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In this paper, we present an adaptive method for the TV-based model of three norms L q ( q = 1 2 , 1 , 2 ) for the image restoration problem. The algorithm with the L2 norm is used in the smooth regions, where the value of |∇u| is small. The algorithm with the L 1 2 norm is applied for the jumps, where the value of |∇u| is large. When the value of |∇u| is moderate, the algorithm with the L1 norm is employed. Thus, the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted. The numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images. Specifically, in contrast to the L1 norm as in the Rudin–Osher–Fatemi model, the L2 norm yields better results in the smooth and flat regions, and the L 1 2 norm is more suitable in regions with strong discontinuities. Therefore, our adaptive algorithm is efficient and robust even for images with large noises.

In this article, first, we establish some compact finite difference schemes of fourth-order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in L ∞ norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in L 2 norm is obtained as well as the maximum estimate, further the convergence in L ∞ norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015

Multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss–Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

Based on some previous work on the connection between image restoration and fluid dynamics, we apply a two-step algorithm for image denoising. In the first step, using a splitting scheme to study a nonlinear Stokes equation, tangent vectors are obtained. In the second step, an image is restored to fit the constructed tangent directions. We apply a fixed point iteration to solve the total variation-based image denoising problem, and use algebraic multigrid method to solve the corresponding linear equations. Numerical results demonstrate that our algorithm is efficient and robust, and boundary conditions are satisfactory for image denoising.

In this article, three difference schemes of the Ginzburg-Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time-splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011

In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter functionθ. The numerical experiments demonstrate that our compound al- gorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and cor- responding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the cor- responding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.

This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge-Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit.

In this paper, we propose a new (mean) boundary conditions (BCs) for the total variation-based image restoration problem. We present a proof of the convergence of our difference schemes. An algebraic multigrid method and Krylov subspace acceleration are used when we solve the corresponding linear equations. The results from our new BCs are compared with the results from the other BCs introduced by several image researchers by simple and significant 2D numerical experiments. Experimental results demonstrate that our new BCs can get better restored images than the existing BCs.

This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the Boussinesq equations with and without dissipative term. Linear stability of the exact solitary-wave solutions is shown for the two Boussinesq equations mentioned above, when small disturbance in travelling wave form satisfies certain conditions.

In this paper, we propose a new time dependent model for solving total variation (TV) minimization problems in image restoration. We present a proof of the existence, uniqueness and stability of the viscosity solution of our model. The results from our new model by explicit numerical schemes are compared with the results from those models introduced by several image researchers by simple and significant 2D numerical experiments. Experimental results demonstrate that our new model can get better results than the existing models.

We introduce the convergence of algebraic multigrid in the form of matrix decomposition. The convergence is proved in block versions of the multi-elimination incomplete LU (BILUM) factorization technique and the approximation of their inverses to preserve sparsity. The convergence theorem can be applied to general interpolation operator. Furthermore, we discuss the error caused by the error matrix.

In this paper, the Boussinesq-like equations with fully nonlinear dispersion, B ( m , n ) equations: u tt − ( u m ) xx + ( u n ) xxxx = 0, are investigated. Exact solitary solutions of the equations are obtained by using the decomposition method. The two special cases, B (2, 2) and B (3, 3), are chosen to illustrate the concrete scheme of the decomposition method in B ( m , n ) equations. General formulas for the solutions of B ( m , n ) equations are established.

In this paper, a new algorithm for calculating Adomian polynomials for nonlinear operators will be established by parametrization. The algorithm requires less formula than the previous method developed by Adomian [Nonlinear Stochastic Operator Equations, Academic Press, 1986, G. Adomian, R. Rach, On composite nonlinearities and decomposition method. J. Math. Anal. Appl. 113 (1986) 504-509, G. Adomian, Applications of Nonlinear Stochastic Systems Theory to Physics, Kluwer, 1988]. Many forms of nonlinearity will be studied to illustrate the new algorithm. The new algorithm will be extended to calculate Adomian polynomials for nonlinearity of several variables.

The nonlinear dispersive Boussinesq-like B ( m , n ) equations, u tt − ( u m ) xx − ( u n ) xxxx = 0, which exhibit compactons: solitons with compact support, are investigated. Exact solitary solutions with compact support are developed. The specific cases, B (2, 2) and B (3, 3), are chosen to illustrate the concrete scheme of the decomposition method in B ( m , n ) equations. General formulas for the solutions of B ( m , n ) equations are established.

New simple proofs of unique continuation of solutions for the Stokes equation and Navier-Stokes equations is presented under weaker conditions.

In this paper, we present a semi-implicit numerical scheme for the model of image processing with time-delay regularization, and prove its convergence by the related theory of viscosity solution. Based on the principle of contraction mapping and discrete functional analysis, we also establish the iterative convergence in the sense of L"~ norm. Numerical experiments with Gauss-Seidel iterative method of sparse matrix show that, compared with the explicit scheme, our numerical method make denoised image sharper.

In this paper, the pseudospectral method to solve the dissipative Zakharov equations is used. Its convergence is proved by priori estimates. The existence of the global attractors and the estimates of dimension are presented. A class of steady state solutions is also disscussed. The numerical results show that if the steady state solutions satisfy some special conditions, they become unstable and limit cycles and strange attractors will occur for very small perturbations. The largest Lyapunov exponent and analysis of the linearized system are applied to explain these phenomena.

In this paper, the initial-boundary value problem of a class of nonlinear Schrodinger equation with wave operator is considered. An explicit and efficient finite difference scheme is presented. This is a scheme of four levels with a discrete conservative law. Convergence and stability are proved.

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein–Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.

This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations by general linear methods. We prove that algebraic stability together with strict stability at infinity implies B-convergence for strictly dissipative systems and that the order of B-convergence of a method is equal to the generalized stage order, where the generalized stage order is not less than the stage order, which extends the relevant results on Runge-Kutta methods. As applications of this result, B-convergence results of some classes of multistep Runge-Kutta methods are obtained.

In this paper, we apply a fixed point method to solve the total variation-based image denoising problem. An algebraic multigrid method is used to solve the corresponding linear equations. Krylov subspace acceleration is adopted to improve convergence in the fixed point iteration. A good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid. Numerical experiments demonstrate that this method is efficient and robust even for images with large noise-to-signal ratios.

In this paper, we consider compound KdV-type and KdV–Burgers-type equations with nonlinear terms of any order. The explicit exact solitary-wave solutions for the equations are obtained by means of proper transformation, which degrades the order of nonlinear terms, and an undetermined coefficient method. A solitary-wave solution with negative velocity for the generalized KdV–Burgers equation ut+upux−αuxx+uxxx=0 is found.

In this paper, efficient algorithms for the algebraic multigrid (AMG) method are advanced, and AMG methods are proposed for improving the convergence rate. A new convergence theorem of AMG is given, and a complete theoretical study of new algorithms is subsequently presented. Various numerical results demonstrate the efficiency of the new approaches.

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.

We study the solutions of block Toeplitz systems Amnu = b by the multigrid method (MGM). Here the block Toeplitz matrices Amn are generated by a nonnegative function f (x,y) with zeros. Since the matrices Amn are ill-conditioned, the convergence factor of classical iterative methods will approach 1 as the size of the matrices becomes large. These classical methods, therefore, are not applicable for solving ill-conditioned systems. The MGM is then proposed in this paper. For a class of block Toeplitz matrices, we show that the convergence factor of the two-grid method (TGM) is uniformly bounded below 1 independent of mn and the full MGM has convergence factor depending only on the number of levels. The cost per iteration for the MGM is of O(mn log mn) operations. Numerical results are given to explain the convergence rate.

The Ginzburg–Landau equation with small complex coefficients is discussed in this paper. A transformation is introduced to change the equation into a three order, ordinary differential equation and the existence of the homoclinic orbit for this system has been proved by analytical methods.

This paper studies finite difference schemes for solving the generalized nonlinear Schrodinger (GNLS) equationiut?uxx+q(|u|2)u=f(x,t)u. A new linearlized Crank?Nicolson-type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the GNLS equation. Several schemes, including Crank?Nicolson-type schemes, Hopscotch-type schemes, split step Fourier scheme, and pseudospectral scheme, are adopted for solving three model problems of GNLS equation which arise from many physical problems. withq(s)=s2,q(s)=ln(1+s), andq(s)=?4s/(1+s), respectively. The numerical results demonstrate that the linearized Crank?Nicolson scheme is efficient and robust.

A modified algebraic multigrid (AMG) method for queueing networks is presented. The method keeps the singularity of queueing networks in the coarse grid by modifying the restriction operators. Numerical results demonstrate that this method is more efficient and robust than conventional AMG method.

The Persistence of Homoclinic orbits for perturbed nonlinear Schrodinger equation with five degree term under even periodic boundary conditions is considered. The existences of the homoclinic orbits for the truncation equation is established by Melnikov's analysis and geometric singular perturbation theory.

In this paper, we consider solutions of Toeplitz systems Anu = b where the Toeplitz matrices An are generated by nonnegative functions with zeros. Since the matrices An are ill conditioned, the convergence factor of classical iterative methods, such as the damped Jacobi method, will approach one as the size n of the matrices becomes large. Here we propose to solve the systems by the multigrid method. The cost per iteration for the method is of O(n log n) operations. For a class of Toeplitz matrices which includes weakly diagonally dominant Toeplitz matrices, we show that the convergence factor of the two-grid method is uniformly bounded below one independent of n, and the full multigrid method has convergence factor depending only on the number of levels. Numerical results are given to illustrate the rate of convergence.

We present a fully implicit and discrete energy conserving finite difference scheme for the solution of an initial-boundary value problem of the nonlinear Klein-Gordon equation. A theoretical analysis is performed, and it has been demonstrated that the numerical scheme is particularly attractive when long time solutions are sought.

New formulations for the algebraic multigrid (AMG) method are presented. A new interpolation operator is developed, in which the weighting could be negative. Numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications. New approaches to construct the restriction operator and the coarse-grid equations are discussed. Two new AMG methods are proposed. Theoretical study and convergence analysis of the AMG methods are presented. The main contributions of this paper are to improve the convergence rate and to extend the range of applications of an AMG method. Numerical experiments are reported for matrix computations that resulted from partial differential equations, signal processing, and queueing network problems. The success of the proposed new AMG algorithms is clearly demonstrated by applications to non-diagonally dominant matrix problems for which the standard AMG method fails to converge.

A new conservative difference scheme is presented for the periodic initial-value problem of Zakharov equations. The scheme can be implicit or semi-explicit, depending on the choice of a parameter. The discretization of the initial condition is of second-order accuracy, which is consistent with the accuracy of the scheme. On the basis of a priori estimates and an inequality about norms, convergence of the difference solutions is proved in the energy norm. Numerical experiments with the schemes are done for several test cases. Computational results demonstrate that the new semi-explicit scheme with a new initial approximation is more accurate and computationally efficient.

In this paper, new interpolation formulas for using geometric assumptions in the algebraic multigrid (AMG) method are reported. The theoretical and convergence analysis will be presented. The effectiveness and robustness of these interpolation formulas are demonstrated by numerical experiments. Not only is a rapid rate of convergence achieved, but the AMG algorithm used in conjunction with these formulas can also be used to solve various ill-conditioned systems of equations. The principal contribution of the present method is to extend the range of applications of the AMG method developed by Ruge and Stuben.

This paper deals with the asymptotic stability of multistep multiderivative methods for systems of delay differential equations. In particular, it is shown that A-stability of multistep multiderivative methods for ordinary differential equations is equivalent to P-stability of the induced methods for delay differential equations.

The discrete Sobolev's inequalities inLp norm are proved for three-dimensional spherical and cylindrical coordinates, by using discrete Holder inequality, property of the triangle functions and complicated deduction.

A conservative difference scheme is given for a model of nonlinear dispersive waves. Convergence and stability of the scheme are proved. By means of this scheme, we explore numerically the relationship between the boundary data and the amplitudes and number of solitary waves it produces.

In this paper, we consider solutions of Toeplitz systems Au = b where the Toeplitz matrices A are generated by nonnegative functions with zeros. Since the matrices A are ill-conditioned, the convergence factor of classical iterative methods, such as the Richardson method, will approach 1 as the size n of the matrices becomes large. In [1,2], convergence of the two-grid method with Richardson method as smoother was proved for band τ matrices and it was conjectured that this convergence result can be carried to Toeplitz systems. In this paper, we show that the two-grid method with Richardson smoother indeed converges for Toeplitz systems that are generated by functions with zeros, provided that the order of the zeros are less than or equal to 2. However, we illustrate by examples that the convergence results of the two-grid method cannot be readily extended to multigrid method for n that are not of the form 2 l − 1.

In this paper, a conservative difference scheme for generalized nonlinear Schrodinger equations is given. We apply multigrid method and adaptive algorithm to solve the equations. Numerical results are presented and compared. They demonstrate that the multigrid and adaptive algorithm are efficient and can considerably relax the restrict on step size of time, which is caused by nonlinear iteration.