27 research outputs found
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
Performance of a parallel code for the Euler equations on hypercube computers
The performance of hypercubes were evaluated on a computational fluid dynamics problem and the parallel environment issues were considered that must be addressed, such as algorithm changes, implementation choices, programming effort, and programming environment. The evaluation focuses on a widely used fluid dynamics code, FLO52, which solves the two dimensional steady Euler equations describing flow around the airfoil. The code development experience is described, including interacting with the operating system, utilizing the message-passing communication system, and code modifications necessary to increase parallel efficiency. Results from two hypercube parallel computers (a 16-node iPSC/2, and a 512-node NCUBE/ten) are discussed and compared. In addition, a mathematical model of the execution time was developed as a function of several machine and algorithm parameters. This model accurately predicts the actual run times obtained and is used to explore the performance of the code in interesting but yet physically realizable regions of the parameter space. Based on this model, predictions about future hypercubes are made
Non-invasive multigrid for semi-structured grids
Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to
promote the efficient utilization of high performance computer architectures.
These HHG meshes are constructed by uniformly refining a relatively coarse
fully unstructured mesh. While HHG meshes provide some flexibility for
unstructured applications, most multigrid calculations can be accomplished
using efficient structured grid ideas and kernels. This paper focuses on
generalizing the HHG idea so that it is applicable to a broader community of
computational scientists, and so that it is easier for existing applications to
leverage structured multigrid components. Specifically, we adapt the structured
multigrid methodology to significantly more complex semi-structured meshes.
Further, we illustrate how mature applications might adopt a semi-structured
solver in a relatively non-invasive fashion. To do this, we propose a formal
mathematical framework for describing the semi-structured solver. This
formalism allows us to precisely define the associated multigrid method and to
show its relationship to a more traditional multigrid solver. Additionally, the
mathematical framework clarifies the associated software design and
implementation. Numerical experiments highlight the relationship of the new
solver with classical multigrid. We also demonstrate the generality and
potential performance gains associated with this type of semi-structured
multigrid
Graph Neural Networks and Applied Linear Algebra
Sparse matrix computations are ubiquitous in scientific computing. With the
recent interest in scientific machine learning, it is natural to ask how sparse
matrix computations can leverage neural networks (NN). Unfortunately,
multi-layer perceptron (MLP) neural networks are typically not natural for
either graph or sparse matrix computations. The issue lies with the fact that
MLPs require fixed-sized inputs while scientific applications generally
generate sparse matrices with arbitrary dimensions and a wide range of nonzero
patterns (or matrix graph vertex interconnections). While convolutional NNs
could possibly address matrix graphs where all vertices have the same number of
nearest neighbors, a more general approach is needed for arbitrary sparse
matrices, e.g. arising from discretized partial differential equations on
unstructured meshes. Graph neural networks (GNNs) are one approach suitable to
sparse matrices. GNNs define aggregation functions (e.g., summations) that
operate on variable size input data to produce data of a fixed output size so
that MLPs can be applied. The goal of this paper is to provide an introduction
to GNNs for a numerical linear algebra audience. Concrete examples are provided
to illustrate how many common linear algebra tasks can be accomplished using
GNNs. We focus on iterative methods that employ computational kernels such as
matrix-vector products, interpolation, relaxation methods, and
strength-of-connection measures. Our GNN examples include cases where
parameters are determined a-priori as well as cases where parameters must be
learned. The intent with this article is to help computational scientists
understand how GNNs can be used to adapt machine learning concepts to
computational tasks associated with sparse matrices. It is hoped that this
understanding will stimulate data-driven extensions of classical sparse linear
algebra tasks
ML 3.0 smoothed aggregation user's guide.
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
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A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations.
In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code