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...

The solution of large systems of linear equations is typically achieved by iterative methods. The rate of convergence of these methods can be substantially improved by the use of preconditioners, which can be either applied in a black-box fashion to the linear system, or exploit properties specific to the underlying problem for maximum efficiency. However, with the shift towards multiand many-c...

An iterative algorithm for solving a mixed finite element formulation of NavierStokes equations on a distributed memory computer is presented. The solver is a Krylov subspace method with a parallel preconditioner suitable for high latency clusters. Nodes are pivoted to minimize the number of synchronization points in each solver iteration. An unstructured mesh is decomposed into non-overlapping...

We present a new parallelizable preconditioner that is used as the local component for a two-level preconditioner similar to BPS. On 2D model problems that exhibit either high anisotropy or discontinuity, we demonstrate its attracting numerical behaviour and compare it to the regular BPS. Finally, to alleviate the construction cost of this new preconditioner, that requires the explicit computat...

We study the computation of the orthogonal spline collocation solution of a linear Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form Lu = ∑ aij(x)uxixj + ∑ bi(x)uxi + c(x)u. We apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator, and prove that the res...

The problem of recovering a parameter function based on measurements of solutions of a system of partial diierential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature , the constraints are eliminated and the resulting uncons...

A method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix A is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations. It is proved that in exact arithmetic the preconditioner is well defined if A is an H-matrix. The results of numerical experiments are pre...

Numerical performance of two different preconditioning approaches, modified SSOR (MSSOR) preconditioner and incomplete factorization with zero fill-in (ILU0) preconditioner, is compared for the iterative solution of symmetric indefinite linear systems arising from finite element discretization of the Biot’s consolidation equations. Numerical results show that the nodal ordering affect the perfo...

ILU(k) is an important preconditioner widely used in many linear algebra solvers for sparse matrices. Unfortunately, there is still no highly scalable parallel ILU(k) algorithm. This paper presents the first such scalable algorithm. For example, the new algorithm achieves 50 times speedup with 80 nodes for general sparse matrices of dimension 160,000 that are diagonally dominant. The algorithm ...