Jonathan Joseph Hu

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.

ML 3.0 Smoothed Aggregation User's Guide

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 Maxwells 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.

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.

An Improved Algebraic Multigrid Method for Solving Maxwell's Equations

We propose two improvements to the Reitzinger and Schoberl algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwells equations. The main focus in the Reitzinger/Schoberl method is to maintain null space properties of the weak

A new smoothed aggregation multigrid method for anisotropic problems

\nabla \times \nabla \times

Toward an h-Independent Algebraic Multigrid Method for Maxwell's Equations

operator on coarse grids. While these null space properties are critical, they are not enough to guarantee h-independent convergence of the overall multigrid method. We illustrate how the Reitzinger/Schoberl AMG method loses h-independence due to the somewhat limited approximation property of the grid transfer operators. We present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates. The first improvement is based on smoothing the Reitzinger/Schoberl grid transfer operators. The second improvement is obtained by using higher order nodal interpolation to derive the corresponding AMG interpolation operators. While not completely h-independent, the resulting AMG/CG method demonstrates improved convergence behavior while maintaining low operator complexity.

An Algebraic Multigrid Approach Based on a Compatible Gauge Reformulation of Maxwell's Equations

A new prolongator is proposed for smoothed aggregation (SA) multigrid. The proposed prolongator addresses a limitation of standard SA when it is applied to anisotropic problems. For anisotropic problems, it is fairly standard to generate small aggregates (used to mimic semi-coarsening) in order to coarsen only in directions of strong coupling. Although beneficial to convergence, this can lead to a prohibitively large number of non-zeros in the standard SA prolongator and the corresponding coarse discretization operator. To avoid this, the new prolongator modifies the standard prolongator by shifting support (non-zeros within a prolongator column) from one aggregate to another to satisfy a specified non-zero pattern. This leads to a sparser operator that can be used effectively within a multigrid V-cycle. The key to this algorithm is that it preserves certain null space interpolation properties that are central to SA for both scalar and systems of partial differential equations (PDEs). We present two-dimensional and three-dimensional numerical experiments to demonstrate that the new method is competitive with standard SA for scalar problems, and significantly better for problems arising from PDE systems.

Spatially adaptive stochastic methods for fluid-structure interactions subject to thermal fluctuations in domains with complex geometries

We propose a new algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwells equations. This AMG method has its roots in an algorithm proposed by Reitzinger and Schoberl. The main focus in the Reitzinger and Schoberl method is to maintain null-space properties of the weak

TOWARDS EXTREME-SCALE SIMULATIONS FOR LOW MACH FLUIDS WITH SECOND-GENERATION TRILINOS

\nabla \times \nabla \>{\times}

Design considerations for a flexible multigrid preconditioning library

operator on coarse grids. While these null-space properties are critical, they are not enough to guarantee