A comparative study of numerical integration based on Haar wavelets and hybrid functions

A quadrature rule based on uniform Haar wavelets and hybrid functions is proposed to find approximate values of definite integrals. The wavelet-based algorithm can be easily extended to find numerical approximations for double, triple and improper integrals. The main advantage of this method is its efficiency and simple applicability. Error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method.     

Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.     

基于提升方案的冗余Haar小波变换与时间序列预测

针对小波分析存在的边界问题，提出一种基于提升方案的冗余Haar小波变换(Haar_RLWT)。使用该方法得到的系数序列，在具备时移不变性的同时，消除了右侧边界存在数据畸变的现象，使小波分析技术结合神经网络等传统预测模型的方法应用于时间序列预测任务具备可行性。同时为进一步提高预测效果，引入神经网络集成技术以改善网络泛化能力。实验表明，该综合预测模型预测效果与稳定性优于传统预测模型。     

Design of hybrid differential evolution and group method of data handling networks for modeling and prediction

This paper proposes a hybrid modeling approach based on two familiar non-linear methods of mathematical modeling; the group method of data handling (GMDH) and differential evolution (DE) population-based algorithm. The proposed method constructs a GMDH self-organizing network model of a population of promising DE solutions. The new hybrid implementation is then applied to modeling tool wear in milling operations and also applied to two representative time series prediction problems of exchange rates of three international currencies and the well-studied Box-Jenkins gas furnace process data. The results of the proposed DE–GMDH approach are compared with the results obtained by the standard GMDH algorithm and its variants. Results presented show that the proposed DE–GMDH algorithm appears to perform better than the standard GMDH algorithm and the polynomial neural network (PNN) model for the tool wear problem. For the exchange rate problem, the results of the proposed DE–GMDH algorithm are competitive with all other approaches except in one case. For the Box-Jenkins gas furnace data, the experimental results clearly demonstrates that the proposed DE–GMDH-type network outperforms the existing models both in terms of better approximation capabilities as well as generalization abilities. Consequently, this self-organizing modeling approach may be useful in modeling advanced manufacturing systems where it is necessary to model tool wear during machining operations, and in time series applications such as in prediction of time series exchange rate and industrial gas furnace problems.     

Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods

Assumed stress hybrid methods are known to improve the performance of standard displacement-based finite elements and are widely used in computational mechanics. The methods are based on the Hellinger–Reissner variational principle for the displacement and stress variables. This work analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian and Sumihara [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg. 20 (9) (1984) 1685–1695] and due to Xie and Zhou [X.P. Xie, T.X. Zhou, Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Methods Engrg. 59 (2004) 293–313], which behave robustly in numerical benchmark tests. For the finite elements, the isoparametric bilinear interpolation is used for the displacement approximation, while different piecewise-independent 5-parameter modes are employed for the stress approximation. We show that the two schemes are free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant λ. We also establish the equivalence of the methods to two assumed enhanced strain schemes. Finally, we derive reliable and efficient residual-based a posteriori error estimators for the stress in L²-norm and the displacement in H¹-norm, and verify the theoretical results by some numerical experiments.     

Coupling strategy for matching different methods in solving singularity problems

For solving elliptic boundary value problems with singularities, we have proposed the combined methods consisting of the Ritz-Galerkin method using singular (or analytic) basic functions for one part,S ₂, of the solution domainS, where there exist singular points, and the finite element method for the remaining partS ₁ ofS, where the solution is smooth enough. In this paper, general approaches using additional integrals are presented to match different numerical methods along their common boundary Г₀. Errors and stability analyses are provided for such a general coupling strategy. These analyses are important because they form a theoretical basis for a number of combinations between the Ritz-Galerkin and finite element methods addressed in [7], and because they can lead to new combinations of other methods, such as the combined methods of the Ritz-Galerkin and finite difference methods. Moreover, the analyses in this paper can be applied or extended to solve general elliptic boundary value problems with angular singularities, interface singularity or unbounded domain.     

量子门Elman神经网络及其梯度扩展的量子反向传播学习算法

针对Elman神经网络的学习速度和泛化性能, 提出一种具有量子门结构的新型Elman神经网络模型及其梯度扩展反向传播(Back-propagation)学习算法, 新模型由量子比特神经元和经典神经元构成. 新网络结构采用量子映射层以确保来自上下文单元的局部反馈与隐藏层输入之间的模式一致; 通过量子比特神经元输出与相关量子门参数的修正互补关系以提高网络更新动力. 新学习算法采用搜索然后收敛的策略自适应地调整学习率参数以提高网络学习速度; 通过将上下文单元的权值扩展到隐藏层的权值矩阵, 使其在与隐藏层权值同步更新过程中获取时间序列的额外信息, 从而提高网络上下文单元输出与隐藏层输入之间的匹配程度. 以峰值检波为例的数值实验结果显示, 在量子反向传播学习过程中, 量子门Elman神经网络具有较快的学习速度和良好的泛化性能.     

Hybrid Difference Methods for PDEs

We propose the hybrid difference methods for partial differential equations (PDEs). The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the interface finite difference (interface FD) on edges of cells. The interface finite difference is obtained from continuity of some physical quantities. The main advantages of this new approach are that the method can applied to non-uniform grids, retaining the optimal order of convergence and stability of the numerical method for the Stokes equations is obtained without introducing staggered grids.     

State analysis of linear time delayed systems via Haar wavelets

State analysis of time delayed systems via Haar wavelets are proposed in this paper. Based upon some useful properties of Haar functions, a special product matrix and a related coefficient matrix are applied to solve the time-delayed systems. The unknown Haar coefficient matrix is solved via the Kronecker product method. The high accuracy and the wide applicability of Haar approach will be demonstrated with numerical examples.     

Heuristic configuration of single hidden-layer feed-forward neural networks

For optimum statistical classification and generalization with single hidden-layer neural network models, two tasks must be performed: (a) learning the best set of weights for a network of k hidden units and (b) determining k, the best complexity fit. We contrast two approaches to construction of neural network classifiers: (a) standard back-propagation as applied to a series of single hidden-layer feed-forward nerual networks with differing number of hidden units and (b) a heuristic cascade-correlation approach that quickly and dynamically configures the hidden units in a network and learns the best set of weights for it. Four real-world applications are considered. On these examples, the back-propagation approach yielded somewhat better results, but with far greater computation times. The best complexity fit, k, for both approaches were quite similar. This suggests a hybrid approach to constructing single hidden-layer feed-forward neural network classifiers in which the number of hidden units is determined by cascade-correlation and the weights are learned by back-propagation.     

Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau method

A new hybrid computational technique based on Ortiz' recursive formulation of the Tau method is introduced in this paper and applied to some model singular boundary value problems which are relevant to fracture mechanics (modes I and III). This technique, which we call Tau-lines, combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of a given partial differential equation; the latter is used to find an accurate approximation of the solution of such a system which involves no further discretization.Recent theoretical results on the Tau method show that its error is optimal in the sense that, for a given degree n, it has the same order of error as the best uniform approximation of the exact solution by algebraic polynomials of degree n.The present work may be considered as an encouraging first step towards the development of the Tau-lines approach into a useful and efficient computational tool for the numerical treatment of problems in fracture mechanics.     

Performance enhancement of extreme learning machine for multi-category sparse data classification problems

This paper presents a performance enhancement scheme for the recently developed extreme learning machine (ELM) for multi-category sparse data classification problems. ELM is a single hidden layer neural network with good generalization capabilities and extremely fast learning capacity. In ELM, the input weights are randomly chosen and the output weights are analytically calculated. The generalization performance of the ELM algorithm for sparse data classification problem depends critically on three free parameters. They are, the number of hidden neurons, the input weights and the bias values which need to be optimally chosen. Selection of these parameters for the best performance of ELM involves a complex optimization problem.In this paper, we present a new, real-coded genetic algorithm approach called ‘RCGA-ELM’ to select the optimal number of hidden neurons, input weights and bias values which results in better performance. Two new genetic operators called ‘network based operator’ and ‘weight based operator’ are proposed to find a compact network with higher generalization performance. We also present an alternate and less computationally intensive approach called ‘sparse-ELM’. Sparse-ELM searches for the best parameters of ELM using K-fold validation. A multi-class human cancer classification problem using micro-array gene expression data (which is sparse), is used for evaluating the performance of the two schemes. Results indicate that the proposed RCGA-ELM and sparse-ELM significantly improve ELM performance for sparse multi-category classification problems.     

Evolutionary extreme learning machine

Extreme learning machine (ELM) [G.-B. Huang, Q.-Y. Zhu, C.-K. Siew, Extreme learning machine: a new learning scheme of feedforward neural networks, in: Proceedings of the International Joint Conference on Neural Networks (IJCNN2004), Budapest, Hungary, 25-29 July 2004], a novel learning algorithm much faster than the traditional gradient-based learning algorithms, was proposed recently for single-hidden-layer feedforward neural networks (SLFNs). However, ELM may need higher number of hidden neurons due to the random determination of the input weights and hidden biases. In this paper, a hybrid learning algorithm is proposed which uses the differential evolutionary algorithm to select the input weights and Moore-Penrose (MP) generalized inverse to analytically determine the output weights. Experimental results show that this approach is able to achieve good generalization performance with much more compact networks.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

New high-order convergence iteration methods without employing derivatives for solving nonlinear equations

A family of new iteration methods without employing derivatives is proposed in this paper. We have proved that these new methods are quadratic convergence. Their efficiency is demonstrated by numerical experiments. The numerical experiments show that our algorithms are comparable to well-known methods of Newton and Steffensen. Furthermore, combining the new method with bisection method we construct another new high-order iteration method with nice asymptotic convergence properties of the diameters (b_n − a_n).     

Loop-free Gray code algorithm for the e-restricted growth functions

The subject of Gray codes algorithms for the set partitions of {1,2,…,n} had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of Knuth?s algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of {1,2,…,n}, Ruskey and Savage (1994) [9] generalized Ehrlich?s results and give two Gray codes for the set of partitions of {1,2,…,n}, and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques.In this paper, we introduce the set of e-restricted growth functions (a generalization of restricted growth functions) and extend the aforementioned results by giving a Gray code with distance one for this set; and as a particular case we obtain a new Gray code for set partitions in restricted growth function representation. Our Gray code satisfies some prefix properties and can be implemented by a loop-free generating algorithm using classical techniques; such algorithms can be used as a practical solution of some difficult problems. Finally, we give some enumerative results concerning the restricted growth functions of order d.     

A robust conjugate-gradient algorithm which minimizes L-functions

A new conjugate-gradient algorithm which minimizes a function of n variables is given. The algorithm performs n orthogonal searches in each stage and hence has the property that it is robust, i.e. it will not easily fail on functions which have a large number of variables (n ? 10) nor on functions which have ‘ridge-like’ properties. A general class of functions called L-functions, which includes the class of quadratic functions as a special case, is defined, and it is shown that the algorithm has the property that it will converge to the minimum of an L-function in n (or less) 1-dimensional minimizations and (n ? 1) (or less) 1-dimensional pseudo-minimizations. Numerical experiments are included for systems of the second to the fortieth order, and based on these experiments, (assuming that the gradients are calculated numerically), the new algorithm appears to be more robust than Powell's [10], Fletcher-Powell's [11], and Jacobson-Oksman's [14] methods, faster than Rosenbrock's [9] method, and especially effective on high dimensional problems.     

An Implementation of Fast Wavelet Galerkin Methods for Integral Equations of the Second Kind

We present a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets. We focus on addressing critical issues for the numerical implementation of such a method. They include a choice of practical truncation strategy, numerical integration of weakly singular integrals and the error control of the numerical quadrature. We also implement a multiscale iteration method for solving the resulting compressed linear system. Numerical examples are given to demonstrate the proposed ideas and methods.     

An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals

In this paper, a robust method is presented for numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals, which exist in the Cauchy principal value sense in two- and three-dimensional problems. In this method, the singularities involved in integration kernels are analytically removed by expressing the non-singular parts of the integration kernels as power series in the local distance ρ of the intrinsic coordinate system. For three-dimensional boundary integrals, the radial integration method [1] is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals. Some examples are provided to verify the correctness and robustness of the presented method.