Object-oriented numerical integration—a template scheme for FEM and BEM applications

An object-oriented numerical integration template implementation is presented on the basis of the C++ programming language. Aiming its straightforward application in finite and boundary element methods, the design supports integrand objects of scalar, vector or matrix types, so that a single programming statement is able to integrate element matrices and vectors. The integrand can contain singularities like the ones typically found in boundary element methods, allowing the evaluation of both regular and singular integrals under the same programming structure. The use of the proposed design is illustrated through some elementary applications as well as finite element and boundary element code excerpts.     

Fuzzified Choquet Integral with a Fuzzy-valued Integrand and its application on temperature prediction.

In this paper, the original Choquet integral is generalized as a Fuzzified Choquet Integral with a Fuzzy-valued Integrand (FCIFI), which supports a fuzzy-valued integrand and an integration result. The calculation of the FCIFI is established on the Choquet integral with an interval-valued integrand (CIII). The definitions, properties, and calculation algorithms of the CIII and the FCIFI are discussed and proposed in this paper. As a specific application scheme, we designed a CIII regression model for the regression problems involving interval-valued data. This CIII regression model has a self-learning ability through a double genetic algorithm. Finally, a daily temperature predictor based on the CIII regression model is discussed, where a series of experiments is implemented to validate the performance of the predictor by real weather records from the Hong Kong Observatory.     

A boundary integral method applied to plates of arbitrary plan form and arbitrary boundary conditions

An integral equation method which has been applied to thin elastic clamped plates of arbitrary plan form[1] has been extended to include arbitrary boundary conditions. Numerical difficulties which arise in the case of free boundaries have been avoided by defining an integration contour which differs from the actual plate boundary. Thus second order singularities which arise in the integrand of the equations are avoided.The method is applied to two rectangular plates with mixed boundary conditions (i.e. clamped, simply-supported and free edges). Average errors are 1.5% for displacements and 3% for bending moments.     

Classification of Heterogeneous Fuzzy Data by Choquet Integral With Fuzzy-Valued Integrand

As a fuzzification of the Choquet integral, the defuzzified choquet integral with fuzzy-valued integrand (DCIFI) takes a fuzzy-valued integrand and gives a crisp-valued integration result. In this paper, the DCIFI acts as a projection to project high-dimensional heterogeneous fuzzy data to one-dimensional crisp data to handle the classification problems involving different data forms, such as crisp data, interval values, fuzzy numbers, and linguistic variables, simultaneously. The nonadditivity of the signed fuzzy measure applied in the DCIFI can represent the interaction among the measurements of features towards the discrimination of classes. Values of the signed fuzzy measure in the DCIFI are considered to be unknown parameters which should be learned before the classifier is used to classify new data. We have implemented a genetic algorithm (GA)-based adaptive classifier-learning algorithm to optimally learn the signed fuzzy measure values and the classified boundaries simultaneously. The performance of our algorithm has been tested both on synthetic and real data. The experimental results are satisfactory and outperform those of existing methods, such as the fuzzy decision trees and the fuzzy-neuro networks.     

一种基于泛函网络求数值积分方法研究

提出了一种基于泛函网络求教值积分新方法,给出了一种泛函网络模型及学习算法,并将该模型用于求任意函数的数值积分,理论上证明了泛函网络用于逼近数值积分定理.最后通过5个数值积分算例,并与传统计算方法作了比较分析,仿真结果表明,提出的数值积分方法精度高,适应性强,且不需要确定被积函数的原函数,因此该方法在工程技术中有较大的应用价值.     

Algorithm 32 automatic computation of integrals with singular integrand,over a finite or an infinite range

A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range. Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at. The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand. Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaner's sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.     

Nonlinear integrals with polynomial kernel and its applications

Nonlinear integrals (NIs) are useful integration tools. It can get a set of virtual values by projecting original data onto a virtual space for classification purpose using NIs. The classical NIs implement projection along a line with respect to the features. But, in many cases, the linear projection cannot achieve good performance for classification or regression due to the limitation of the integrand. The linear function used for the integrand is just a special type of function with respect to the features. In this paper, we propose a nonlinear integrals with polynomial kernel (NIPK). A polynomial function with respect to the features is used as the integrand of NIs. It enables the projection to be along different types of curves to the virtual space so that the virtual values gotten by NIs can be better regularized and have higher separation power for classification. We use genetic algorithm to learn the fuzzy measures so that a larger solution space can be searched. To test the capability of the NIPK, we apply it to classification on several benchmark datasets and a bioinformatics project. Experiments show that there is evident improvement on performance for the NIPK compared to classical NIs. © 2011 Wiley Periodicals, Inc.     

Bidirectional Ray Tracing for the Integration of Illumination by the Quasi-Monte Carlo Method

Algorithms used to generate physically accurate images are usually based on the Monte Carlo methods for the forward and backward ray tracing. These methods are used to numerically solve the light energy transport equation (the rendering equation). Stochastic methods are used because the integration is performed in a high-dimensional space, and the convergence rate of the Monte Carlo methods is independent of the dimension. Nevertheless, modern studies are focused on quasi-random samples that depend on the dimension of the integration space and make it possible to achieve, under certain conditions, a high rate of convergence, which is necessary for interactive applications. In this paper, an approach to the development of an algorithm for the bidirectional ray tracing is suggested that reduces the overheads of the quasi-Monte Carlo integration caused by the high effective dimension and discontinuity of the integrand in the rendering equation. The pseudorandom and quasi-random integration methods are compared using the rendering equations that have analytical solutions.     

应用Laguerre小波算子矩阵求二重数值积分

基于Laguerre小波函数及其对应的积分算子矩阵给出了一个求二重积分的数值方法。该方法通过对被积函数进行恰当的离散化,将二重数值积分问题转化为矩阵运算,从而易于求解、方便计算。该方法不仅适用于积分区域是矩形区域,也适用于二重变限积分的情况。数值算例验证了该方法的可行性及有效性。     

Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals

The Clenshaw-Curtis method for numerical integration is extended to semi-infinite ([0, ] and infinite [-∞, ] intervals. The common framework for both these extensions and for integration on a finite interval is to (1) map the integration domain tol [0,], (2) compute a Fourier sine or cosine approximation to the transformd integrand via interpolation, and (3) integrate the approximation. The interpolation is most easily performed via the sine or cosine cardinal functions, which are discussed in the appendix. The algorithm is mathematically equivalent to expanding the integrand in (mapped or unmapped) Chebyshev polynomials as done by Clenshaw and Curtis, but the trigonometric approach simplifies the mechanics. Like Gaussian quadrature, the error for the change-of-coordinates Fourier method decreases exponentially withN, the number of grid points, but the generalized Curtis-Clenshaw algorithm is much easier to program than Gaussian quadrature because the abscissas and weights are given by simple, explicit formulas.     

Toward Symbolic Integration of Elliptic Integrals

A method is proposed by which elliptic integrals can be integrated symbolically without information regarding limits of integration and branch points of the integrand that is required in integral tables using Legendre’s integrals. However, it is assumed that when all polynomials in the integrand have been factored symbolically into linear factors, the exponents of all distinct linear factors are known. The recurrence relations are one-parameter relations, all formulas are given explicitly, and the integral is eventually expressed in terms of canonicalR-functions, with no increase in their number if neither limit of integration is a branch point of the integrand. It is the use ofR-functions rather than Legendre’s integrals that makes it possible to carry out the whole process symbolically. If (possibly complex) numerical values of the symbols are known, there are published algorithms for numerical computation of theR-functions.     

SISAM and MIXIN: Two algorithms for the computation of posterior moments and densities using Monte Carlo integration

Two algorithms, and corresponding Fortran computer programs, for the computation of posterior moments and densities using the principle of importance sampling are described in detail. The first algorithm makes use of a multivariate Student t importance function as approximation of the posterior. It can be applied when the integrand is moderately skew. The second algorithm makes use of a decomposition: a multivariate normal importance function is used to generate directions (lines) and one-dimensional classical quadrature is used to evaluate the integrals defined on the generated lines. The second algorithm can be used in cases where the integrand is possibly very skew in any direction.     

Direct Integration of the Newton Potential over Cubes

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way. Received April 18, 2001; revised September 17, 2001 Published online April 25, 2002     

Two-dimensional numerical integration using a square mesh

A method is presented to compute a surface integral over an arbitrary domain using only the values of the integrand function at the vertices of a square mesh. The method provides suitable weights to be given only to the points of the mesh close to the boundary of the integration domain, and can be embodied in algorithms of any order n, with an error decreasing as hⁿ⁺¹ with the spacing h of the mesh. Because of the square mesh employed, this method can be used to improve the precision of integral transform algorithms like the Fast Fourier Transform with an insignificant overhead in computation time.     

Representativity for robust and adaptive multiple importance sampling

We present a general method enhancing the robustness of estimators based on multiple importance sampling (MIS) in a numerical integration context. MIS minimizes variance of estimators for a given sampling configuration, but when this configuration is less adapted to the integrand, the resulting estimator suffers from extra variance. We address this issue by introducing the notion of "representativity" of a sampling strategy, and demonstrate how it can be used to increase robustness of estimators, by adapting them to the integrand. We first show how to compute representativities using common rendering informations such as BSDF, photon maps, or caches in order to choose the best sampling strategy for MIS. We then give hints to generalize our method to any integration problem and demonstrate that it can be used successfully to enhance robustness in different common rendering algorithms.     

Constructing lattice rules based on weighted degree of exactness and worst case error

Recall that an integration rule is said to have a trigonometric degree of exactness m if it integrates exactly all trigonometric polynomials of degree ≤ m. In this paper we focus on high dimensions, say, d ? 6. We introduce three notions of weighted degree of exactness, where we use weights to characterize the anisotropicness of the integrand with respect to successive coordinate directions. Unlike in the classical unweighted setting, the minimal number of integration points needed to achieve a prescribed weighted degree of exactness no longer grows exponentially with d provided that the weights decay sufficiently fast. We present a component-by-component algorithm for the construction of a rank-1 lattice rule such that (i) it has a prescribed weighted degree of exactness, and (ii) its worst case error achieves the optimal rate of convergence in a weighted Korobov space. Then we introduce a modified, more practical, version of this algorithm which maximizes the weighted degree of exactness in each step of the construction. Both algorithms are illustrated by numerical results.     

Numerical integration of sparsely sampled data

In experimental work as well as in computational applications for which limited computational resources are available for the numerical calculations a coarse mesh problem frequently appears. In particular, we consider here the problem of numerical integration when the integrand is available only at nodes of a coarse uniform computational grid. Our research is motivated by the coarse mesh problem arising in ecological applications such as pest insect monitoring and control. In our study we formulate a criterion for assessing mesh coarseness and demonstrate that the definition of a coarse mesh depends on the integrand function. We then discuss the accuracy of computations on coarse meshes to conclude that the conventional methods used to improve accuracy on fine meshes cannot be applied to coarse meshes. Our discussion is illustrated by numerical examples.     

Replica Exchange Light Transport

We solve the light transport problem by introducing a novel unbiased Monte Carlo algorithm called replica exchange light transport, inspired by the replica exchange Monte Carlo method in the fields of computational physics and statistical information processing. The replica exchange Monte Carlo method is a sampling technique whose operation resembles simulated annealing in optimization algorithms using a set of sampling distributions. We apply it to the solution of light transport integration by extending the probability density function of an integrand of the integration to a set of distributions. That set of distributions is composed of combinations of the path densities of different path generation types: uniform distributions in the integral domain, explicit and implicit paths in light (particle/photon) tracing, indirect paths in bidirectional path tracing, explicit and implicit paths in path tracing, and implicit caustics paths seen through specular surfaces including the delta function in path tracing. The replica‐exchange light transport algorithm generates a sequence of path samples from each distribution and samples the simultaneous distribution of those distributions as a stationary distribution by using the Markov chain Monte Carlo method. Then the algorithm combines the obtained path samples from each distribution using multiple importance sampling. We compare the images generated with our algorithm to those generated with bidirectional path tracing and Metropolis light transport based on the primary sample space. Our proposing algorithm has better convergence property than bidirectional path tracing and the Metropolis light transport, and it is easy to implement by extending the Metropolis light transport.     

On the evaluation of correction terms in Gauss-Legendre quadrature

In the numerical integration of analytic functions, the singularities of the integrand affect the rate of convergence of the quadrature. This convergence can be improved significantly by adding the residue correction terms for the poles of the integrand. But this needs the evaluation of the basis function and its corresponding second kind function with complex arguments. We indicate a simple and accurate method to evaluate the correction term involving the basis and its second kind functions in the case of Gauss-Legendre quadrature. This approach does not call for the evaluation of the hypergeometric functions.     

A node-centric indirect boundary element method: three-dimensional displacement discontinuities

An indirect boundary element formulation based on unknown physical values, defined only at the nodes (vertices) of a boundary discretization of a linear elastic continuum, is introduced. As an adaptation of this general framework, a linear displacement discontinuity density distribution using a flat triangular boundary discretization is considered. A unified element integration methodology based on the continuation principle is introduced to handle regular as well as near-singular and singular integrals. The boundary functions that form the basis of the integration methodology are derived and tabulated in the appendix for linear displacement discontinuity densities. The integration of the boundary functions is performed numerically using an adaptive algorithm which ensures a specified numerical accuracy. The applications include verification examples which have closed-form analytical solutions as well as practical problems arising in rock engineering. The node-centric displacement discontinuity method is shown to be numerically efficient and robust for such problems.