Classical Algebraic Multigrid Research Articles

Learning-based local weighted least squares for algebraic multigrid method

Algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problem-dependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system.

Algebraic Multigrid Block Preconditioning for Multi-Group Radiation Diffusion Equations

The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)\times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical algebraic multigrid is applied to solve the subsystems that arise in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We use representative one-group and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, and scale well both algorithmically and in parallel.

A supplementary strategy for coarsening in algebraic multigrid

Algebraic multigrid (AMG) is an efficient iterative method for solving linear equation systems arising from the elliptic partial differential equations. The coarsening algorithm, which determines the coarse-variable set in the classical AMG, is a critical component. This paper targets at reducing the overall solution time of the classical AMG by improving the quality of the coarse-variable set obtained by the coarsening algorithm. We combine the classical coarsening algorithm with the compatible relaxation (CR)-based coarsening algorithm to construct the coarse-variable set. The combined coarsening algorithm constructs the coarse-variable set within two stages. In the first stage, a basic coarse-variable set is built by the classical coarsening algorithm, e.g., PMIS. In the second stage, the quality of the set is measured based on compatible relaxation, and the variables that converge slowly in the CR relaxation are added into the previous set. We test various model problems, as well as some linear equation systems arising from real applications, to verify the effectiveness of our method.

A fast constrained image segmentation algorithm

Normalized cut or Ncut has been one of the most widely used models for image processing. A constraint can also be included in the framework of Ncut to represent a priori information for an effective image segmentation. This results in the so-called constrained Ncut problem. In this paper we present an observation that the constrained Ncut problem can be formulated as an indefinite system of equations under a mild condition on targeted segmentation results. We then show that the indefinite system can be effectively handled by the Augmented Lagrangian Uzawa iterative method together with a classical Algebraic Multigrid Method. Both mathematically and numerically, we demonstrate that the Augmented Lagrangian Uzawa method achieves a solution in one iteration. We show how the proposed method can be efficiently applied for the newly tested recursive two-way Ncut with constraints, i.e., a new constrained sequential segmentation as well. A number of numerical experiments are presented to confirm our theory and to show the superiority of the proposed method. In particular, numerical experiments show that the speed of our algorithm can be orders of magnitude faster than the previously proposed Projected Power Method, a significant improvement of conventional image segmentation algorithms.

Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with a so-called ideal interpolation that minimizes a weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap AMG setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the bootstrap AMG method with sparse interpolation matrix (and spanning multiple levels) converges faster than the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

Reducing complexity of algebraic multigrid by aggregation

SummaryA typical approach to decrease computational costs and memory requirements of classical algebraic multigrid methods is to replace a conservative coarsening algorithm and short‐distance interpolation on a fixed number of fine levels by an aggressive coarsening with a long‐distance interpolation. Although the quality of the resulting algebraic multigrid grid preconditioner often deteriorates in terms of convergence rates and iteration counts of the preconditioned iterative solver, the overall performance can improve substantially. We investigate here, as an alternative, a possibility to replace the classical aggressive coarsening by aggregation, which is motivated by the fact that the convergence of aggregation methods can be independent of the problem size provided that the number of levels is fixed. The relative simplicity of aggregation can lead to improved solution and setup costs. The numerical experiments show the relevance of the proposed combination on both academic and benchmark problems in reservoir simulation from oil industry. Copyright © 2016 John Wiley & Sons, Ltd.

Algebraic Two-Level Convergence Theory for Singular Systems

We consider the algebraic convergence theory that gives its theoretical foundations to classical algebraic multigrid methods. All the main results constitutive of the approach are properly extended to singular compatible systems, including the recent sharp convergence estimates for both symmetric and nonsymmetric systems. In fact, all results carry over smoothly to the singular case, which therefore does not require a specific treatment (except a proper handling of issues associated with singular coarse grid matrices). Regarding problems with a low-dimensional null space, the presented results thus mainly confirm what has been observed often at a more practical level in many previous works. As illustrated by the discussion of the application to the curl-curl equation, the potential impact is greater for problems with large-dimensional null space. Indeed, it turns out that the design of multilevel methods can then actually be easier for the singular case than it is for nearby regularized problems.

Stabilisation of AMG solvers for saddle‐point stokes problems

SummaryWhen a block factorisation is used to precondition the saddle‐point equations of the discrete Stokes problem, the stability that this gives for the relaxation of residual errors may not be conserved in the coarse‐grid approximations (CGA) of algebraic multi‐grid (AMG) solvers. If the same first‐order interpolation is used in the inter‐grid transfer operators for the scalar and the vector fields, the conditioning degrades with each coarsening step until eventually a critical coarsening is reached beyond which residual errors are no longer damped and will become divergent with any further coarsening. It is shown that by introducing the same block pre‐conditioner as an integral part of the coarsening algorithm, stable smoothing can be maintained at all levels of the CGA. The pre‐conditioning need only be applied at preselected grid levels, one immediately before the critical threshold and others beyond that level if required. Excessive complexity in the CGA is thereby avoided. The method is purely algebraic and may be used for both classical AMG solvers and for smoothed‐aggregation AMG solvers. It should be applicable to other coupled vector and scalar fields in science and engineering that involve second‐order (block‐diagonal) and first‐order (block‐off‐diagonal) discrete difference operators. Copyright © 2015 John Wiley & Sons, Ltd.

Accelerating algebraic multigrid solvers on NVIDIA GPUs

This paper presents the development of parallel algebraic multigrid solvers on NVIDIA GPUs. A classical algebraic multigrid solver and a smoothed aggregation algebraic multigrid solver are implemented. The W-cycle, F-cycle and V-cycle are investigated. Various coarsening strategies and interpolation operators are implemented. A group of smoothers are also provided. Numerical experiments show that the solution phase of our GPU-based algebraic solvers is up to 13 times faster than that of the sequential CPU-based algebraic multigrid solvers on our workstation.

AmgX: A Library for GPU Accelerated Algebraic Multigrid and Preconditioned Iterative Methods

The solution of large sparse linear systems arises in many applications, such as computational fluid dynamics and oil reservoir simulation. In realistic cases the matrices are often so large that they require large scale distributed parallel computing to obtain the solution of interest in a reasonable time. In this paper we discuss the design and implementation of the AmgX library, which provides drop-in GPU acceleration of distributed algebraic multigrid (AMG) and preconditioned iterative methods. The AmgX library implements both classical and aggregation-based AMG methods with different selector and interpolation strategies, along with a variety of smoothers and preconditioners, including block-Jacobi, Gauss--Seidel, and incomplete-LU factorization. The library contains many of the standard and flexible preconditioned Krylov subspace iterative methods, which can be combined with any of the available multigrid methods or simpler preconditioners. The parallelism in the aggregation scheme exploits parallel graph matching techniques, while the smoothers and preconditioners often rely on parallel graph coloring algorithms. The AMG algorithm implemented in the AmgX library achieves $2$--$5\times$ speedup on a single GPU against a competitive implementation on the CPU. As will be shown in the numerical experiments section, both setup and solve phases scale well across multiple nodes, sustaining this performance advantage.

A GPU accelerated aggregation algebraic multigrid method

We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a “fine-grain” parallel algorithm for finding a maximal independent set from a graph of strong negative connections. The aggregations are combined with a piece-wise constant (unsmooth) interpolation from the coarse grid solution to the fine grid solution, ensuring low setup and interpolation cost. The grid independent convergence is achieved by using recursive Krylov iterations (K-cycles) in the solve stage. An efficient combination of K-cycles and standard multigrid V-cycles is used as a preconditioner for Krylov iterative solvers such as generalized minimal residual and conjugate gradient. We compare the solver performance with other solvers based on smooth aggregation and classical algebraic multigrid methods.

An efficient parallel algebraic multigrid method for 3D injection moulding simulation based on finite volume method

Elapsed time is always one of the most important performance measures for polymer injection moulding simulation. Solving pressure correction equations is the most time-consuming part in the mould filling simulation using finite volume method with SIMPLE-like algorithms. Algebraic multigrid (AMG) is one of the most promising methods for this type of elliptic equations. It, thus, has better performance by contrast with some common one-level iterative methods, especially for large problems. And it is also suitable for parallel computing. However, AMG is not easy to be applied due to its complex theory and poor generality for the large range of computational fluid dynamics applications. This paper gives a robust and efficient parallel AMG solver, A1-pAMG, for 3D mould filling simulation of injection moulding. Numerical experiments demonstrate that, A1-pAMG has better parallel performance than the classical AMG, and also has algorithmic scalability in the context of 3D unstructured problems.

A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

SUMMARYWe propose a multigrid method for solving large‐scale sparse linear systems arising from discretizations of partial differential equations, such as those from finite element and generalized finite difference methods. Our proposed method has the following two characteristics. First, we introduce a hybrid geometric+algebraic multigrid method, or HyGA, to leverage the rigor, accuracy, and efficiency of geometric multigrid (GMG) for hierarchical unstructured meshes, with the flexibility of algebraic multigrid (AMG). Second, we introduce efficient smoothers based on the Chebyshev–Jacobi method for both symmetric and asymmetric matrices. We also introduce a unified derivation of restriction and prolongation operators for weighted‐residual formulations over unstructured hierarchical meshes and apply it to both finite element and generalized finite difference methods. We present numerical results of our method for Poisson equations in both 2‐D and 3‐D and compare our method against the classical GMG and AMG as well as Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd.

Algebraic Multigrid for Moderate Order Finite Elements

We investigate the use of algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is required by standard analyses of classical AMG and aggregation-based AMG methods. A unified approach is presented that allows us to extend these analyses. It uses an intermediate M-matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix. This constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations, provided that jumps are located at elements' boundaries. For two-dimensional problems, it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound. This analysis validates the application of the AMG methods to the considered problems. On the other hand, because the intermediate M-matrix can be computed automatically, an alternative strategy is to define the AMG preconditioners from this matrix, instead of defining them from the original matrix. Numerical experiments are presented that assess both strategies using publicly available state-of-the-art implementations of classical AMG and aggregation-based AMG methods.

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.

Robust and adaptive multigrid methods: comparing structured and algebraic approaches

SUMMARYAlthough there have been significant advances in robust algebraic multigrid (AMG) methods in recent years, numerical studies and emerging hardware architectures continue to favor structured‐grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the black box multigrid (BoxMG) solver, have been shown to be 10 times faster than AMG for three‐dimensional heterogeneous diffusion problems on structured grids. BoxMG offers important features such as operator‐induced interpolation for robustness, while taking advantage of direct data access and bounded complexity in the Galerkin coarse‐grid operator. Moreover, because BoxMG uses a variational framework, it can be used to explore advances of modern adaptive AMG approaches in a structured setting. In this paper, we show how to extend the adaptive multigrid methodology to the BoxMG setting. This extension not only retains the favorable properties of the adaptive framework but also sheds light on the relationship between BoxMG and AMG. In particular, we show how classical BoxMG can be viewed as a special case of classical AMG and how this viewpoint leads to a richer family of adaptive BoxMG approaches. We present numerical results that explore this family of adaptive methods and compare its robustness and efficiency to the classical BoxMG solver.Copyright © 2012 John Wiley & Sons, Ltd.

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.

Efficient Multilevel Eigensolvers with Applications to Data Analysis Tasks

Multigrid solvers proved very efficient for solving massive systems of equations in various fields. These solvers are based on iterative relaxation schemes together with the approximation of the "smooth" error function on a coarser level (grid). We present two efficient multilevel eigensolvers for solving massive eigenvalue problems that emerge in data analysis tasks. The first solver, a version of classical algebraic multigrid (AMG), is applied to eigenproblems arising in clustering, image segmentation, and dimensionality reduction, demonstrating an order of magnitude speedup compared to the popular Lanczos algorithm. The second solver is based on a new, much more accurate interpolation scheme. It enables calculating a large number of eigenvectors very inexpensively.

Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge‐matrices and two‐level convergence

AbstractWe study an algebraic multigrid (AMG) method for solving elliptic finite element equations of linear elasticity problems. In this method, which has been proposed in (Kraus, SIAM J Sci Comput 2008; 30: 505–524), the coarsening is based on the so‐called edge‐matrices, which allows to generalize the concept of strong and weak connections, as used in the classical AMG, to ‘algebraic vertices’ that accumulate the nodal degrees of freedom in case of vector‐field problems. The major contribution of this work is related to the investigation of a measure for the nodal dependence and on the generation of the edge‐matrices, which are the basic building blocks of this method. A natural measure is the cosine of the abstract angle between the two subspaces spanned by the basis functions corresponding to the respective algebraic vertices. Another original contribution of this work is a two‐level convergence analysis of the method. The presented numerical results cover also problems with jumps in Young's modulus of elasticity and orthotropic materials. Copyright © 2010 John Wiley & Sons, Ltd.

Efficient algebraic multigrid for migration–diffusion–convection–reaction systems arising in electrochemical simulations

The article discusses components and performance of an algebraic multigrid (AMG) preconditioner for the fully coupled multi-ion transport and reaction model (MITReM) with nonlinear boundary conditions, important for electrochemical modeling. The governing partial differential equations (PDEs) are discretized in space by a combined finite element and residual distribution method. Solution of the discrete system is obtained by means of a Newton-based nonlinear solver, and an AMG-preconditioned BICGSTAB Krylov linear solver. The presented AMG preconditioner is based on so-called point-based classical AMG. The linear solver is compared to a standard direct and several one-level iterative solvers for a range of geometries and chemical systems with scientific and industrial relevance. The results indicate that point-based AMG methods, carefully designed, are an attractive alternative to more commonly employed numerical methods for the simulation of complex electrochemical processes.