Ray S. Tuminaro

An overview of the Trilinos project

The Trilinos Project is an effort to facilitate the design, development, integration, and ongoing support of mathematical software libraries within an object-oriented framework for the solution of large-scale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software.Trilinos uses a two-level software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common look-and-feel, including configuration, documentation, licensing, and bug-tracking.Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates high-quality software engineering practices that are increasingly required from simulation software.

Parallel multigrid smoothing: polynomial versus Gauss--Seidel

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poissons equation, thin-body elasticity, and eddy current approximations to Maxwells equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.

Block Preconditioners Based on Approximate Commutators

This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier--Stokes equations. We consider the pressure convection--diffusion preconditioners proposed by Kay, Loghin, and Wathen [SIAM J. Sci. Comput., 24 (2002), pp. 237-256] and Silvester, Elman, Kay, and Wathen [J. Comput. Appl. Math., 128 (2001), pp. 261-279]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology. A drawback, however, is that it requires the construction of a convection--diffusion operator (denoted

Parallel Smoothed Aggregation Multigrid : Aggregation Strategies on Massively Parallel Machines

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A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations

) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing

Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations

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A parallel block multi-level preconditioner for the 3D incompressible Navier--Stokes equations

based on purely algebraic considerations. The new methods are closely related to the BFBt preconditioner of Elman [SIAM J. Sci. Comput., 20 (1999), pp. 1299-1316]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection--diffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix

A comparison of preconditioned nonsymmetric Krylov methods on a large-scale MIMD machine

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Parallel sparse matrix vector multiply software for matrices with data locality

automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.

PRESERVING LAGRANGIAN STRUCTURE IN NONLINEAR MODEL REDUCTION WITH APPLICATION TO STRUCTURAL DYNAMICS

Algebraic multigrid methods offer the hope that multigrid convergence can be achieve (for at least some important applications) without a great deal of effort from engineers an scientists wishing to solve linear systems. In this paper we consider parallelization of the smoothe aggregation multigrid methods. Smoothed aggregation is one of the most promising algebraic multigrid methods. Therefore, eveloping parallel variants with both good convergence an efficiency properties is of great importance. However, parallelization is nontrivial due to the somewhat sequential aggregation (or grid coarsening) phase. In this paper, we discuss three different parallel aggregation algorithms an illustrate the advantages an disadvantages of each variant in terms of parallelism an convergence. Numerical results will be shown on the Intel Teraflop computer for some large problems coming from nontrivial codes: quasi-static electric potential simulation an a fluid flow calculation.