The application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a linear system whose coefficient matrix is block tridiagonal and symmetric indefinite. Such a linear system can be solved efficiently by a conjugate residual method, particularly when combined with a good preconditioner. We show that specific incomplete block factorization exi...

Consider the solution of a large linear system when the coe cient matrix is sparse, symmetric, and positive de nite. One approach is the method of \conjugate gradient" (CG) with an incomplete Cholesky (IC) preconditioner (ICCG). A key problem with the design of a parallel ICCG solver is the bottleneck posed by repeated parallel sparse triangular solves to apply the preconditioner. Our work conc...

In this paper, we use the PCG method to solve the linear systems obtained from the continuity equations of semiconductor models. A new kind of preconditioner which is based on the domain decomposition technique is discussed in detail. This kind of preconditioner is suitable for parallel computing and efficient for the linear system with rapidly varied solutions. Since the class of linear system...

In the present study, an advection-diffusion problem has been considered for the numerical solution. The continuum equation is discretized using both upwind and centered scheme. The linear system is solved using the ILU preconditioned BiCGSTAB method. Both Dirichlet and Neumann boundary condition has been considered. The obtained results have been compared for different cases.

Limited-memory incomplete Cholesky factorizations can provide robust preconditioners for sparse symmetric positive-definite linear systems. In this paper, the focus is on extending the approach to sparse symmetric indefinite systems in saddle-point form. A limited-memory signed incomplete Cholesky factorization of the form LDL is proposed, where the diagonal matrix D has entries ±1. The main ad...

The solution of large sparse and ill-conditioned systems of linear equations is a central task in numerical linear algebra. Such systems arise from many applications like the discretization of partial differential equations or image restoration. Herefore, Gaussian elimination or other classical direct solvers can not be used since the dimension of the underlying coefficient matrices is too larg...

SUMMARY Efficient numerical wave modelling is essential for imaging methods such as reverse-time migration (RTM) and full waveform inversion (FWI). In 2D, frequency-domain modelling with LU factorization as a direct solver can outperform time-domain methods by one order. For 3-D problems, the computational complexity of the LU factorization as well as its memory requirements are a disadvantage ...

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using...

In this paper we study the parallel aspects of PCGLS, a basic iterative method whose main idea is to organize the computation of conjugate gradient method with preconditioner applied to normal equations, and Incomplete Modified Gram-Schmidt (IMGS) preconditioner for solving sparse least squares problems on massively parallel distributed memory computers. The performance of these methods on this...

We describe a novel technique for computing a sparse incomplete factorization of a general symmetric positive de nite matrix A. The factorization is not based on the Cholesky algorithm (or Gaussian elimination), but on A-orthogonalization. Thus, the incomplete factorization always exists and can be computed without any diagonal modi cation. When used in conjunction with the conjugate gradient a...