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Abstract: In this paper, we use a projected gradient algorithm to solve a nonlinear operator equation with -norm constraint. Gradient iterations with -norm constraints have been studied recently both in the context of inverse problem and of compressed sensing. In this paper, the constrained gradient iteration is implemented via a projected operator. We establish the -norm convergence of sequence constructed by the constrained gradient iteration when . The performance of the method is testified by a numerical example. [Copyright &y& Elsevier]

Abstract: In this paper, based on GCRS algorithm in Zhang and Zhao (2010) and the ideas in Gu et al. (2007), we present an improved generalized conjugate residual squared (IGCRS) algorithm that is designed for distributed parallel environments. The new improved algorithm reduces two global synchronization points to one by changing the computation sequence in the GCRS algorithm in such a way that all inner products per iteration are independent so that communication time required for inner products can be overlapped with useful computation. Theoretical analysis and numerical comparison of isoefficiency analysis show that the IGCRS method has better parallelism and scalability than the GCRS method, and the parallel performance can be improved by a factor of about 2. Finally, some numerical experiments clearly show that the IGCRS method can achieve better parallel performance with a higher scalability than the GCRS method and the improvement percentage of communication is up to 52.19% averagely, which meets our theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a reweighted nuclear norm minimization algorithm based on the weighted fixed point method (RNNM–WFP algorithm) to recover a low rank matrix, which iteratively solves an unconstrained minimization problem introduced as a nonconvex smooth approximation of the low rank matrix minimization problem. We prove that any accumulation point of the sequence generated by the RNNM–WFP algorithm is a stationary point of the minimization problem. Numerical experiments on randomly generated matrix completion problems indicate that the proposed algorithm has better recoverability compared to existing iteratively reweighted algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method. [Copyright &y& Elsevier]

Abstract: A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA–GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of -error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results. [Copyright &y& Elsevier]

Abstract: The variational iteration method and the homotopy analysis method, as alternative methods, have been widely used to handle linear and nonlinear models. The main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently. This paper deals with the numerical solutions of nonlinear fractional differential equations, where the fractional derivatives are considered in Caputo sense. The main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations. In these algorithms, Legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work. The proposed algorithms are capable of reducing the size of calculations, improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals. Numerical examples are examined to show the efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems. [ABSTRACT FROM AUTHOR]

Abstract: The inexact GMRES algorithm is a variant of the GMRES algorithm where matrix–vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix–vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix–vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix–vector products. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new iterative method (successive projection iterative method) to solve matrix equation , where is a symmetric positive definite (SPD) matrix. Based on this method an algorithm is proposed and proved to be convergent. In addition, analysis of the algorithm and numerical experiments are also given to show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: For nonsmooth convex optimization, Robert Mifflin and Claudia Sagastizábal introduce a -space decomposition algorithm in Mifflin and Sagastizábal (2005) . An attractive property of this algorithm is that if a primal–dual track exists, this algorithm uses a bundle subroutine. With the inclusion of a simple line search, it is proved to be globally and superlinearly convergent. However, a drawback is that it needs the exact subgradients of the objective function, which is expensive to compute. In this paper an approximate decomposition algorithm based on proximal bundle-type method is introduced that is capable to deal with approximate subgradients. It is shown that the sequence of iterates generated by the resulting algorithm converges to the optimal solutions of the problem. Numerical tests emphasize the theoretical findings. [Copyright &y& Elsevier]

Abstract: A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes. [Copyright &y& Elsevier]

Abstract: Ill-posed problems are numerically underdetermined. It is therefore often beneficial to impose known properties of the desired solution, such as nonnegativity, during the solution process. This paper proposes the use of an interior-point method in conjunction with truncated iteration for the solution of large-scale linear discrete ill-posed problems with box constraints. An estimate of the error in the data is assumed to be available. Numerical examples demonstrate the competitiveness of this approach. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of boundary value problems in the reproducing kernel space. The analytical solution and approximate solution are represented in terms of series. For any initial function , we prove , , . Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective. [Copyright &y& Elsevier]

Abstract: The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian''s work and Lev-Ari''s results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case. [Copyright &y& Elsevier]

Abstract: In this paper a globally convergent first-order training algorithm is proposed that uses sign-based information of the batch error measure in the framework of the nonlinear Jacobi process. This approach allows us to equip the recently proposed Jacobi–Rprop method with the global convergence property, i.e. convergence to a local minimizer from any initial starting point. We also propose a strategy that ensures the search direction of the globally convergent Jacobi–Rprop is a descent one. The behaviour of the algorithm is empirically investigated in eight benchmark problems. Simulation results verify that there are indeed improvements on the convergence success of the algorithm. [Copyright &y& Elsevier]

For tridiagonal matrix systems, a simple direct algorithm giving the solution exists, but in the most general case of tridiagonal matrix with fringes, the direct solving algorithms are more complicated. For big systems, direct methods are not well fitted and iterative algorithms are preferable. In this paper a relaxation type iterative algorithm is presented. It is an extension of the backward substitution method used for simple tridiagonal matrix systems. The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR. Its nature suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method. A special emphasis is done on the programming aspect. The solving Fortran subroutines implementing the algorithm have been generated automatically from their specification by using a computer algebra system technique. [Copyright &y& Elsevier]