The application of the finite difference method to discretize the complex Helmholtz equation on a bounded region in the plane produces a linear system whose coefficient matrix is block tridiagonal and is some (complex) perturbation of an M-matrix. The matrix is also complex symmetric, and its real part is frequently indefinite. Conjugate gradient type methods are available for this kind of line...

Interior point methods have been widely used to solve large-scale linear programming problems. The bulk of the work in these methods is computing the search direction by solving one or more linear systems. The most commom approach in interior point solvers uses Cholesky sparse factorization to solve these systems. In some problems this factorization becomes prohibitive due to storage and time l...

To solve saddle point systems efficiently, several preconditioners have been published. There are many methods for constructing preconditioners for linear systems from saddle point problems, for instance, the relaxed dimensional factorization (RDF) preconditioner and the augmented Lagrangian (AL) preconditioner are used for both steady and unsteady Navier-Stokes equations. In this paper we comp...

Funding Information Russian Science Foundation, Grant/Award Number: 14-31-00024 Summary The paper studies numerical properties of LU and incomplete LU factorizations applied to the discrete linearized incompressible Navier–Stokes problem also known as the Oseen problem. A commonly used stabilized Petrov–Galerkin finite element method for the Oseen problem leads to the system of algebraic equati...

When solving a linear system in parallel, a large overhead in using an incomplete LU factorization as a preconditioner may annihilate any gains made from the improved convergence. This overhead is due to the inherently sequential nature of such a preconditioning. Multicoloring of the subdomains assigned to processors is a common remedy for increasing the parallelism of a global ordering. Howeve...

We have recently developed a new program analysis strategy called fractal symbolic analysis that addresses some of limitations of techniques such as dependence analysis. In this paper, we show how fractal symbolic analysis can be used to convert between left-looking and right-looking versions of three kernels of central importance in computational science: Cholesky factorization, LU factorizati...

Incomplete LU factorization (ILU) techniques are a well-known class of preconditioners, often used in conjunction with Krylov accelerators for the iterative solution of linear systems of equations. However, for certain problems, ILU factorizations can yield factors that are unstable, and in some cases quite dense. Reordering techniques based on permuting the matrix prior to performing the facto...

Based on a particular node ordering and corresponding block decomposition of the matrix we analyse an e cient, algebraic multilevel preconditioner for the iterative solution of nite element discretizations of elliptic boundary value problems. Thereby an analysis of a new version of block-factorization preconditioning methods is presented. The approximate factorization requires an approximation ...

Introduction Incomplete factorizations of sparse symmetric positive definite (SSPD) matrices have been used to generate preconditioners for various iterative solvers. These solvers generally use preconditioners derived from the matrix system, , in order to reduce the total number of iterations until convergence. In this report, we investigate the findings of ref. [1] on their method for computi...