using the homotopy continuation concept (Bartoň and Calo, 2016) that transforms optimal quadrature rules from source spaces to target spaces, we derive optimal rules for splines defined on finite domains. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odd-degree space, we derive rules for target spaces of higher continuity. We further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points. We demonstrate it by deriving optimal rules for various odd-degree spline spaces, particularly with non-uniform knot sequences and non-uniform multiplicities. We also discuss convergence of our rules to their asymptotic counterparts, that is, the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains. For spaces of low continuities, we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values.

Gaussian quadrature for splines via homotopy continuation: Rules for [formula omitted] cubic splines

We introduce a new concept for generating optimal quadrature rules for splines. To generate an optimal quadrature rule in a given (target) spline space, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, while preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by changing the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with C1 cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C2 cubic spline spaces where the rule was only conjectured to date. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.

Multiple orthogonality in the space of trigonometric polynomials of semi-integer degree

In this paper we consider multiple orthogonal trigonometric polynomials of semi-integer degree, which are necessary for constructing of an optimal set of quadrature rules with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994) 271-288]. We prove that such multiple orthogonal trigonometric polynomials satisfy certain recurrence relations and present numerical method for their construction, as well as for construction of mentioned optimal set of quadrature rules. Theoretical results are illustrated by some numerical examples.

An isogeometric analysis for elliptic homogenization problems

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rationalB-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method.

Multiphase equilibrium calculations using the semi-continuous thermodynamics of hydrocarbon mixtures

Natural hydrocarbon mixtures are composed of very many components. Usually, several components are lumped into pseudo-components to reduce the dimensionality of the equilibrium flash calculations. An interesting alternative to lumping is the use of semi-continuous thermodynamics, which enables a better characterization of mixture compositions. Traditionally, in the semi-continuous approach, the feed composition is approximated by a standard distribution function. Then, a Gaussian quadrature is used to find discrete carbon numbers representing the “pseudo-components”, for which a discrete flash can be performed. However, the feed composition can be random and/or far from classical distributions. In the quadrature method of moments (QMoM), developed by Lage (Lage, Comput. Chem. Eng. 31 (2007) 782–799), an optimal quadrature rule is used, considering the feed composition distribution as the weight function. In this work, a hidden feature of QMoM is revealed, by identifying a distribution function used to approximate a newly considered dependence, which involves the feed composition and the component liquid mole numbers. This dependence is extremely smooth, and, the most important, the shape of the function remains identical for any composition, pressure and temperature. The smoothness of the distribution enables a really good accuracy in solving the quadrature. The calculation of the quadrature is based on a procedure which avoids problems due to the ill-conditioned nature of the problem, even for a larger number of pseudo-components (unlike the original QMoM formulation). An analytical delumping procedure can be used (with an extremely small computational effort) to recover detailed composition in each equilibrium phase. Comparisons between a full discrete flash and the semi-continuous description proved the QMoM to be highly accurate. Various test problems, including two-phase and multiphase flash calculations, showed that Lage's methodology is generally applicable, even in cases when no distribution function can model the feed composition, or several distribution functions are needed to model different portions of the mixture.

A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis

We develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem. A simple and robust algorithm is developed to determine the rules which are exact for important B-spline spaces of uniform and geometrically stretched knot spacings. We consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems. We find that the rules are almost optimally efficient, but much easier to obtain than optimal rules, which require the solution of global nonlinear problems that are often ill-posed.