Conjugate Gradient Method For Systems Research Articles

Enhanced Dai–Liao conjugate gradient methods for systems of monotone nonlinear equations

In this paper, we propose two conjugate gradient methods for solving large-scale monotone nonlinear equations. The methods are developed by combining the hyperplane projection method by Solodov and Svaiter (Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Springer, pp 355–369, 1998) and two modified search directions of the famous Dai and Liao (Appl Math Optim 43(1): 87–101, 2001) method. It is shown that the proposed schemes satisfy the sufficient descent condition. The global convergence of the methods are established under mild conditions, and computational experiments on some benchmark test problems show that the methods are promising.

Descent Perry conjugate gradient methods for systems of monotone nonlinear equations

In this paper, we present a family of Perry conjugate gradient methods for solving large-scale systems of monotone nonlinear equations. The methods are developed by combining modified versions of Perry (Oper. Res. Tech. Notes 26(6), 1073–1078, 1978) conjugate gradient method with the hyperplane projection technique of Solodov and Svaiter (1998). Global convergence and numerical results of the methods are established and preliminary numerical results shows that the proposed methods are promising and more effective compared to some existing methods in the literature.

Restrictively preconditioned conjugate gradient methods for systems of linear equations

A restrictively preconditioned conjugate gradient method is presented for solving a large sparse system of linear equations. This new method originates from the classical conjugate gradient method and its restrictively preconditioned variant, and covers many standard Krylov subspace iteration methods such as the conjugate gradient, conjugate residual, CGNR, CGNE and the corresponding preconditioned variants. In particular, the new method is applied to solve the linear systems of block two‐by‐two structures, and consequently, special but quite efficient preconditioned conjugate gradient‐like methods are obtained for these problems. Numerical computations show that the new methods are more effective than some standard preconditioned Krylov subspace methods like the preconditioned conjugate gradient and the preconditioned MINRES.

Parallel Multigrid Preconditioning of the Conjugate Gradient Method for Systems of Subsurface Hydrology

Parallel preconditioners are considered for improving the convergence rate of the conjugate gradient method for solving sparse symmetric positive definite systems generated by finite element models of subsurface flow. The difficulties of adapting effective sequential preconditioners to the parallel environment are illustrated by our treatment of incomplete Cholesky preconditioning. These difficulties are avoided with multigrid preconditioning, which can be extended naturally to many processors so that the preconditioner remains global and effective.The coarse grid correction which defines the multigrid preconditioner is outlined and its parallel implementation with the distributed finite element data structure is presented, along with some examples of its use as a parallel preconditioner.

Conjugate gradient method for systems of nonlinear equations

One presents an iteration method for solving nonlinear algebraic systems, based on the ideas of the conjugate gradient method. One proves the convergence of the method and one obtains estimates for the rate of convergence.