Maintaining convergence properties of BiCGstab methods in finite precision arithmetic

It is well-known that Bi-CG can be adapted so that hybrid methods with computational complexity almost similar to Bi-CG can be constructed, in which it is attempted to further improve the convergence behavior. In this paper we will study the class of BiCGstab methods.In many applications, the speed of convergence of these methods appears to be determined mainly by the incorporated Bi-CG process, and the problem is that the Bi-CG iteration coefficients have to be determined from the BiCGstab process. We will focus our attention to the accuracy of these Bi-CG coefficients, and how rounding errors may affect the speed of convergence of the BiCGstab methods. We will propose a strategy for a more stable determination of the Bi-CG iteration coefficients and by experiments we will show that this indeed may lead to faster convergence.     

BiCR variants of the hybrid BiCG methods for solving linear systems with nonsymmetric matrices

We propose Bi-Conjugate Residual (BiCR) variants of the hybrid Bi-Conjugate Gradient (BiCG) methods (referred to as the hybrid BiCR variants) for solving linear systems with nonsymmetric coefficient matrices. The recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid BiCG method, but the recurrence coefficients are different; they are determined so as to compute the coefficients of the residual polynomial of BiCR. From our experience it appears that the hybrid BiCR variants often converge faster than their BiCG counterpart. Numerical experiments show that our proposed hybrid BiCR variants are more effective and less affected by rounding errors. The factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid BiCR variants and the hybrid BiCG methods.     

Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell-Whitehead-Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.     

一种新的修正SR1更新公式及其算法收敛性

SR1更新公式对比其他的拟牛顿更新公式,会更加简单且每次迭代需要更少的计算量。但是一般SR1更新公式的收敛性质是在一致线性无关这一很强的条件下证明的。基于前人的研究成果,提出了一种新的修正SR1公式,并分别证明了其在一致线性无关和没有一致线性无关这两个条件下的局部收敛性,最后通过数值实验验证了提出的更新公式的有效性,以及所作出假设的合理性。根据实验数据显示,在某些条件下基于所提出更新公式的拟牛顿算法会比基于传统的SR1更新公式的算法收敛效果更好一些。     

CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

1. Illtroductioncrust region method is a well-accepted technique in nonlinear optindzation to assure globalconvergence. One of the adVantages of the model is that it does not require the objectivefunction to be convex. Many differellt versions have been suggested in using trust regiontechnique. For each iteration, suppose a current iterate point, a local quadratic model of thefunction and a trust region with center at the point and a certain radius are given. A point thatminimizes the model f…     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

Implementation and efficiency analysis of an adaptive <Emphasis Type="Italic">hp</Emphasis>-finite element method for solving boundary value problems for the stationary reaction–diffusion equation

An iterative process implementing an adaptive hp-version of the finite element method (FEM) previously proposed by the authors for the approximate solution of boundary value problems for the stationary reaction–diffusion equation is described. The method relies on piecewise polynomial basis functions and makes use of an adaptive strategy for constructing a sequence of finite-dimensional subspaces based on the computation of correction indicators. Singularly perturbed boundary value test problems with smooth and not very smooth solutions are used to analyze the efficiency of the method in the situation when an approximate solution has to be found with high accuracy. The convergence of the approximate solution to the exact one is investigated depending on the value of the small parameter multiplying the highest derivative, on the family of basis functions and the quadrature formulas used, and on the internal parameters of the method. The method is compared with an adaptive h-version of FEM that also relies on correction indicators and with its nonadaptive variant based on the bisection of grid intervals.     

Exponential fitting Direct Quadrature methods for Volterra integral equations

This paper is the first approach to the solution of Volterra integral equation by exponential fitting methods. We have developed a Direct Quadrature method, which uses a class of ef-based quadrature rules adapted to the current problem to solve. We have analyzed the convergence of the method and have found different formulas for the coefficients, which limit rounding errors for small stepsizes. Numerical experiments for comparison with other DQ methods are presented.     

An extension of the conjugate residual method to nonsymmetric linear systems

The Conjugate Gradient (CG) method and the Conjugate Residual (CR) method are Krylov subspace methods for solving symmetric (positive definite) linear systems. To solve nonsymmetric linear systems, the Bi-Conjugate Gradient (Bi-CG) method has been proposed as an extension of CG. Bi-CG has attractive short-term recurrences, and it is the basis for the successful variants such as Bi-CGSTAB. In this paper, we extend CR to nonsymmetric linear systems with the aim of finding an alternative basic solver. Numerical experiments show that the resulting algorithm with short-term recurrences often gives smoother convergence behavior than Bi-CG. Hence, it may take the place of Bi-CG for the successful variants.     

线性不等式约束优化的弧线路径信赖域算法

提供了弧线路径结合仿射内点信赖域策略的非单调回代算法解线性不等式约束的优化问题.基于仿射投影的信赖域子问题获得新的搜索方向,采用弧线路径的近似信赖域和线搜索结合技术得到回代步,获得新的步长.通过证明所提供的弧线路径具有一系列良好性质,从而在合理的条件下,证明所提供的算法不仅具有整体收敛性,而且保持算法的局部超线性收敛速率.数值测试表明了算法的有效性与可靠性.     

The numerical solution of Fredholm integral equations using product type quadrature formulas

Product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations. A convergence theorem, and several numerical examples which demonstrate the efficacy of the technique, are presented.     

Novel numerical techniques for the finite moment log stable computational model for European call option

Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.     

The block diagram method for designing the particle swarm optimization algorithm

In this paper, the block diagram method of the dispersed control system is proposed for designing or improving the normal particle swarm optimization algorithms (PSO), that is, it uses the Jury-test of the control theory to compare the block diagrams getting from existing particle swarm optimization methods and finds out some defects of the existing particle swarm optimization methods, for example, the premature convergence of PSO algorithm, and so on. Thus a new particle swarm algorithm is also proposed for improving these defects, that is, the speed iteration and position iteration formulas of PSO are revised for both adjusting its convergence speed and jumping out of the local minimum points. To show effectiveness of the proposed method, the simulations of 13 benchmark examples are carried out, as a result, it indicates that the proposed method is very useful.     

Efficient computation of a mathematical model for the damping of pressure waves in tubes of circular form

A computational method for a system of partial integrodifferential equations of Volterra type is developed. These equations describe the damping of pressure waves in tubes of circular form, taking into account the nonstationary increase of friction. The analytical solution of the model equations is given. A rapidly convergent expansion as well as sharp error estimates for the approximate solutions are obtained. Finally an efficient algorithm using recursion formulas is presented and the algorithm is compared numerically with the method of characteristics. © 1995 John Wiley & Sons, Inc.     

A new family of global methods for linear systems with multiple right-hand sides

The global bi-conjugate gradient (Gl-BCG) method is an attractive matrix Krylov subspace method for solving nonsymmetric linear systems with multiple right-hand sides, but it often show irregular convergence behavior in many applications. In this paper, we present a new family of global A-biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version. Finally, numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

A Finite Smoothing Algorithm for Quantile Regression

This article introduces a new method for computing regression quantile functions. This method applies a finite smoothing algorithm based on smoothing the nondifferentiable quantile regression objective function ρ_τ. The smoothing can be done for all τ ∈ (0, 1), and the convergence is finite for any finite number of τ_i ∈ (0, 1), i = 1,…,N. Numerical comparison shows that the finite smoothing algorithm outperforms the simplex algorithm in computing speed. Compared with the powerful interior point algorithm, which was introduced in an earlier article, it is competitive overall; however, it is significantly faster than the interior point algorithm when the design matrix in quantile regression has a large number of covariates. Additionally, the new algorithm provides the same accuracy as the simplex algorithm. In contrast, the interior point algorithm gives only the approximate solutions in theory, and rounding may be necessary to improve the accuracy of these solutions in practice.     

A Posteriori Estimates for Modelling Errors of the Stationary Diffusion Equation with Complicated Coefficients

In this paper, we suggest a new computational strategy for diffusion type problems with coefficients that may sharply change values and have a complicated distribution over the domain. We solve models, which approximate the coefficients, numerically and estimate the corresponding approximation errors by the a posteriori estimates of functional type. We show that the modelling error can be also explicitly calculated. This allows to obtain numerical solutions of complicated diffusion problems by solving much simpler problems that do not require full detailization of the structure of coefficients. Balancing modelling and discretization errors provides an economical way of getting an approximate solution with an a priori given accuracy. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

A posteriori error estimation of goal-oriented quantities for elliptic type BVPs

In practice the process or object under analysis is usually modelled by means of a selected mathematical model, whose approximate solution is computed with a help of a certain computer code. This approximate solution necessarily includes various errors related to the approximation itself, special features of the particular method used, round-off errors, etc. Therefore, it inevitably rises the question about the reliability of the computed approximations. In the present paper we describe and test numerically the new effective computational technology designed for a control of the accuracy of approximate solutions in terms of goal-oriented quantities (or goal-oriented criteria). Such quantities are to be chosen by a user depending on solution properties that present a special interest. The technology proposed is applicable to the elliptic type boundary-value problems and leads to effective computer codes aimed to control errors of approximate solutions obtained by the finite element method which presents nowadays the main computational tool in industrial software. Various numerical tests confirming high effectivity of this technology are presented.