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In this paper, we study orthogonal nonnegative matrix factorization. We demonstrate the coefficient matrix can be sparse and low-rank in the orthogonal nonnegative matrix factorization. By using these properties, we propose to use a sparsity and nuclear norm minimization for the factorization and develop a convex optimization model for finding the coefficient matrix in the factorization. Numerical examples including synthetic and real-world data sets are presented to illustrate the effectiveness of the proposed algorithm and demonstrate that its performance is better than other testing methods. [ABSTRACT FROM AUTHOR]

The uniqueness of multilinear PageRank vectors is discussed, and the new uniqueness condition is given. The new results are better than the one given in the work of Gleich et al. published in SIAM J Matrix Anal Appl. 2015;36;1409-1465. Numerical examples are given to demonstrate the new theoretical results. [ABSTRACT FROM AUTHOR]

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor A. The main contribution of this paper is to show that whenAis perturbed to a nonnegative tensor à by ΔA, the absolute difference between the spectral radii of A and à is bounded by the largest magnitude of the ratio of the ith component of ΔAxm-1 and the ith component xm-1, where x is an eigenvector associated with the largest eigenvalue of A in magnitude and its entries are positive. We further derive the bound in terms of the entries of Aonly when x is not known in advance. Based on the perturbation analysis, wemake use of the NQZ algorithmto estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward errormatrix ΔA and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to develop an algorithm for blending of multiple images in the image-stitching process. Our idea is to propose a variational method containing an energy functional to determine both a stitched image and weighting mask functions of multiple input images for image blending. The existence of the solution of the proposed energy functional is shown. We also present an alternative minimizing algorithm to solve the proposed model numerically and show the convergence of this algorithm. Experimental results show that the proposed model works effectively and efficiently and that the proposed method is competitive with the tested existing methods under noisy conditions. [ABSTRACT FROM AUTHOR]

SUMMARY A generalized refined Arnoldi method based on the weighted inner product is presented for computing PageRank. The properties of the generalized refined Arnoldi method were studied. To speed up the convergence performance for computing PageRank, we propose to change the weights adaptively where the weights are calculated based on the current residual corresponding to the approximate PageRank vector. Numerical results show that the proposed Arnoldi method converges faster than existing methods, in particular when the damping factor is large. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo . Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity. [Copyright &y& Elsevier]

In the image processing community, there have recently been many restoration and reconstruction problems that can be reformulated into linearly constrained convex programming models whose objective functions have separable structures. These favorable reformulations have promoted impressive applications of the alternating direction method (ADM) in the field of image processing. At each iteration, the computation of ADM is dominated by solving two subproblems exactly. However, in many restoration and reconstruction applications, it is cither impossible or extremely expensive to obtain exact solutions of these ADM subproblems. This fact urges the development on inexact versions of ADM, which allow the generated ADM subproblems to be solved approximately subject to certain inexactness criteria. In this paper, we develop some truly implementable inexact ADMs whose inexactness criteria controlling the accuracy of the ADM subproblems are easily implementable. The convergence of the new inexact ADMs will be proved. Numerical results on several image processing problems will be given to illustrate the effectiveness of the proposed inexact ADMs. [ABSTRACT FROM AUTHOR]

In this article, we consider a bilaterally constrained optimization model arising from the semisupervised multiple-class image segmentation problem. We prove that the solution of the corresponding unconstrained problem satisfies a discrete maximum principle. This implies that the bilateral constraints are satisfied automatically and that the solution is unique. Although the structure of the coefficient matrices arising from the optimality conditions of the segmentation problem is different for different input images, we show that they are M-matrices in general. Therefore, we study several numerical methods for solving such linear systems and demonstrate that domain decomposition with block relaxation methods are quite effective and outperform other tested methods. We also carry out a numerical study of condition numbers on the effect of boundary conditions on the optimization problems, which provides some insights into the specification of boundary conditions as an input knowledge in the learning context. © 2010 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 20, 191–201, 2010 [ABSTRACT FROM AUTHOR]

Abstract: In a radar detecting system, the radar cross section (RCS) of a target is an important application in electromagnetics computation. Cavity prediction is usually required in the computation process. When wavelengths of electromagnetic field are large, cavity calculation is a challenging problem. The main aim of this paper is to employ the preconditioned GMRES method to solve linear systems arising in cavity calculation of the transverse magnetic and transverse electric problems. We develop new preconditioners based on fast transforms and Toeplitz solvers. Our numerical results show that the proposed preconditioners are quite efficient and effective, especially when the wavelength is large. [Copyright &y& Elsevier]

This paper presents a coordinate gradient descent approach for minimizing the sum of a smooth function and a nonseparable convex function. We find a search direction by solving a subproblem obtained by a second-order approximation of the smooth function and adding a separable convex function. Under a local Lipschitzian error bound assumption, we show that the algorithm possesses global and local linear convergence properties. We also give some numerical tests (including image recovery examples) to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Image restoration problems are often solved by finding the minimizer of a suitable objective function consisting of a data-fitting term and a regularization term. In this paper, we consider the data-fitting term measured in the ℓ1 norm to handle non-Gaussian additive noise and the regularization term given by the total variation (TV) to restore image edges. We propose a new algorithm for this image restoration problem by making use of new variables to modify the data-fitting term and the TV regularization term. An alternating minimization method based on the new formulation is employed to restore blurred and noisy images. Our experimental results show that the quality of restored images by the proposed method is competitive with those restored by the other tested methods. We also show the convergence of the alternating minimization algorithm and demonstrate that the proposed algorithm is very efficient. [ABSTRACT FROM AUTHOR]

In this paper, we present an agglomerative fuzzy K-Means clustering algorithm for numerical data, an extension to the standard fuzzy K-Means algorithm by introducing a penalty term to the objective function to make the clustering process not sensitive to the initial cluster centers. The new algorithm can produce more consistent clustering results from different sets of initial clusters centers. Combined with cluster validation techniques, the new algorithm can determine the number of clusters in a data set, which is a well-known problem in K-Means clustering. Experimental results on synthetic data sets (2 to 5 dimensions, 500 to 5,000 objects and 3 to 7 clusters), the BIRCH two-dimensional data set of 20,000 objects and 100 clusterOand the WINE data set of 178 objects, 17 dimensions, and 3 clusters from UCI have demonstrated the effectiveness of the new algorithm in producing consistent clustering results and determining the correct number of clusters in different data sets, some with overlapping inherent clusters. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in HSS preconditioners for saddle point linear systems with a nonzero (2,2)-th block. We study an approximation of the spectra of HSS preconditioned matrices and use these results to illustrate and explain the spectra obtained from numerical examples, where the previous spectral analysis of HSS preconditioned matrices does not cover. [ABSTRACT FROM AUTHOR]

Abstract: We present new preconditioners based on matrix splittings for the saddle point problems. The spectral property of one of the preconditioned matrix is studied in detail. Numerical examples are used to illustrate the efficiency of the new preconditioners. [Copyright &y& Elsevier]

Abstract: Preconditioned iterative methods are described for the solution of an elliptic partial differential equation over an unit square region with Robbins boundary conditions. Transform based preconditioners are constructed and analyzed. The motivation is to exploit the fast inversion of transform based systems via the fast transform. We prove that transform based preconditioners can be chosen so that the condition number of the preconditioned system can be reduced from O(n 2) to O(1). Numerical results are reported to illustrate the effectiveness of the preconditioners. [Copyright &y& Elsevier]

A matrix A is said to possess an UV-displacement structure if rank(AU – VA) is small compared with the rank of A. Estimates for the rank AgV – UAg are presented, where Ag is the group inverse of A. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The symmetric Sinc–Galerkin method developed by Lund (Math. Comput. 1986; 47:571–588), when applied to second-order self-adjoint boundary value problems on d dimensional rectangular domains, gives rise to an N × N positive definite coefficient matrix which can be viewed as the sum of d Kronecker products among d - 1 real diagonal matrices and one symmetric Toeplitz-plus-diagonal matrix. Thus, the resulting coefficient matrix has a strong structure so that it can be advantageously used in solving the discrete system. The main contribution of this paper is to present and analyse a multigrid method for these Sinc–Galerkin systems. In particular, we show by numerical examples that the solution of a discrete symmetric Sinc–Galerkin system can be obtained in an optimal way only using O(N log N) arithmetic operations. Numerical examples concerning one- and two-dimensional problems show that the multigrid method is practical and efficient for solving the above symmetric Sinc–Galerkin linear system. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We modify the algorithm of [1], based on Newton''s iteration and on the concept of ε-displacement rank, to the computation of the Moore-Penrose inverse of a rank-deficient Toeplitz matrix. Numerical results are presented to illustrate the effectiveness of the method. [Copyright &y& Elsevier]

We study efficient iterative methods for Toeplitz systems based on the circulant and skew-circulant splitting (CSCS) of the Toeplitz matrix. Theoretical analysis show that if the circulant and the skew-circulant splitting matrices are positive definite, then the CSCS method converges to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew-circulant matrices involved. Numerical examples are presented to demonstrate the method. [Copyright &y& Elsevier]

This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems. The high-resolution color images are reconstructed from multiple undersampled, shifted, degraded color frames with subpixel displacements. The resulting degradation matrices are spatially variant. To capture the changes of reflectivity across color channels, the weighted H1 regularization functional is used in the Tikhonov regularization. The Neumann boundary condition is also employed to reduce the boundary artifacts. The preconditioners are derived by taking the cosine transform approximation of the degradation matrices. Numerical examples are given to illustrate the fast convergence of the preconditioned conjugate gradient method. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms. We show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges. We make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained. Also we consider tensor numerical ranges based on inner products, however, they may not be convex in general. [ABSTRACT FROM AUTHOR]

In this paper, we study and analyze absolute and relative condition numbers of the spectral projections for regular periodic eigenproblems. The main contribution is to derive explicit expressions of the condition numbers of the j-th left and right spectral projections. Numerical examples are given to illustrate the proposed condition numbers for the spectral projections associated with periodic eigenproblems. [ABSTRACT FROM AUTHOR]