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In this paper, we consider nonseparable convex minimization models with quadratic coupling terms arised in many practical applications. We use a majorized indefinite proximal alternating direction method of multipliers (iPADMM) to solve this model. The indefiniteness of proximal matrices allows the function we actually solved to be no longer the majorization of the original function in each subproblem. While the convergence still can be guaranteed and larger stepsize is permitted which can speed up convergence. For this model, we analyze the global convergence of majorized iPADMM with two different techniques and the sublinear convergence rate in the nonergodic sense. Numerical experiments illustrate the advantages of the indefinite proximal matrices over the positive definite or the semi-definite proximal matrices.

In this paper, we propose a linearly implicit Fourier pseudo-spectral scheme, which preserves the total mass and energy conservation laws for the damped nonlinear Schrodinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal L2-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.

Based on the local energy dissipation property of the molecular beam epitaxy (MBE) model with slope selection, we develop three, second order fully discrete, local energy dissipation rate preserving (LEDP) algorithms for the model using finite difference methods. For periodic boundary conditions, we show that these algorithms are global energy dissipation rate preserving (GEDP). For adiabatic, physical boundary conditions, we construct two GEDP algorithms from the three LEDP ones with consistently discretized physical boundary conditions. In addition, we show that all the algorithms preserve the total mass at the discrete level as well. Mesh refinement tests are conducted to confirm the convergence rates of the algorithms and two benchmark examples are presented to show the accuracy and performance of the methods.

In this paper, we propose a modification of the accelerated projective steepest descent method for solving nonlinear inverse problems with an $\ell_1$ constraint on the variable, which was recently proposed by Teschke and Borries (2010 Inverse Problems 26 025007). In their method, there are some parameters need to be estimated, which is a difficult task for many applications. We overcome this difficulty by introducing a self-adaptive strategy in choosing the parameters. Theoretically, the convergence of their algorithm was guaranteed under the assumption that the underlying mapping $F$ is twice Frechet differentiable together with some other conditions, while we can prove weak and strong convergence of the proposed algorithm under the condition that $F$ is Frechet differentiable, which is a relatively weak condition. We also report some preliminary computational results and compare our algorithm with that of Teschke and Borries, which indicate that our method is efficient.

In order to improve the quality of the image, many works impose explicit priors on the solution to regularize the ill-posed inverse problem. In this paper, we propose a hybrid variational model which takes advantages of the wavelet tight frame model and the total variation model for image restoration. The core of the method is a new, nonconvex penalty function that is designed for efficient minimization by means of the firm thresholding and soft shrinkage operations. We address the proposed optimization problem by converting it to a constrained problem with variable splitting and using the alternating direction method of multipliers. Numerical examples for image restoration are given to show that the proposed method outperforms some existing methods in terms of the peak signal-to-noise ratio and structural similarity index.

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions., Comment: 23 pages, 47 figures

In this paper, we propose an efficient parameterized Uzawa waveform relaxation method for solving a class of initial value problems of differential-algebraic equations (DAEs). The semi-convergence of this method is studied. Furthermore, the optimal iteration parameters and the corresponding optimal factor for the parameterized Uzawa waveform relaxation method are determined. Two numerical examples are given to illustrate the theoretical results.

In this paper, we propose a fast and efficient way to restore blurred and noisy images with a high-order total variation minimization technique. The proposed method is based on an alternating technique for image deblurring and denoising. It starts by finding an approximate image using a Tikhonov regularization method. This corresponds to a deblurring process with possible artifacts and noise remaining. In the denoising step, a high-order total variation algorithm is used to remove noise in the deblurred image. We see that the edges in the restored image can be preserved quite well and the staircase effect is reduced effectively in the proposed algorithm. We also discuss the convergence of the proposed regularization method. Some numerical results show that the proposed method gives restored images of higher quality than some existing total variation restoration methods such as the fast TV method and the modified TV method with the lagged diffusivity fixed-point iteration.

This paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations. Using the computer algebra system Mathematica, we construct an iterative scheme and discuss the conditions to obtain fourth-order methods from it. All the presented fourth-order methods require one-function and two-derivative evaluation per iteration, and are optimal higher-order iterative methods for obtaining multiple roots. We present some special methods from the iterative scheme, including some known already. Numerical examples are also given to show their performance.

Linear systems with M-matrices often appear in a wide variety of areas. In this paper, we give general preconditioners for solving the systems with nonsingular M-matrix. We show that our preconditioners increase the convergence rate of AOR iterative methods. Numerical results for corresponding preconditioned GMRES methods are also given.

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.

Comparison theorems for spectral radii of two block USSOR methods, two block SSOR methods, two block AOR methods, two block SOR methods with respect to a partition π and the corresponding finer partition π* are given, respectively, whenever A is an M-matrix or an H-matrix.

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type II response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

When A∈ R n×n is an M -matrix we prove the following inequality: q(A∘A −1 ) =1 for n=2, > 2 n for n>2, where A∘B is the Hadamard product of two matrices A and B , q(B)=[p(B −1 )] −1 and p(C) denotes the Perron eigenvalue of a nonnegative matrix C . This also gives a positive answer to the conjecture posed by Fiedler and Markham.

In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix A is a singular M-matrix with ‘property c’ and an irreducible singular M-matrix, respectively. Since the (generalized, block) JOR and AOR methods are respectively the extrapolated methods of the (generalized, block) Jacobi and SOR methods, so the semiconvergence of the (generalized, block) JOR and AOR methods for solving general singular systems are proved. Furthermore, the semiconvergence of the extrapolated power method, the (block) JOR, AOR and SOR methods for solving Markov chains are discussed.

Due to their nanometer-scale dimensions, high mechanicalstrength, and very high thermal and electrical conductivity,carbon nanotubes (CNTs) have attracted much attention tobe recognized as an excellent material choice for nano-composites, in which even a very small amount of CNTscan induce significant changes in the material’s properties[1–4].Since carbon and carbon–carbon composites have manyadvantages, such as excellent mechanical behavior at hightemperatures, low reactivity, high heat capacity, and so on,modern advanced technologies make wide use of thesematerials in different fields, including nuclear reactorwalls, rocket nozzles, battery electrodes, and seal andfriction materials [5–7]. However, the conventional tech-niques to prepare carbon and carbon–carbon compositesare complicated and costly due to the time-consumingprocess, such as liquid phase impregnation and chemicalvapor deposition which are indispensable to obtain high-performance carbon and carbon–carbon composites. Mucheffort has been taken to develop new methods to simplifythe processing and shorten the time of preparing carbon–carbon composite. Most significantly, Naphthalene-derivedsynthetic mesophase pitch (MP) have been recognized asan excellent precursor to the high-performance carbon andgraphite due to their high carbon yield [8–11].However, few researchers try to fabricate the MP basedcomposites reinforced with CNTs because of the difficul-ties in homogeneously distributing CNTs in a MP matrixby traditional methods. In this communication, CNTsreinforced oxidized MP based composites were preparedby one-step self-sintering. The influence of CNTs on themechanical and physical properties on obtained CNTs/carbon derived from oxidized MP composites.The CNTs (multi-walled carbon nanotubes, purity99 wt.%, diameter 20–40 nm) synthesized by chemicalvapor deposition were supplied from Wuhan University ofScience and Technology, China. MP was obtained byMitsubishi Gas Chemical Co., Japan. MWNTs were soni-cated in a bath containing toluene, and then the milled MPwas added to the suspension and hot-mixed at 333 K in theother bath containing toluene under moderate stirring for2 h. Subsequently, the system was moved to a plate anddried at 373 K. The dried compound was ground by a ball-mill into grains of particle size in the range of 2.8–8.2

We consider the generalized AOR method presented by K. R. James. We give some convergence theorems for various kinds of matrices, namely when the coefficient matrix of the linear system is positive definite, an H -, L -, or M -matrix, or strictly or irreducibly diagonally dominant.

Let A and B be two n × n matrices with spectra λ ( A )={ λ 1 ,…, λ n } and λ ( B )={ μ 1 ,…, μ n }. Suppose that the nonsingular matrix Q satisfies Q −1 AQ =diag( J 1 ,…, J p ), where each submatrix J i , i=1,…,p , is a Jordan block. Then there exists a permutation π of {1,…, n } such that ∑ j=1 n μ π(j) −λ j 2 ⩽ n (1+ n−p ) max ∥Q −1 (B−A)Q∥ F , ∥Q −1 (B−A)Q∥ F m and for j =1,…, n , μ π(j) −λ j ⩽ n (1+ n−p ) max n ∥Q −1 (B−A)Q∥ 2 , n ∥Q −1 (B−A)Q∥ 2 m , where m is the order of the largest Jordan block of A and ∥ ∥ F and ∥ ∥ 2 denote, respectively, the Frobenius norm and the spectral norm.

In order to solve a linear system Ax = b , Hadjidimos et al. (1992) defined a class of modified AOR (MAOR) method, whose special case implies the MSOR method. In this paper, some sufficient and/or necessary conditions for convergence of the MAOR and MSOR methods will be achieved, when A is a two-cyclic matrix and when A is a Hermitian positive-definite matrix, an H -, L - or M -matrix, and a strictly or irreducibly diagonally dominant matrix. The convergence results on the MSOR method are better than some known theorems. The optimum parameters and the optimum spectral radii of the MAOR and MSOR methods are obtained, which also answers the open problem given by Hadjidimos et al.

The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors. In this article, some new comparison theorems for two nonnegative splittings (a splitting A = M − N is nonnegative if M −1 exist and M −1 N is nonnegative) are derived. They extend the known results in the literature. In addition, we also point out three incorrect conditions in a paper by Beauwens. Furthermore, we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates. This also answers the open question which additional and appropriate conditions should be imposed on Miller-Neumann splittings to obtain strict inequality. Finally, some applications to a class of generalized AOR, SOR, and JOR iterative methods whose special cases imply block (also point) AOR, SOR, and JOR iterative methods for solving linear systems are discussed.

For an n × n matrix A = (aij) ⩾ 0, we prove the Perron root r(A) satisfies the inequality r(A) ⩾ ∏ 1 ⩽i (a ij a ji ) x i x j ∏ i= 1 n (a ii +q) x i 2 ∑ i= 1 n x i 2 −q for any x=(x1,…,xn)T⩾0 with ∑ i= 1 n x i = 1 and for any q⩾–min{aii}. Using this, a necessary condition for a real n × n matrix A with aii>0 and aij⩽0, i≠j, i, j = 1,…,n, to be an M-matrix is presented, viz. ∏ 1 ⩽i a ij a ji a ii a jj x i x j 1 +q)q − ∑ i= 1 n x i 2 ∑ i= 1 n x i 2 for any x=(x1,…,xn)T⩾0 with ∑ i= 1 n x i = 1 and for any q>0.

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.

A new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375–1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.