On properties of the iterative maximum likelihood reconstruction method

In this paper, we continue our investigations⁶ on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback–Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementary proof of convergence of the iterative method is given for the discrete case. Numerical examples have already been given in References 6. Here, an example is considered which can be worked out analytically and which demonstrates fundamental properties of the algorithm.     

On nonlinear Fredholm integral equations with non-differentiable Nemystkii operator

From decomposition method for operators, we consider a Newton-Steffensen iterative scheme for approximating a solution of nonlinear Fredholm integral equations with non-differentiable Nemystkii operator. By means of a convergence study of the iterative scheme applied to this type of nonlinear Fredholm integral equations, we obtain domains of existence and uniqueness of solution for these equations. In addition, we illustrate this study with a numerical experiment.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems

The integral equations of acoustic and electromagnetic scattering generate large dense systems of linear equations. These systems are efficiently solved with iterative methods where the matrix-vector multiplication is computed using a special fast method, such as the fast Fourier transform or the fast multipole method (FMM). In this paper, the so called diagonal forms of the translation operators for the fast multipole method are derived starting from integral representations of certain special functions. Error analysis of the FMM is given, considering both the truncation error of potential expansions and the errors from the use of numerical integration in the diagonal translation theorem. The implications of the error bounds on the FMM algorithm are discussed.This work has been financially supported by the Jenny and Antti Wihuri Foundation and by the Cultural Foundation of Finland.     

A Steffensen's type method in Banach spaces with applications on boundary-value problems

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.     

类Hartree-Fock方程的数值方法

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系（含几十到几百个电子）的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法（ACE）,投影的C-DⅡS方法（PC-DⅡS）方法,以及插值可分密度近似方法（ISDF）,我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

极正交各向异性圆板非线性弯曲的定性分析及单调迭代解

本文对极正交各向异性圆板在任意轴对称载荷和边界条件下的非线性弯曲问题进行了较为系统的研究.首先,将边值问题归结为等价的积分方程,并且借助于广义函数得到了线性问题的一般解答.其次,对导出的非线性积分方程解的性质作了较为细致的讨论,例如边缘皱褶,非负性和奇性等.然后,构造了解的双边单调迭代格式,并给出了迭代格式的收敛性判据和误差估计,同时还讨论了解的全局存在唯一性.最后,给出了一个数值例子来说明本文方法和结论的应用.本文某些结果是由作者新得到的.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.     

Error correction iterative method for the stationary incompressible MHD flow

A new iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations is derived in this paper. The method consists of two steps at each iteration step, we need first to solve the MHD equations by the Oseen-type iterative scheme, and then an error correction strategy is applied to control the error arising from the linearization of the nonlinear MHD equations. The new method not only maintains the advantage of the standard Oseen-type scheme but also possesses a rapid rate of convergence. It is proved that the convergence rate of the proposed method is increased greatly under the uniqueness condition. The uniform stability and convergence of the new scheme are analyzed. Ample numerical experiments are performed to validate the accuracy and the efficiency of the new numerical scheme.     

Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a systemin some special cases leading to Bessel type differential equations. Next, a finite difference scheme in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.     

Block by block method for the systems of nonlinear Volterra integral equations

The approach given in this paper leads to numerical methods for solving system of Volterra integral equations which avoid the need for special starting procedures. The method has also the advantages of simplicity of application and at least four order of convergence which is easy to achieve. Also, at each step we get four unknowns simultaneously. A convergence theorem is proved for the described method. Finally numerical examples presented to certify convergence and accuracy of the method.     

A fifth-order iterative method for solving nonlinear equations

The object of this paper is to construct a new efficient iterative method for solving nonlinear equations. This method is mainly based on Javidi paper [1] by using a new scheme of a modified homotopy perturbation method. This new method is of the fifth order of convergence, and it is compared with the second-, third-, fifth-, and sixth-ordermethods. Some numerical test problems are given to show the accuracy and fast convergence of the method proposed.     

Convergence and spectral analysis of the Frontini-Sormani family of multipoint third order methods from quadrature rule

Point of attraction theory is an important tool to analyze the local convergence of iterative methods for solving systems of nonlinear equations. In this work, we prove a generalized form of Ortega-Rheinbolt result based on point of attraction theory. The new result guarantees that the solution of the nonlinear system is a point of attraction of iterative scheme, especially multipoint iterations. We then apply it to study the attraction theorem of the Frontini-Sormani family of multipoint third order methods from Quadrature Rule. Error estimates are given and compared with existing ones. We also obtain the radius of convergence of the special members of the family. Two numerical examples are provided to illustrate the theory. Further, a spectral analysis of the Discrete Fourier Transform of the numerical errors is conducted in order to find the best method of the family. The convergence and the spectral analysis of a multistep version of one of the special member of the family are studied.     

Newton methods for a class of nonlinear hypersingular integral equations

Different iterative schemes based on collocation methods have been well studied and widely applied to the numerical solution of nonlinear hypersingular integral equations (Capobianco et al. 2005). In this paper we apply Newton’s method and its modified version to solve the equations obtained by applying a collocation method to a nonlinear hypersingular integral equation of Prandtl’s type. The corresponding convergence results are derived in suitable Sobolev spaces. Some numerical tests are also presented to validate the theoretical results.     

求解一类特殊非光滑极大值函数方程的光滑保守DPRP共轭梯度法

对一类特殊极大值函数非光滑方程问题的方法进行了研究, 利用极大值函数和绝对值函数的光滑函数对提出的非光滑方程问题进行转化, 提出了一种光滑保守DPRP共轭梯度法. 在一般的条件下, 给出了光滑保守DPRP共轭梯度法的全局收敛性, 最后给出相关的数值实验表明方法的有效性.     

Theoretical investigation on error analysis of Sinc approximation for mixed Volterra-Fredholm integral equation

In this study, we propose one of the new techniques used in solving numerical problems involving integral equations known as the Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this article, a mixed Volterra-Fredholm integral equation which has been appeared in many science an engineering phenomena is discredited by using some properties of the Sinc-collocation method and Sinc quadrature rule to reduce integral equation to some algebraic equations. Then exponential convergence rate of this numerical technique is discussed by preparing a theorem. Finally, some numerical examples are included to demonstrate the validity and applicability of the convergence theorem and numerical scheme.