An approximation method based on MRA for the quasi-Plateau problem

In this paper, we discuss how to obtain the parametric surface of minimal area defined on a rectangular parameter domain among all the surfaces with prescribed entire or partial borders, i.e, the quasi-Plateau problem. An approximation method is presented based on the Multiresolution Analysis (MRA) using B-splines. Four algorithms are proposed according to different cases of the boundary conditions. The solution surfaces can be obtained by solving a system of linear equations of simple and sparse structure. Examples are provided to illustrate that the proposed method is effective and flexible.     

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.     

Iterative solutions to the Kalman-Yakubovich-conjugate matrix equation

Two operations are introduced for complex matrices. In terms of these two operations an infinite series expression is obtained for the unique solution of the Kalman-Yakubovich-conjugate matrix equation. Based on the obtained explicit solution, some iterative algorithms are given for solving this class of matrix equations. Convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices.     

Efficient analytic method for solving nonlinear fractional differential equations

In this paper, we propose a new approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of generalized differential transform (GDT) associated with the solution in the given fractional equation. However, the recurrence relations of complex nonlinear functions such as exponential, logarithmic and trigonometry functions have not been derived before in GDTM. We propose new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve nonlinear fractional differential equations. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed technique is robust and accurate for solving fractional differential equations.     

SEQUENTIAL SYSTEMS OF LINEAR EQUATIONS ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS-INEQUALITY CONSTRAINED PROBLEMS

AbstractIn this paper, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints is proposed. Since the new algorithm only needs to solve several systems of linear equations having a same coefficient matrix per iteration, the computation amount of the algorithm is much less than that of the existing SQP algorithms per iteration. Moreover, for the SQP type algorithms, there exist so-called inconsistent problems, i.e., quadratic programming subproblems of the SQP algorithms may not have a solution at some iterations, but this phenomenon will not occur with the SSLE algorithms because the related systems of linear equations always have solutions. Some numerical results are reported.     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Efficient Trefftz collocation algorithms for elliptic problems in circular domains

We consider the numerical solution of certain elliptic boundary value problems in disks and annuli using the Trefftz collocation method. In particular we examine boundary value problems for the Laplace, Helmholtz, modified Helmholtz and biharmonic equations in such domains. It is shown that this approach leads to systems in which the matrices possess specific structures. By exploiting these structures we propose efficient algorithms for the solution of the systems. The proposed algorithms are applied to standard test problems.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Multilevel algorithms for ill-posed problems

Summary In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.Dedicated to Jim Bramble on the occasion of his sixtieth birthday     

Multi-grid methods for Hamilton-Jacobi-Bellman equations

Summary In this paper we develop multi-grid algorithms for the numerical solution of Hamilton-Jacobi-Bellman equations. The proposed schemes result from a combination of standard multi-grid techniques and the iterative methods used by Lions and mercier in [11]. A convergence result is given and the efficiency of the algorithms is illustrated by some numerical examples.     

电力系统中地网腐蚀诊断的一种新的数学模型及其仿真计算

对电力系统中具有重大应用价值的地网腐蚀诊断问题抽象出仿真求解的一种新的数学模型:即求解带约束的非线性隐式方程组模型.但由于问题本身的物理特性决定了所建立的数学模型具有以下特点:一是非线性方程组为欠定方程组,而且非线性程度非常高;二是方程组的所有函数均为隐函数;三是方程组附加若干箱约束条件.这种特性给模型分析与算法设计带来巨大困难.对于欠定方程组的求解,文中根据工程实际背景,尽可能地扩充方程的个数,使之成为超定方程组,然后对欠定方程组和超定方程组分别求解并进行比较.将带约束的非线性隐函数方程组求解问题,转化为无约束非线性最小二乘问题,并采用矩阵求导等技术和各种算法设计技巧克服隐函数的计算困难,最后使用拟牛顿信赖域方法进行计算.大量的计算实例表明,文中所提出的数学模型及求解方法是可行的.与目前广泛采用的工程简化模型相比较,在模型和算法上具有很大优势.     

一类Riccati方程组对称自反解的两种迭代算法

针对源于Markov跳变线性二次控制问题中的一类对偶代数Riccati方程组,分别采用修正共轭梯度算法和正交投影算法作为非精确Newton算法的内迭代方法,建立求其对称自反解的非精确Newton-MCG算法和非精确Newton-OGP算法.两种迭代算法仅要求Riccati方程组存在对称自反解,对系数矩阵等没有附加限定.数值算例表明,两种迭代算法是有效的.     

Complexity of dense-linear-system solution on a multiprocessor ring

Different algorithms, based on Gaussian elimination, for the solution of dense linear systems of equations are discussed for a multiprocessor ring. The number of processors is assumed not to exceed the problem size. A fairly general model for data transfer is proposed, and the algorithms are analyzed with respect to their requirements of arithmetic as well as communication times.     

New algorithms for the evaluation of complex bifurcation points in ordinary differential equations. a comparative numerical study

Four iterative algorithms (two of them new) for the evaluation of complex (Hopf) bifurcation points in ordinary differential equations are compared. A comparison of effectiveness of the proposed algorithms is made for two examples taken from chemical reaction engineering. The applicability of the algorithms for the solution of various types of problems is also discussed.     

Semi-analytic time-marching algorithms for semi-linear parabolic equations

New time marching algorithms for numerical solution of semi-linear parabolic equations are described. They are based on the approximation method proposed by the first author. An important feature of the algorithms is that they are both explicit and stable under mild restrictions to the time step, which come from the non-linear part of the equation.     

Partitioned treatment of surface-coupled problems with application to the fluid-porous-media interaction

Employing a decoupled solution strategy for the numerical treatment of the set of governing equations describing a surface-coupled phenomenon is a common practice. In this regard, many partitioned solution algorithms have been developed, which usually either belong to the family of Schur-complement methods or to the group of staggered integration schemes. To select a decoupled solution strategy over another is, however, a case-dependent process that should be done with special care. In particular, the performances of the algorithms from the viewpoints of stability and accuracy of the results on the one hand, and the solution speed on the other hand should be investigated. In this contribution, two strategies for a partitioned treatment of the surface-coupled problem of fluid-porous-media interaction (FPMI) are considered. These are one parallel solution algorithm, which is based on the method of localised Lagrange multipliers (LLM), and one sequential solution method, which follows the block-Gauss-Seidel (BGS) integration strategy. In order to investigate the performances of the proposed schemes, an exemplary initial-boundary-value problem is considered and the numerical results obtained by employing the solution algorithms are compared. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

Some algorithms for approximate solution of convolution integral equations

We study algorithms for approximate solution of convolution integral equations of first kind in which an approximate solution of the original equations can be constructed using the solution of equations having simpler structure or certain necessary properties, for example regularized equations. We obtain a number of algorithms for recursive refinement of the solution, where if the a priori information is given by linear algorithms, such algorithms are pseudo-recursive. The results may have application in problems of secondary processing of the data of indirect measurement. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.