Conjugate gradient solvers

A solution is to use Krylov subspace methods, such as the conjugate gradient (CG) method, the biconjugate gradient (BiCG) 

These matrix multiplications can be avoided using a combination of the CG and BiCG algorithms to bypass construction of K [19]. In the first stage, the BiCG algorithm is used to solve the equation 

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In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

In contrast to the conjugate gradient method, the multigrid method is rather a general framework for iterative solvers than a specific method. The multigrid method exploits the fact that the iteration error... 

P 77] Simulation was done using CFD-ACE+ [55,163], The physical properties of water were assumed. Discretization with structured grids of 15 pm length was used. The first-order upwind scheme and conjugate gradient with preconditioning solver were applied. 

Algebraic equation solvers conjugate gradient Acceleration tools multigrid/block correction Parallelization technology speed-up efficiency... 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

The final system of equations is S3munetric, positive definite, hence the solution of such a system can be obtained in an efficient way using standard matrix solvers like conjugate gradient method. 

The system of N(M + 1) equations is then solved by using standard nonlinear solvers such as the Newton-Raphson method or the conjugate-gradient minimization algorithm, both of which are described in Press et al. (1992). 

In our contribution, we address this aspect and describe numerical methods based on the use of efficient iterative solvers, which exploit the conjugate gradient (CG) method, its generalization and the space decomposition preconditioners. The efficiency of these solvers will be illustrated by the solution of elasticity and thermo-elasticity problems arising from the finite element analysis of selected benchmarks with computations performed on a PC cluster. The introduced ideas could be useful also for the solution of more complicated coupled problems. 

M-Shake This is Newton-iteration-based implementation of SHAKE, using (4.18)-(4.19) to solve Eqs. (4.27)-(4.29). Methods like this were first proposed by Ciccotti and Ryckaert [84] in the context of rigid body molecular dynamics. An extended discussion of such methods with reference to their convergence, implementation, in particular linear system solvers, and variants such as SHAKE-SOR (which uses the successive over-relaxation method) can be found in [25]. A conjugate gradient method can also be used [392]. 

Used as a direct problem solver to work with simplified conjugate-gradient method optimizer to solve for optimal gas channel width fraction, gas channel height, and thickness of gas diffusion layer... 

The Estimates, Derivatives and Search parameters can be changed to optimize the solution process. The Search parameter specifies which gradient search method to use the Newton method requires more memory but fewer iterations, the Conjugate method requires less memory but more iterations. The Derivatives parameter specifies how the gradients for the search are calculated the Central derivatives method requires more calculations but may be helpful if the Solver reports that it is unable to find a solution. The Estimates parameter determines the method by which new estimates of the coefficients are obtained from previous values the Quadratic method may improve results if the system is highly nonlinear. 

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