Block Preconditioning for the Conjugate Gradient Method

Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

“…In particular, this is the case of the matrices related to the discretization important classes of time-dependent partial differential equations, see [4,5]. This basic constraint suggested to Bai, Golub and Ng to use a natural splitting of A n in terms of the Hermitian part and of the skew-Hermitian part of A n (see also [12,13] for related splittings). More precisely, for a given matrix A n , there exists a unique Hermitian/skew-Hermitian decomposition We are interested in the case where the real part Re(A n ), which is Hermitian by definition, is positive definite as well.…”

Multigrid preconditioners for symmetric Sinc systems

“…Considering that the computational complexity of solving Bv = r is still quite expensive, Ng [9] further adopted the minv [3] as a second-step preconditioner to precondition the matrix B in numerical experiments. Numerical results showed that the preconditioned conjugate gradient method with the two-step preconditioner composed of the banded matrix B and its minv factorization converges quite rapidly.…”

Mixed finite elements for accurate particle tracking in saturated groundwater flow

“…), such that <p = gD on ef!D and Problem (3.4) has a unique solution (see [8: (3.5). A variety of choices for the preconditioning matrix have been discussed in the literature (see, e.g., [2], [9]). Popular methods for computing the preconditioning matrix are the incomplete Cholesky decomposition (see [19]) and the modified incomplete Cholesky decomposition ([see [14], [3: Section 1.4]).…”