Numerical Integration Formulas Research Articles

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

This article deals with the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.

Numerical integration rules with improved accuracy close to discontinuities

This work is devoted to the construction and analysis of a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depend on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. We include several numerical experiments that confirm the theoretical conclusions reached in this article.

Online deep Bingham network for probabilistic orientation estimation

AbstractOrientation estimation is one of the core problems in several computer vision tasks. Recently deep learning techniques combined with the Bingham distribution have attracted considerable interest towards this problem when considering ambiguities and rotational symmetries of objects. However, existing works suffer from two issues. First, the computational overhead for calculating the normalisation constant of the Bingham distribution is relatively high. Second, the choice of loss functions is uncertain. In light of these problems, we present an online deep Bingham network to estimate the orientation of objects. We sharply reduce the computational overhead of the normalisation constant by directly applying a numerical integration formula. Additionally, we are the first to give theorems on the convexity and Lipschitz continuity of the Bingham distribution's negative log‐likelihood, which formally indicates that it is a proper choice of the loss function. We test our method on three public datasets, namely the UPNA, the T‐LESS and Pascal3D+, showing that our method outperforms the state‐of‐the‐art in terms of orientation accuracy and time efficiency, which can reduce the runtime by more than 6 h compared to the offline methods. The ablation experiments further demonstrate the effectiveness and robustness of our model.

COMPUTATIONAL FEATURES OF CALCULAT-ING EXTENDED CONDUCTOR ANTENNAS BASED ON THE INTEGRAL EQUATIONS METHOD

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

New 2D numerical integration formula based on the Legendre wavelets

Based on Legendre wavelets, a new efficient numerical integration method is proposed to estimate double integrals. This new method is expressed in terms of the operational integration matrices. Those which were introduced by Parsian, are calculated in an approximate way. In this work, we also propose the exact computation of these matrices. Better precision of the new method is observed during a comparative numerical study with the composite Simpson method.

A Novel Ergodic Cellular Automaton Model of Gene-Protein Network: Theoretical Nonlinear Analyses and Efficient FPGA Implementation

A novel ergodic cellular automaton model of gene-protein network is presented. It is shown that the presented model can predict occurrences of typical nonlinear phenomena of a conventional ordinary differential equation gene-protein network model. In addition, theoretical analysis methods of the presented model are proposed. Using the analysis methods, an important advantage of the presented model is revealed: the ergodic cellular automaton is better suited to predict the occurrences of the nonlinear phenomena of the differential equation gene-protein network model compared to a regular (standard) cellular automaton. Furthermore, the presented model is implemented by a field programmable gate array and experiments validate its operations. It is then revealed that the presented model is much more hardware-efficient compared to a standard numerical integration formula of the differential equation model.

Iterative Numerical Methods for a Fredholm–Hammerstein Integral Equation with Modified Argument

Iterative processes are a powerful tool for providing numerical methods for integral equations of the second kind. Integral equations with symmetric kernels are extensively used to model problems, e.g., optimization, electronic and optic problems. We analyze iterative methods for Fredholm–Hammerstein integral equations with modified argument. The approximation consists of two parts, a fixed point result and a quadrature formula. We derive a method that uses a Picard iterative process and the trapezium numerical integration formula, for which we prove convergence and give error estimates. Numerical experiments show the applicability of the method and the agreement with the theoretical results.

Estimates on some quadrature rules via weighted Hermite-Hadamard inequality

In this article new estimates on some quadrature rules are given using weighted Hermite-Hadamard inequality for higher order convex functions and weighted version of the integral identity expressed by w-harmonic sequences of functions. Obtained results are applied to weighted one-point formula for numerical integration in order to derive new estimates of the definite integral values.

On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

In the present paper, we establish an efficient numerical scheme based on weakly L‐stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burgers' equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomial of order five and backward explicit Taylor's series approximation of order six to derive the numerical integration formula for the initial value problem (IVP) . We combine this method with the NSFD scheme and convert the problem into the system of algebraic equations. We discuss the convergence, truncation error, and stability of the developed method. To demonstrate the efficiency of the developed method, we compare the numerical results with some existing numerical results and exact solutions.

Effect of numerical integration on a new rotated nonconforming quadrilateral element

A new nonparametric nonconforming quadrilateral finite element is used to approximate the general second-order elliptic problem in two dimensions. Some optimal numerical integration formulas are presented and analyzed. These formulas are derived on a reference quadrilateral which can be linearly mapped to a physical quadrilateral. The novelty of these formulas is that they only involve two quadrature nodes which excludes even a Q1 unisolvent set, and they are not required to be exact for all the shape functions. Numerical tests show that the presented quadrature rules can also be used coupled with other low-order elements.

Vibration effect correction method of inclinometer in intermittent transonic wind tunnel

In order to solve the problem that the servo accelerometer-based inclinometer is always affected by vibration of the aerodynamic test model when measuring the angle of attack in the intermittent transonic wind tunnel, a vibration effect correction method based on two tri-axial accelerometers is proposed in this paper. In this method, the tri-axial accelerometers are symmetrically mounted on the front and rear sides of the inclinometer along the sensitive axis, and then the centrifugal accelerations caused by different vibration modes and different directions are calculated by time-frequency analysis, numerical integration and equivalent formula. Finally, the total angle measurement error can be accumulated by the angle measurement error caused by each vibration mode. Experimental results of intermittent transonic wind tunnel show that the difference between the proposed method and the Elastic Angle Correction method is less than 0.05 degree when the vibration peak is less than 5 g.

Maximum likelihood estimation of the Fisher–Bingham distribution via efficient calculation of its normalizing constant

This paper proposes an efficient numerical integration formula to compute the normalizing constant of Fisher–Bingham distributions. This formula uses a numerical integration formula with the continuous Euler transform to a Fourier-type integral representation of the normalizing constant. As this method is fast and accurate, it can be applied to the calculation of the normalizing constant of high-dimensional Fisher–Bingham distributions. More precisely, the error decays exponentially with an increase in the integration points, and the computation cost increases linearly with the dimensions. In addition, this formula is useful for calculating the gradient and Hessian matrix of the normalizing constant. Therefore, we apply this formula to efficiently calculate the maximum likelihood estimation (MLE) of high-dimensional data. Finally, we apply the MLE to the hyperspherical variational auto-encoder (S-VAE), a deep-learning-based generative model that restricts the latent space to a unit hypersphere. We use the S-VAE trained with images of handwritten numbers to estimate the distributions of each label. This application is useful for adding new labels to the models.

APPLICATION OF THE THEORY OF NEW INFORMATION OPERATORS IN CONDUCTING RESEARCH IN THE FIELD OF INFORMATION TECHNOLOGIES

Due to the rapid development and implementation of the latest information technologies in many fields of science and technology significant changes have taken place. In particular, new mathematical theories have emerged; they can be effectively used in building and improving existing mathematical models of various phenomena and objects. Mathematical modeling is currently one of the main methods in scientific research of technological, social, economic processes and natural phenomena. No research thesis is complete without the use of mathematical methods and algorithms. In view of this it is important to acquaint young scientists with modern trends in mathematical modeling during their training for PhD degree, in particular. The development of information technology has contributed to the emergence of new approaches to obtaining, processing and analyzing information in scientific research. In this way, the theory of new information operators, which is widely used in mathematical modeling, was created. The analysis of the existing research reveals the need to consider the theoretical basics of new information operators and their application to the numerical integration of several variables functions. The example of the problem of numerical integration of two variables functions shows the way how to create cubature formulas depending on the type of the input information about function. The paper presents new cubature formulas for the approximate calculation of double integrals, in the case when the information is given by the set of traces on lines and values of the function at points. The proposed formulas have high accuracy but use much less input information compared to classical formulas for numerical integration of double integrals. All the results are supported by numerous examples made in the computer mathematics system Mathcad. The implementation of the developed methodology of using new information operators in scientific research has shown its efficacy during seminars with Master and PhD students.

Analysis of time-varying meshing stiffness of helical gear pair with tooth surface spalling

Time-varying meshing stiffness (TVMS) provides significant information about the health of gear systems. Tooth spalling will change the TVMS. Considering the contact state of gear pair, the purpose of this study is to investigate the influence of tooth surface spalling on the TVMS of helical gears. Firstly, the meshing stiffness analysis algorithm of helical gear pair with tooth surface spalling is proposed, based on the potential energy method, slicing method and numerical integration formula. Then the influence of different spalling parameters on the TVMS of helical gear pair is analysed by considering spalling length, width and the oblique cylinder bottom shape. Finally, the results show that the contact line falls into the defect area, the spalling has the most significant effect on the stiffness. The influence of oblique cylinder shape at the bottom of spalling is mainly affected after the contact line passes through the spalling area. This study improve the theoretical basis for spalling fault of helical geared mesh.

Adaptive numerical integration of exponential finite elements for a phase field fracture model

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.

Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

Energy consumption forecasting using improved non-homogeneous exponential grey model

The classical non-homogeneous exponential grey model is a valid tool for accurate prediction with small samples. But the classical non-homogeneous exponential grey model needs to be improved for higher accuracy. In this paper, a new discrete non-homogeneous exponential grey model is proposed, and the background value of the new model is given by Simpson numerical integration formula. Further, the electricity consumption is given to validate the accuracy of the new model and the result illustrates the new model provides accurate prediction. In the example, the mean absolute percentage error is 2.14%. Finally, the new model is used to forecast the consumption of energy for China in 2025 and the result is 1076.08 million tons of standard coal. It is of great significance to make scientific decisions and plans on energy development.

Transmutation operators and a new representation for solutions of perturbed Bessel equations

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form are obtained. The integral kernel is represented as a Fourier‐Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones previously obtained by the authors for large values of ω and ℓ and for noninteger values of ℓ. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley‐Wiener theorem, and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.

Applications of numerical integration in geodesy and geophysics

In this paper, two applications of numerical integration in geodesy and geophysics are presented. In the first application, the Molodenskij truncation coefficients for the Abel-Poisson kernel are computed using eleven different numerical integration procedures, namely two-, three-, four-, and five-point Gaussian, Gauss–Kronrod, trapezoidal rule, Simpson and its adaptive mode, Romberg, Lobatto, and Sard’s approximating functional numerical integration methods. The coefficients are computed for truncation degree 90, and truncation radius $$6^\circ$$ . The results are then compared with an independent method for calculating these coefficients. It is shown that numerical integration methods represent better accuracy. In the second application, the gravity accelerations at sea surface in Qeshm in southern Iran are calculated using the spherical spline numerical integration method. The formulae for spherical spline numerical integration in two different modes weighted and without weight are derived. The special case when the weight of the integral is the so-called Stokes’ kernel is thoroughly investigated. Then, the results are used to generate gravity accelerations. First, the geoid height from the sum of the mean sea level and sea surface topography is calculated. Then, a spherical spline analytical representation—with unknown coefficients—is considered for the gravity anomaly. In the next step, using the Stokes’ formula for the integral relation between geoid height and gravity anomaly, the unknown coefficients in the previous step are calculated and subsequently the gravity anomalies are derived. Adding the gravity of the reference ellipsoid to the gravity anomalies, the actual gravity accelerations at sea surface in Qeshm are calculated. To analyze the accuracy, the derived values are compared with the values observed by shipborne gravimetry. It is shown that using Bernstein polynomials as basis function for calculating numerical integration has a better accuracy than other numerical integration methods of the same degree.

Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations

In this article, a new collocation technique for numerical solution of Fredholm, Volterra and mixed Volterra-Fredholm integral equations of the second kind is introduced and also developed a numerical integration formula on the basis of linear Legendre multi-wavelets. We also use the linear Legendre multi-wavelets basis for the proposed method. In this technique, the unknown function is approximated by truncated linear Legendre multi-wavelets series. The newly developed numerical technique is applied to both linear and nonlinear benchmark test models from the literature including models with discontinues and non-differentiable exact solutions. The numerical results are compared with the other existing numerical techniques from literature. The comparison among the discussed methods with Monte Carlo method, rationalized Haar functions method, Operational matrix with block-pulse functions method, Bernstein operational matrices method and P-order spline direct method divulge that the present technique is authentic and valid for other physical and engineering problems. The proposed numerical method handle the discontinuity and non-differentiability in the exact solutions very well, whereas the other existence numerical methods are failed to capture these problems. Further, more the comparison of exact and approximate in term of L∞ norms which demonstrate the accuracy, flexibility and robustness of the newly proposed numerical method.