## An overview of PyTrilinos

Proposed for publication in ACM Transactions on Mathematical Software.

Abstract not provided.

Proposed for publication in ACM Transactions on Mathematical Software.

Abstract not provided.

Wereportonthedesignofgeneral,flexible,consistentandefficientinterfacestodirectsolveralgorithmsforthesolutionofsystemsoflinearequations.Wesupposethatsuchalgorithmsareavailableinformofsoftwarelibraries,andweintroduceaframeworktofacilitatetheusageoftheselibraries.Thisframeworkiscomposedbytwocomponents:anabstractmatrixinterfacetoaccessthelinearsystemmatrixelements,andanabstractsolverinterfacethatcontrolsthesolutionofthelinearsystem.Wedescribeaconcreteimplementationoftheproposedframework,whichallowsahigh-levelviewandusageofmostofthecurrentlyavailablelibrariesthatimplementsdirectsolutionmethodsforlinearsystems.Wecommentontheadvantagesandlimitationoftheframework.3

Proposed for publication in the SIAM Journal on Matrix Analysis and Applications.

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 O(H/{delta} + H{sub 0}/{delta}), where H and H{sub 0} are the diameters of the subdomains and the aggregates, respectively, and {delta} is the overlap among the subdomains and the aggregates. This extends the bounds presented in [C. Lasser and A. Toselli, Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces, in Recent Developments in Domain Decomposition Methods, Lecture Notes in Comput. Sci. Engrg. 23, L. Pavarino and A. Toselli, eds., Springer-Verlag, Berlin, 2002, pp. 95-117; M. Sala, Domain Decomposition Preconditioners: Theoretical Properties, Application to the Compressible Euler Equations, Parallel Aspects, Ph.D. thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 2003; M. Sala, Math. Model. Numer. Anal., 38 (2004), pp. 765-780]. Numerical experiments on a model problem are reported to illustrate the performance of the proposed preconditioner.

Abstract not provided.

IFPACK provides a suite of object-oriented algebraic preconditioners for the solution of preconditioned iterative solvers. IFPACK constructors expect the (distributed) real sparse matrix to be an Epetra RowMatrix object. IFPACK can be used to define point and block relaxation preconditioners, various flavors of incomplete factorizations for symmetric and non-symmetric matrices, and one-level additive Schwarz preconditioners with variable overlap. Exact LU factorizations of the local submatrix can be accessed through the AMESOS packages. IFPACK , as part of the Trilinos Solver Project, interacts well with other Trilinos packages. In particular, IFPACK objects can be used as preconditioners for AZTECOO, and as smoothers for ML. IFPACK is mainly written in C++, but only a limited subset of C++ features is used, in order to enhance portability.

Abstract not provided.

Abstract not provided.

Proposed for publication in International Journal for Numerical Methods in Engineering.

This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton-Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behavior of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non-zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two-level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large-scale distributed-memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two- and three-level preconditioners are demonstrated to be scalable to 1024 processors.

ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package or to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the Aztec 2.1 and AztecOO iterative package [16]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and nonsymmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.

Abstract not provided.

Abstract not provided.

ML development was started in 1997 by Ray Tuminaro and Charles Tong. Currently, there are several full- and part-time developers. The kernel of ML is written in ANSI C, and there is a rich C++ interface for Trilinos users and developers. ML can be customized to run geometric and algebraic multigrid; it can solve a scalar or a vector equation (with constant number of equations per grid node), and it can solve a form of Maxwell's equations. For a general introduction to ML and its applications, we refer to the Users Guide [SHT04], and to the ML web site, http://software.sandia.gov/ml.

This document describes the main functionalities of the Amesos package, version 1.0. Amesos, available as part of Trilinos 4.0, provides an object-oriented interface to several serial and parallel sparse direct solvers libraries, for the solution of the linear systems of equations A X = B where A is a real sparse, distributed matrix, defined as an EpetraRowMatrix object, and X and B are defined as EpetraMultiVector objects. Amesos provides a common look-and-feel to several direct solvers, insulating the user from each package's details, such as matrix and vector formats, and data distribution.

The Trilinos Project is an effort to facilitate the design, development, integration and ongoing support of mathematical software libraries. The goal of the Trilinos Project is to develop parallel solver algorithms and libraries within an object-oriented software framework for the solution of large-scale, complex multiphysics engineering and scientific applications. The emphasis is on developing robust, scalable algorithms in a software framework, using abstract interfaces for flexible interoperability of components while providing a full-featured set of concrete classes that implement all the abstract interfaces. This document introduces the use of Trilinos, version 4.0. The presented material includes, among others, the definition of distributed matrices and vectors with Epetra, the iterative solution of linear systems with AztecOO, incomplete factorizations with IF-PACK, multilevel and domain decomposition preconditioners with ML, direct solution of linear system with Amesos, and iterative solution of nonlinear systems with NOX. The tutorial is a self-contained introduction, intended to help computational scientists effectively apply the appropriate Trilinos package to their applications. Basic examples are presented that are fit to be imitated. This document is a companion to the Trilinos User's Guide [20] and Trilinos Development Guides [21,22]. Please note that the documentation included in each of the Trilinos' packages is of fundamental importance.

ML is a multigrid preconditioning package intended to solve linear systems of equations Az = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package or to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the AZTEC 2.1 and AZTECOO iterative package [15]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and non-symmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.

This document introduces the use of Trilinos, version 3.1. Trilinos has been written to support, in a rigorous manner, the solver needs of the engineering and scientific applications at Sandia National Laboratories. Aim of this manuscript is to present the basic features of some of the Trilinos packages. The presented material includes the definition of distributed matrices and vectors with Epetra, the iterative solution of linear system with AztecOO, incomplete factorizations with IFPACK, multilevel methods with ML, direct solution of linear system with Amesos, and iterative solution of nonlinear systems with NOX. With the help of several examples, some of the most important classes and methods are detailed to the inexperienced user. For the most majority, each example is largely commented throughout the text. Other comments can be found in the source of each example. This document is a companion to the Trilinos User's Guide and Trilinos Development Guides. Also, the documentation included in each of the Trilinos' packages is of fundamental importance.

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