Search

In this paper, we study a class of efficient iterative algorithms for the large sparse nonsingular saddle point problems based on the upper and lower triangular (ULT) splitting of the coefficient matrix. We call these algorithms ULT methods. First, the ULT algorithm is established and the characteristic of eigenvalues of the iteration matrix of these new methods is analyzed. Then we give the sufficient and necessary conditions for the convergence of these ULT methods. Moreover, the optimal iteration parameters and the corresponding convergence factors for some special cases of the ULT methods are presented. Numerical experiments on a few model problems are presented to support the theoretical results and examine the numerical effectiveness of these new methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems, etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result. [ABSTRACT FROM AUTHOR]

In this paper, we establish a class of accelerated Uzawa (AU) algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique. This extrapolation technique is based on the eigenvalues of the iterative matrix. These AU algorithms involve two iteration parameters whose special choices can cover the known classical Uzawa method, as well as yield new ones. Firstly, the accelerated model for the Uzawa algorithm is established and the detail algorithm description of AU method is presented. Then the convergence analyse of the AU method is given. Moreover, theoretical analyses show that the AU algorithm converges faster than some Uzawa-type methods (the Uzawa method is also included in) when the eigenvalues of the iterative matrix and the parameter τ satisfy some conditions. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the AU method. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method. [ABSTRACT FROM PUBLISHER]