In this report we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems. In earlier works, the local Schur complements were computed exactly using a sparse direct solver. The robustness of the preconditioner comes at the price of this memory and time intensive computation that is t...

The scalability and robustness of a class of non-overlapping domain decomposition preconditioners using 2-way nested dissection reordering is studied. In particular, three methods are considered: a nested symmetric successive over-relaxation (NSSOR), a nested version of modified ILU with rowsum constraint (NMILUR), and nested filtering factorization (NFF). The NMILUR preconditioner is exact wit...

This thesis presents a parallel resolution method for sparse linear systems which combines effectively techniques of direct and iterative solvers using a Schur complement approach. A domain decomposition is built ; the interiors of the subdomains are eliminated by a direct method in order to use an iterative method only on the interface unknowns. The system on the interface (Schur complement) i...

We present a new preconditioner for linear systems arising from finite-elements discretizations of scalar elliptic partial differential equations (pde’s). The solver splits the collection {Ke} of element matrices into a subset E(t) of matrices that are approximable by diagonally-dominant matrices and a subset of matrices that are not approximable. The approximable Ke’s are approximated by diago...

A key ingredient in the solution of a large, sparse system of linear equations by an iterative method like conjugate gradients is a preconditioner, which is in a sense an approximation to the matrix of coefficients. Ideally, the iterative method converges much faster on the preconditioned system at the extra cost of one solve against the preconditioner per iteration. We survey a little-known te...

Iterative solvers for sparse linear systems often benefit from using preconditioners. While there are implementations for many iterative methods that leverage the computing power of accelerators, porting the latest developments in preconditioners to accelerators has been challenging. In this paper we develop a selfadaptive multi-elimination preconditioner for graphics processing units (GPUs). T...

Two methods for solving the generalized Stokes problems that occur in viscous, incompressible ows are described and tested. Both are based on some type of linear algebraic orthogonaliza-tion process. The rst, EMGS, is a preconditioner derived from an incomplete Gram{Schmidt factorization, and it is proven to exist whenever the matrix being preconditioned can be factored using Gaussian eliminati...

Incomplete factorization is one of the most effective general-purpose preconditioning methods for Krylov subspace solvers for large sparse systems of linear equations. Two techniques for enhancing the robustness and performance of incomplete LU factorization for sparse unsymmetric systems are described. A block incomplete factorization algorithm based on the Crout variation of LU factorization ...

We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE’s). The solver splits the collection {Ke} of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable Ke’s are approximated by diagonally ...