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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 of Tai et al., the modified TV-Stokes models for image denoising are considered. In this paper, we present some new fast algorithms for the modified TV-Stokes models. In the first step, we use the dual formulation of denoising and multigrid method to get a fast algorithm. Another new imcompressibility-preserved algorithm is proposed for the tangent field smoothing. In the second step, completely new algorithms are presented by us. A function g fitting the smoothing normal field is computed by the Poisson equation. Then, restored image is obtained efficiently by using the algorithm of Jia and Zhao. This new method is very fast such that the CPU time in the reconstruction step is less than 0.1 s. Numerical results demonstrate that our new algorithms are efficient and robust. The restored images are better than the general denoising methods.

In this paper, we propose an efficient combined algorithm of split Bregman method, algebraic multigrid (AMG) method and Krylov acceleration method for deblurring and denoising. The split Bregman method is used to convert nonlinear TV model into three linear systems. But the linear system with blur operator is difficult to solve. We add an auxiliary linear stabilizing term to the linear system, then apply an AMG method and Krylov acceleration method to solve the new linear system. Various numerical experiments and comparisons demonstrate that the combined algorithm is efficient, fast, comparable to several existing algorithms, and robust over a wide range of parameters.

In this paper, we construct a Lattice Boltzmann scheme to simulate the well known total variation based restoration model, that is, ROF model. The advantages of the Lattice Boltzmann method include the fast computational speed and the easily implemented fully parallel algorithm. A conservative property of the LB method is discussed. The macroscopic PDE associated with the LB algorithm is derived which is just the ROF model. Moreover, the linearized stability of the method is analyzed. The numerical computations demonstrate that the LB algorithm is efficient and robust. Even though the quality of the restored images is slightly lower than those by using the ROF model, the restored images of the LB method are satisfactory. Furthermore, computational speed of the LB method is much faster than ROF model. In general, CPU time of the LB method for restored images is about one tenth of ROF model.

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, 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 full waveform inversion, the lack of low frequency information in the inversion results has been a long standing problem. In this work, we show that by using mixed basis functions this problem can be resolved satisfactorily. Examples of full waveform inversion on layered systems, using surface reflection data from point sources, have shown excellent results nearly indistinguishable from the target model. Our method is robust against additive white noise (up to 20\% of the signal) and can resolve layers that are comparable to or smaller than a wavelength in thickness. Physical reason for the success of our approach is illustrated through a simple example.

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.

In this Letter we study the limit behavior of solution of multi-dimensional Landau–Lifshitz equation with external magnetic field and construct some exact solutions via an explicit transform between Landau–Lifshitz equation with external magnetic field and the one without external magnetic field.

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 improve the Brauer-type inclusion interval and present a modified Brauer-type theorem for singular values. Numerical examples show that our estimation for singular values is more precise than those corresponding results in recent literature. Moreover, we also consider the marginal singular value on the Brauer-type inclusion interval. Some applications of our results are presented.

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of linear multistep methods, one-leg methods, and Runge-Kutta methods is considered. The emphasis is on the linear stability of numerical methods. It is proved that A -stable methods can inherit the asymptotic stability of underlying linear systems. Some general results of stability on explicit and implicit methods are also given.

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.

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.

In this paper, multigrid methods for solving the biharmonic equation using some nonconforming plate elements are considered. An average algorithm is applied to define the transfer operator. A general analysis of convergence is given.

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.