Marzio Sala

A New Petrov-Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems

We propose a new variant of smoothed aggregation (SA) suitable for nonsymmetric linear systems. The new algorithm is based on two key generalizations of SA: restriction smoothing and local damping. Restriction smoothing refers to the smoothing of a tentative restriction operator via a damped Jacobi-like iteration. Restriction smoothing is analogous to prolongator smoothing in standard SA and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric. Local damping refers to damping parameters used in the Jacobi-like iteration. In standard SA, a single damping parameter is computed via an eigenvalue computation. Here, local damping parameters are computed by considering the minimization of an energy-like quantity for each individual grid transfer basis function. Numerical results are given showing how this method performs on highly nonsymmetric systems.

Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling

In this study results are presented for the large-scale parallel performance of an algebraic multilevel preconditioner for solution of the drift-diffusion model for semiconductor devices. The preconditioner is the key numerical procedure determining the robustness, efficiency and scalability of the fully-coupled Newton-Krylov based, nonlinear solution method that is employed for this system of equations. The coupled system is comprised of a source term dominated Poisson equation for the electric potential, and two convection-diffusion-reaction type equations for the electron and hole concentration. The governing PDEs are discretized in space by a stabilized finite element method. Solution of the discrete system is obtained through a fully-implicit time integrator, a fully-coupled Newton-based nonlinear solver, and a restarted GMRES Krylov linear system solver. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the nonzero block structure of the Jacobian matrix. Representative performance results are presented for various choices of multigrid V-cycles and W-cycles and parameter variations for smoothers based on incomplete factorizations. Parallel scalability results are presented for solution of up to 10^8 unknowns on 4096 processors of a Cray XT3/4 and an IBM POWER eServer system.

An Improved Convergence Bound for Aggregation-Based Domain Decomposition Preconditioners

In this paper we present a two-level overlapping domain decomposition preconditioner for the finite-element discretization of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction, based on aggregation techniques, is added. Our definition of the coarse space does not require the introduction of a coarse grid. We consider a set of assumptions on the coarse basis functions to bound the condition number of the resulting preconditioned system. These assumptions involve only geometrical quantities associated with the aggregates and the subdomains. We prove that the condition number using the two-level additive Schwarz preconditioner is

Domain Decomposition Techniques

\mathcal{O}(H/\delta + H_0/\delta)

Analysis of two-level domain decomposition preconditioners based on aggregation

, where

ROBUST ALGEBRAIC PRECONDITIONERS USING IFPACK 3.0.

H

An object-oriented framework for the development of scalable parallel multilevel preconditioners

and

Algebraic coarse grid operators for domain decomposition based preconditioners

H_0

A parallel adaptive newton-krylov-schwarz method for 3D compressible inviscid flow simulations

are the diameters of the subdomains and the aggregates, respectively, and

Algebraic Multilevel Preconditioners for Nonsymmetric PDEs on Stretched Grids

\delta