42 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
AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport
We develop a reduction multigrid based on approximate ideal restriction (AIR)
for use with asymmetric linear systems. We use fixed-order GMRES polynomials to
approximate and we use these polynomials to build grid
transfer operators and perform F-point smoothing. We can also apply a fixed
sparsity to these polynomials to prevent fill-in.
When applied in the streaming limit of the Boltzmann Transport Equation
(BTE), with a P angular discretisation and a low-memory spatial
discretisation on unstructured grids, this "AIRG" multigrid used as a
preconditioner to an outer GMRES iteration outperforms the lAIR implementation
in hypre, with two to three times less work. AIRG is very close to scalable; we
find either fixed work in the solve with slight growth in the setup, or slight
growth in the solve with fixed work in the setup when using fixed sparsity.
Using fixed sparsity we see less than 20% growth in the work of the solve with
either 6 levels of spatial refinement or 3 levels of angular refinement. In
problems with scattering AIRG performs as well as lAIR, but using the full
matrix with scattering is not scalable.
We then present an iterative method designed for use with scattering which
uses the additive combination of two fixed-sparsity preconditioners applied to
the angular flux; a single AIRG V-cycle on the streaming/removal operator and a
DSA method with a CG FEM. We find with space or angle refinement our iterative
method is very close to scalable with fixed memory use
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Developments in the Extended Finite Element Method and Algebraic Multigrid for Solid Mechanics Problems Involving Discontinuities
In this dissertation, some contributions related to computational modeling and solution of solid mechanics problems involving discontinuities are discussed. The main tool employed for discrete modeling of discontinuities is the extended finite element method and the primary solution method discussed is the algebraic multigrid. The extended finite element method has been shown to be effective for both weak and strong discontinuities. With respect to weak discontinuities, a new approach that couples the extended finite element method with Monte Carlo simulations with the goal of quantifying uncertainty in homogenization of material properties of random microstructures is presented. For accelerated solution of linear systems arising from problems involving cracks, several new methods involving the algebraic multigrid are presented.
In the first approach, the Schur complement of the linear system arising from XFEM is used to develop a Hybrid-AMG method such that crack-conforming aggregates are formed. Another alternative approach involves transforming the original linear system into a modified system that is amenable for a direct application of algebraic multigrid. It is shown that if only Heaviside-enrichments are present, a simple transformation based on the phantom-node approach is available, which decouples the linear system along the discontinuities such that crack conforming aggregates are automatically generated via smoother aggregation algebraic multigrid. Various numerical examples are presented to verify the accuracy of the resulting solutions and the convergence properties of the proposed algorithms. The parallel scalability performance of the implementation are also discussed
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Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies
The application of multilevel methods to solving large algebraic systems obtained by discretization of PDEs has seen great success. However, these methods often perform sub-optimally when treating problems with anisotropies. For problems posed over unstructured meshes, optimal automatic multigrid coarsening is not a fully solved problem for the smoothed aggregation multigrid.
The focus of this thesis is on enhancing robustness of the coarsening in the Smoothed Aggregation (SA) multigrid. We focus on improving the standard detection of coupling, on which the coarsening decisions in SA are based. Our approach takes the form of a two-pass test which allows for a more robust local control over the coupling detection, as well as added exibility permitting utilization of new coupling detection measures in a more systematic way.
For isotropic problems, smoothed aggregation coarsening is known to offer very favorable operator complexity, but achieving similar behavior in the presence of anisotropy is more challenging. Special attention is paid to addressing the issue of controlling the complexity of the method. We discuss several existing approaches to curbing coarse-level operator fill-in, and offer generalizations and improvements.
Numerical experiments are provided to demonstrate the performance of the improved coarsening on model examples of anisotropic problems featuring both cases where anisotropies are aligned with the grid, as well as cases where they are not
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On Some Versions of the Element Agglomeration AMGe Method
The present paper deals with element-based AMG methods that target linear systems of equations coming from finite element discretizations of elliptic PDEs. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate element based AMG method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom (dofs) is based on solving large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V-cycle utilizing simple interpolation matrices and nested conjugate gradient (CG) based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods
BootCMatch: A software package for bootstrap AMG based on graph weighted matching
This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process.
Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software
Multi-GPU aggregation-based AMG preconditioner for iterative linear solvers
We present and release in open source format a sparse linear solver which
efficiently exploits heterogeneous parallel computers. The solver can be easily
integrated into scientific applications that need to solve large and sparse
linear systems on modern parallel computers made of hybrid nodes hosting NVIDIA
Graphics Processing Unit (GPU) accelerators.
The work extends our previous efforts in the exploitation of a single GPU
accelerator and proposes an implementation, based on the hybrid MPI-CUDA
software environment, of a Krylov-type linear solver relying on an efficient
Algebraic MultiGrid (AMG) preconditioner already available in the BootCMatchG
library. Our design for the hybrid implementation has been driven by the best
practices for minimizing data communication overhead when multiple GPUs are
employed, yet preserving the efficiency of the single GPU kernels. Strong and
weak scalability results on well-known benchmark test cases of the new version
of the library are discussed. Comparisons with the Nvidia AmgX solution show an
improvement of up to 2.0x in the solve phase