<p>The conjugate gradient methods are noted to be exceedingly valuable for solving large-scale unconstrained optimization problems since it needn't the storage of matrices. Mostly parameter is focus methods. The current paper proposes new type solve optimization. A Hessian approximation in a diagonal matrix form on basis second and third-order Taylor series expansion was employed this stu...

Discretizing the elliptic Poisson equation with homogeneous Dirichlet boundary conditions by the finite difference method results in a system of linear equations with a large, sparse, highly structured system matrix. It is a classical test problem for comparing the performance of direct and iterative linear solvers. We compare in this report Gaussian elimination applied to a dense system matrix...

In this preliminary work, left and right conjugate vectors are deened for nonsymmetric, nonsin-gular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method reduces to the usual conjugate gradient method when A is symmetric positive deenite. A nite termination property of th...

A new approach was presented in [11] for construcling preconditioners through a function approximation for the domain decomposi~ion-basedpreconditioned conjugate gradient method. This work extends the approach to more general cases where grids may be nonuniform; elliptic operatoI'B may have variable coefficients (but are separable and self-adjoint); and geometric domains may be nonrectangular. ...

We present a domain decomposition method suitable for the drift-diiusion equations. The new scheme is applied to the linear systems resulting from Gummel's method and corresponds to a preconditioned conjugate gradient technique for the Schur complement of the linear systems. In designing the preconditioner two problems are addressed : anisotropic phenomena and large scale variations in the magn...

We propose a generalization of the conjugate gradient method that uses multiple preconditioners, combining them automatically in an optimal way. The derivation is described in detail, and analytical observations are made. A short recurrence relation does not hold in general for this new method, but in at least one case such a relation is satisfied: for two symmetric positive definite preconditi...

AbRtract This paper deals with the domain decomposition-based preconditioned conjugate gradient method. The Schur complement is expressed as a function of & simple interface matrix. This function is approximated by a simple rational function to generate a simple matrix that is then used 8.8 & preconditioner for the Schur complement. Extensive experiments are performed to examine the effectivene...

The concept of multilevel filtering (MF) preconditioning applied to second-order selfadjoint elliptic problems is briefly reviewed. It is then shown how to effectively apply this concept to other elliptic problems such as the second-order anisotropic problem, biharmonic equation, equations on locally refined grids and interface operators arising from domain decomposition methods. Numerical resu...

This study develops and analyzes preconditioned Krylov subspace methods to solve linear systems arising from discretization of the time-independent space-fractional models. First, we apply shifted Grunwald formulas to obtain a stable finite difference approximation to fractional advection-diffusion equations. Then, we employee two preconditioned iterative methods, namely, the preconditioned gen...