Multilevel augmentation methods for nonlinear ill-posed problems

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.     

Maximum likelihood estimation of linear stochastic systems in the class of sequential square-root orthogonal filtering methods

Theoretical and applied aspects are considered for development of numerically stable adaptive methods of the parametric identification of linear discrete stochastic systems in the space of states. The unknown system parameters to be estimated can enter into any matrices specifying a system and into initial conditions. The class of gradient methods is first suggested, which is developed on the basis of orthogonal square-root implementations of the discrete Kalman filter with the use of the technology of sequential data processing. It is shown that the algorithms of the given type can be effectively used to solve ill-conditioned problems of parametric identification. A test example is drawn. The practical significance of the suggested class of methods is illustrated by an example of the solution for one of the financial mathematics problems — the identification of a multidimensional model of stochastic volatility.     

A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations

We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.     

基于最大加权投影求解的彩色图像灰度化对比度保留算法

针对目前彩色图像灰度化难以充分保留原彩色图像对比度的问题,本文提出了基于最大加权投影求解的彩色图像灰度化模型及算法.首先,在好的彩色图像灰度化算法应使灰度化图像具有最大对比度的假设下,本模型提出最大加权投影的目标优化函数,并且将原始彩色图像梯度权重引入到最大化函数中,使得原彩色图像中对比度较小的区域也能够在灰度化后的图像中得到保持.每个彩色通道梯度的高斯加权系数反映灰度图像的对比度和原彩色图像的颜色顺序.其次,对所提模型使用参数离散搜索策略求解,通过对线性离散参数模型产生的候选图像进行搜索,由于只有几个算术运算,计算速度较快.最后,为评价所提出算法在复杂场景下图像灰度化对比度保持性能,本文对Cadik、CSDD和COLOR250数据集分别进行灰度化实验.定性和定量实验结果表明,所提算法相比于其他算法能较好地保留原彩色图像颜色对比度,同时具有对噪声鲁棒和运算速度快的优势.     

A family of simultaneous zero finding methods

A class of iterative methods with arbitrary high order of convergence for the simultaneous approximation of multiple complex zeros is considered in this paper. A special attention is paid to the fourth order method and its modifications because of their good computational efficiency. The order of convergence of the presented methods is determined. Numerical examples are given.     

DCT- and DST-based splitting methods for Toeplitz systems

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 706–734]. Theoretical analysis shows that new splitting iterative methods converge to the unique solution of a symmetric Toeplitz linear system. Moreover, an upper bound of the contraction factor of our new splitting iterations is derived. Numerical examples are reported to illustrate the effectiveness of new splitting iterative methods.     

Gradient methods exploiting spectral properties

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73–88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, R-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.     

Locality preserving and global discriminant projection with prior information

Existing supervised and semi-supervised dimensionality reduction methods utilize training data only with class labels being associated to the data samples for classification. In this paper, we present a new algorithm called locality preserving and global discriminant projection with prior information (LPGDP) for dimensionality reduction and classification, by considering both the manifold structure and the prior information, where the prior information includes not only the class label but also the misclassification of marginal samples. In the LPGDP algorithm, the overlap among the class-specific manifolds is discriminated by a global class graph, and a locality preserving criterion is employed to obtain the projections that best preserve the within-class local structures. The feasibility of the LPGDP algorithm has been evaluated in face recognition, object categorization and handwritten Chinese character recognition experiments. Experiment results show the superior performance of data modeling and classification to other techniques, such as linear discriminant analysis, locality preserving projection, discriminant locality preserving projection and marginal Fisher analysis.     

图最优化线性鉴别投影及其在图像识别中的应用

在图最优化局部保持投影(GoLPP)算法的基础上,本文充分利用数据的类别信息,提出一种新的特征抽取算法——图最优化线性鉴别投影(GoLDP)。与GoLPP类似,GoLDP的邻接图是通过最优化一个目标函数创建的;与GoLPP不同,GoLDP利用数据的类别信息创建两幅最优邻接图——最优内在图和最优惩罚图,由这两幅最优邻接图求得最优投影矩阵。FERET与YALE人脸数据库以及PolyU掌纹数据库上的实验结果证明了GoLDP算法的有效性。     

Asymptotic stability of block boundary value methods for delay differential-algebraic equations

Block boundary value methods are applied to solve a class of delay differential-algebraic equations. We focus on the asymptotic stability of the numerical methods for linear delay differential-algebraic equations with multiple delays. It is shown that A-stable block boundary value methods satisfying a restrictive condition can preserve the asymptotic stability of the analytical solution. Numerical experiments further confirm the effectiveness and stability of the methods.     

Convergence of memory gradient methods

In this paper we present a new class of memory gradient methods for unconstrained optimization problems and develop some useful global convergence properties under some mild conditions. In the new algorithms, trust region approach is used to guarantee the global convergence. Numerical results show that some memory gradient methods are stable and efficient in practical computation. In particular, some memory gradient methods can be reduced to the BB method in some special cases.     

组合2DFLDA监督的非负矩阵分解和独立分量分析的特征提取方法*

通过对投影非负矩阵分解(NMF)和二维Fisher线性判别的分析,针对NMF的特征提取存在无监督学习以及特征维数高的问题,提出了组合2DFLDA监督的非负矩阵分解和独立分量分析(SPGNMFICA)的特征提取方法。首先对样本进行投影梯度的非负矩阵分解,将得到的NMF子图像进行二维Fisher线性判别,主要反映类间差异信息构建子空间;对子空间的向量进行独立分量分析(ICA),得到独立分量特征空间;其次将样本在独立分量特征空间上进行投影;最后使用径向基网络对投影系数进行识别。通用人脸库ORL和YALE的识别实验证明,该算法是一种有效的特征提取和识别方法。     

Kernel class-wise locality preserving projection

In the recent years, the pattern recognition community paid more attention to a new kind of feature extraction method, the manifold learning methods, which attempt to project the original data into a lower dimensional feature space by preserving the local neighborhood structure. Among them, locality preserving projection (LPP) is one of the most promising feature extraction techniques. However, when LPP is applied to the classification tasks, it shows some limitations, such as the ignorance of the label information. In this paper, we propose a novel local structure based feature extraction method, called class-wise locality preserving projection (CLPP). CLPP utilizes class information to guide the procedure of feature extraction. In CLPP, the local structure of the original data is constructed according to a certain kind of similarity between data points, which takes special consideration of both the local information and the class information. The kernelized (nonlinear) counterpart of this linear feature extractor is also established in the paper. Moreover, a kernel version of CLPP namely Kernel CLPP (KCLPP) is developed through applying the kernel trick to CLPP to increase its performance on nonlinear feature extraction. Experiments on ORL face database and YALE face database are performed to test and evaluate the proposed algorithm.     

Numerical methods for elastic structural stability analysis

The Gauss-Seidel, the successive overrelaxation, the conjugate gradient, the Cholesky's square root and the Cholesky's decomposition methods for solving systems of linear algebraic equations encountered in structural analysis are considered. Theorems for these methods establishing necessary and sufficient conditions for the matrix of the coefficients to be positive definite are converted into criteria for elastic structural stability. Numerical procedures, by using these methods in conjunction with their corresponding stability criteria, for checking the elastic stability and determining the elastic critical load of a structure are proposed. Application of these procedures to the stability analysis of plane frameworks leads to the conclusion that the proposed direct schemes are more efficient than the iterative ones.     

Boundary value methods: The third way between linear multistep and Runge-Kutta methods

It is well known that the approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a k^th-order discrete one. Such transformation has the undesired effect of introducing spurious, or parasitic, solutions to be kept under control. It is such control which is responsible of the main drawbacks (e.g., the two Dahlquist barriers) of the classical LMF with respect to Runge-Kutta methods. It is, however, less known that the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem. Starting from such an idea, a new class of multistep methods, called Boundary Value Methods (BVMs), has been proposed and analyzed in the last few years. Of course, they are free of barriers. Moreover, a block version of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods. In this paper, the recent results on the subject are reviewed.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

A new class of projection and contraction methods for solving variational inequality problems

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods.     

Discrete weighted residual methods A survey

This paper introduces find exemplifies discrete weighted residual methods (DWRM's) for the approximate solution of discrete boundary-value problems in ordinary and partial difference equations. The solution of the discrete boundary-value problem is approximated by a linear combination of known functions with undetermined coefficients. DWRM's specify procedures for determining these coefficients as the solution of a. system of algebraic equations

This paper develops the discrete analogue of the continuous weighted residual methods. In so doing, the differences arising from this development are delineated and resolved. The convergence of DWRM's is demonstrated and the monotone decreasing properties of the root-mean-square error are noted. The DWRM's surveyed are: the collocation technique, the subdomain method, the Galerkin procedure, and the method of least-squares

Numerical results are presented to illustrate the efficacy of DWRM's.     

A Weak Galerkin Finite Element Method for the Maxwell Equations

This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either \(H^1\)-like or \(L^2\) and \(L^2\)-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.     

Convergence properties of a class of nonlinear conjugate gradient methods

Conjugate gradient methods are a class of important methods for unconstrained optimization problems, especially when the dimension is large. In this paper, we study a class of modified conjugate gradient methods based on the famous LS conjugate gradient method, which produces a sufficient descent direction at each iteration and converges globally provided that the line search satisfies the strong Wolfe condition. At the same time, a new specific nonlinear conjugate gradient method is constructed. Our numerical results show that the new method is very efficient for the given test problems by comparing with the famous LS method, PRP method and CG-DESCENT method.