Iterative methods for systems of equations with interval coefficients and linear form

We introduce iterative methods for systems of equations with interval coefficients and linear form by suitable matrix splittings. When compared to the iterative methods for systems amenable to iteration introduced in [1], improved convergence and inclusion properties can be proved under suitable conditions. The method can also be used in the solution of specific nonlinear systems of equations by interval arithmetic methods.     

Probability distribution functions for servicing two types of requests

A method for calculating the parameters of queuing systems with heterogeneous requests was obtained in our paper [1]. In [2], the accuracy of this method for a network of queuing systems was studied. This method is applicable for the case of the estimation of queuing systems with respect to an average value (averaged over the parameters of the distribution averages). This paper develops the method for calculating the parameters of queuing systems by using the integral characteristics (distribution functions of random quantities). This approach allows one to use more accurate methods for calculating the characteristics of queuing systems with different types of requests. In addition, this method provides more detailed specification of the characteristics of a queuing service system, in particular, the examination of the characteristics of output streams.     

Higher-order derivatives in linear and quadratic programming

An interior method for linear and quadratic programming which makes use of higher derivatives of the logarithm barrier function is presented. A convergence analysis is considered too. Our computational experience shows that the method considered performs quite well and seems to be more reliable and robust than the standard method.Scope and purposeLinear programming problems were, for many years, the exclusive domain of the simplex algorithm developed by G.B. Dantzing in 1947. With the introduction of a new algorithm, developed by N.K. Karmarkar in 1984, an alternative computational approach became available for solving such problems. This algorithm established a new class of algorithms: interior point methods for linear programming. In this paper we introduce a barrier method for solving a linear and quadratic programming problem which [9], [10], [11], [12], [13], [14], [15] makes use of higher-order derivatives. We note that a different approach used to construct higher-order interior point methods is presented in [1], [2], [3], [4]. We think that making use of an approximation of higher-order we may obtain a faster convergence and an algorithm more robust than a method obtained using a second-order approximation.     

Control performance assessment for nonlinear systems

Assessing the quality of existing industrial control loops, or comparing between two alternative controller designs is becoming an important routine auditing task for the control engineer. While most of the research and commercial activity in CPA has been applied to linear systems to date, those researchers investigating nonlinear systems fall into one of two groups. The first group focussed on the diagnosis of a common specific nonlinearity, namely valve stiction [1], [2], [3], while the second group tried to establish the minimum variance performance lower bound (MVPLB) [4], [5], [6], [7], [8]. In this paper we will propose a new CPA performance index for general nonlinear models based on an ANOVA-like variance decomposition method. The results of two simulation examples illustrate that the proposed methodology is efficient and accurate.     

On absolute stability of discrete systems

Modernization is considered of the criterion of absolute stability of discrete systems [1, 2], which uses the method of quadratic transformation of the state vector. The modernization aim is to obtain the criterion form that does not explicitly depend on the notions of this method. The new formulation permits performing all calculations in the space of the initial dimension, which reduces the complexity of the problem solution. This work represents the development of the approach suggested in [3] for continuous systems.     

线性离散时延系统的辨识

本文根据逐步回归思想,提出了一种辨识多输入-单输出线性时延系统(MISO-D)的新算 法.本算法可推广应用于一类极其广泛的多输入-多输出时延系统的辨识,与迭代法相比[1-4], 计算量小,应用范围广.     

A finite difference method for a Stefan problem

Many physical problems involve initial boundary value problems for parabolic differential equations in which part of the boundary is not given a priori but is found as part of the solution. These problems have been considered under the name of «Stefan problems». Stefan problems occur in such physical processes as the melting of solids and the crystallizing of liquids. Existence and uniqueness for various one dimensional Stefan problems have been shown by several authors ([1], [4], [5], [8], [9]). Some multidimensional Stefan problems were considered by ([6], [11]). The purpose of this paper is to approximate the solution of a Stefan problem using an implicit finite difference analog for the heat equation and an explicit finite difference analog for the differential equation describing the free boundary. Also, we shall show that the finite difference solution we obtain converges uniformly to the actual solution. Numerous numerical schemes for various Stefan problems have been successfully employed by several authors ([2], [3], [10], [12], [14], [16], [17]). In Chapter I we formulate a continuous time Stefan problem, and in Chapter II we describe a finite difference scheme for approximating results. Chapter IV contains the main result which shows that the finite difference solution converges to the actual solution. Finally, in Chapter V we give a numerical example.     

A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.     

Models and data engineering

Data and models are two well established communities that are continuously contributing in new challenges in different research domains including cyber-physical systems  [1], cloud computing [2], service oriented applications, social networks [3], big data (with its five Vs characteristics: Volume, Variety, Velocity, Veracity and Value)  [4], etc. The success story of data and models communities is mainly based on the availability of foundations relying on formal methods [5], modelling methods  [6], storage systems and platforms  [7], advanced optimization structures, benchmarking, scalability, etc. These foundations are usually associated with tools and commercial and academic systems.The selected papers for this special issue address a variety of topics and concerns in models and data fields, including advanced databases, engagement systems, embedded and complex systems, etc.     

Higher order finite volume methods on selfadaptive grids for convection dominated reactive transport problems in porous media

In this paper we study first and higher order finite volume methods for the approximation of reactive systems of transport equations. For the first order scheme an a posteriori error estimate is derived which generalizes the result of [24], as also nonlinear adsorption is taken into account. Based on the a posteriori result, we define strategies for grid adaptation for both methods. Numerical results show the efficiency of the adaptive higher order method, in particular in comparison with first order schemes on uniform computational grids.     

一类高度非线性网格生成方程的迭代求解方法

在[1]中第170页指出,对于变分方法导出的一类高度非线性的守恒型网格生成方程,采用通常的Picard迭代方法无法正确求解.本文构造了一种新的Picard迭代求解方法,数值结果表明这一方法较好地解决了此类方程的求解问题.     

On solving nonlinear least-squares problems in case of rankdeficient Jacobians

Some real and interval algorithms for the numerical solution of a class of nonlinear algebraic equations are described. These algorithms are based upon the symmetric single-step [2] and Newton [10], [11], [8] methods. Convergence theorems and numerical results which illustrate the effectiveness of the algorithms are given.     

Konvergenz impliziter Mehrschrittverfahren zur numerischen Lösung quasilinearer Anfangswertaufgaben

Generalizing the theory of linear innitial-value problems, formulated by Lax and Richtmyer [7], necessary conditions for solvability and convergence of a large class of implicit finite-difference methods for the numerical solution of certain quasilinear initial-value problems are given. In addition an equivalence theorem for these methods is proved.     

Improved predictor-corrector method for solving fuzzy initial value problems

In this paper, an improved predictor-corrector (IPC) method is proposed to solve the “fuzzy initial value problem”. The IPC method is generated by combining an explicit three-step method and an implicit two-step method. The methods are compared with the methods discussed in [T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences 177(7) (2007) 1633-1647], and are shown to be more accurate. The convergence and stability of the proposed methods are also presented in detail. These methods are illustrated by solving some examples.     

小样本的线性连续时间系统的辨识

本文简要地讨论了各种辨识线性连续时间系统的方法,并根据J.Aitchison提出的预报拟 合优度准则[1],建立了一种新的连续时间系统在小样本条件下的辨识方法.模拟仿真实验的 结果表明:当量测数据很少且噪声较大时,该方法明显优于其它方法.     

An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees

In recent years, several quite successful attempts have been made to solve systems of polynomial constraints, using geometric design tools, exploiting the availability of subdivision-based solvers [7], [11], [12], [15]. This broad range of methods includes both binary domain subdivision as well as the projected polyhedron method of Sherbrooke and Patrikalakis [15]. A prime obstacle in using subdivision solvers is their scalability. When the given constraint is represented as a tensor product of all its independent variables, it grows exponentially in size as a function of the number of variables. In this work, we show that for many applications, especially geometric ones, the exponential complexity of the constraints can be reduced to a polynomial by representing the underlying structure of the problem in the form of expression trees that represent the constraints. We demonstrate the applicability and scalability of this representation and compare its performance to that of tensor product constraint representation through several examples.     

Enclosing the solutions of nonlinear systems of equations

The quadratically convergent Newton-type methods and its variants are generally used for solving the nonlinear systems of equations. Most of these methods use the convexity conditions [7] of the involved bounded linear operators for their convergence. Alefeld and Herzberger [1] have proposed a quadratically convergent iterative method for enclosing the solutions of the special type of nonlinear system of equations arising from the discretization of nonlinear boundary value problems which do not require the convexity conditions but uses the subinverses of the bounded linear operators. In this paper, we have proposed a modification of this method which takes it further faster. The proposed method uses bom the superinverses and subinverses of bounded linear operators. At the expense of slightly more computations than used in [1], the rate of convergence of our method enhances from quadratic to cubic. Finally, the method is tested on a numerical example.     

Iterative solvers of Ax = b derived from ode's numerical integration methods

Although iterative methods for solving linear systems has been the subject of study for a long time, the acceleration of such methods is still object of interest, research focusing in improvements of already known methods as well as on new, faster ones. In this sense we can cite among several other authors, for example, the works of Martins [8], Hadjidimos [9] and Evans [7]. The purpose of this paper is twofold. First we show that the numerical integration methods for Ordinary Differential Equations, obtained by Taylor expansions, result in a B-extrapolation method [3] for iteratively solving linear algebraic systems. Second, we compare the best rates of convergence of the algorithms developed here, with the best rate of convergence of the Jacobi Over-Relaxation method, (JOR), proving that depending on the choice of the step of integration and the behavior of the spectrum of the matrix A of the original system Ax = b the third order methods derived from the Taylor expansions can be better than the JOR. The procedure used here with the splitting A – I – J can also be easely applied to other splittings, resulting in a comparison of the convergence of the present method with the SOR or iterations methods or any other linear iterative methods of degree one.     

Extended sufficient semilocal convergence for the Secant method

We establish new sufficient convergence conditions for the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new semilocal convergence analysis of the Secant method. Moreover, our sufficient convergence conditions expand the applicability of the Secant method in cases not covered before (Dennis, 1971 [9], Hernández et al., 2005 [8], Laasonen, 1969 [15], Ortega and Rheinboldt, 1970 [11], Potra, 1982 [5], Potra, 1985 [7], Schmidt, 1978 [18], Yamamoto, 1987 [12], Wolfe, 1978 [19]). Numerical examples are also provided in this study.