Classical Algebraic Multigrid Research Articles

Performance of algebraic multigrid methods for non‐symmetric matrices arising in particle methods

AbstractLarge linear systems with sparse, non‐symmetric matrices are known to arise in the modeling of Markov chains or in the discretization of convection–diffusion problems. Due to their potential of solving sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non‐symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M‐matrix.In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non‐symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M‐matrix structure, an approach based on linear optimization yields optimally sparse M‐matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an algebraic multilevel iteration (AMLI) type method. While in the considered test problems, the M‐matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity. Copyright © 2010 John Wiley & Sons, Ltd.

Operator‐based interpolation for bootstrap algebraic multigrid

AbstractBootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax=b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax=0, and are then possibly improved later using a multilevel eigensolver. This paper introduces a flexible variant of BAMG that determines the interpolation weights indirectly by ‘collapsing’ the unwanted connections in ‘operator interpolation’. Compared to BAMG, this indirect BAMG approach (iBAMG) is more in the spirit of classical AMG, which collapses unwanted connections in operator interpolation based on the (restrictive) assumption that smooth error is locally constant. This paper studies the numerical performance of iBAMG and establishes an equivalence, under certain assumptions, between it and a slightly modified (standard) BAMG scheme. To focus on camparing BAMG and iBAMG and exposing their behavior as the problem size grows, the numerical experiments concentrate on Poisson‐like scalar problems. Copyright © 2010 John Wiley & Sons, Ltd.

Improving algebraic multigrid interpolation operators for linear elasticity problems

AbstractLinear systems arising from discretizations of systems of partial differential equations can be challenging for algebraic multigrid (AMG), as the design of AMG relies on assumptions based on the near‐nullspace properties of scalar diffusion problems. For elasticity applications, the near‐nullspace of the operator includes the so‐called rigid body modes (RBMs), which are not adequately represented by the classical AMG interpolation operators. In this paper we investigate several approaches for improving AMG convergence on linear elasticity problems by explicitly incorporating the near‐nullspace modes in the range of the interpolation. In particular, we propose two new methods for extending any initial AMG interpolation operator to exactly fit the RBMs based on the introduction of additional coarse degrees of freedom at each node. Though the methodology is general and can be used to fit any set of near‐nullspace vectors, we focus on the RBMs of linear elasticity in this paper. The new methods can be incorporated easily into existing AMG codes, do not require matrix inversions, and do not assume an aggregation approach or a finite element framework. We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems. Copyright © 2009 John Wiley & Sons, Ltd.

Notes on convergence of an algebraic multigrid method

The convergence theory for algebraic multigrid (AMG) algorithms proposed in Chang and Huang [Q.S. Chang, Z.H. Huang, Efficient algebraic multigrid algorithms and their convergence, SIAM J. Sci. Comput. 24 (2002) 597–618] is further discussed and a smaller and elegant upper bound is obtained. On the basis of element-free AMGe [V.E. Henson, P.S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23(2) (2001) 629–650] we rewrite the interpolation operator for the classical AMG (cAMG), present a uniform expression and then, by introducing a sparse approximate inverse in the Frobenius norm, give a general convergence theorem which is suited for not only cAMG but also AMG for finite elements and element-free AMGe.

Algebraic multigrid and incompressible fluid flow

AbstractThis paper is concerned with the development of algebraic multigrid (AMG) solution methods for the coupled vector–scalar fields of incompressible fluid flow. It addresses in particular the problems of unstable smoothing and of maintaining good vector–scalar coupling in the AMG coarse‐grid approximations. Two different approaches have been adopted. The first is a direct approach based on a second‐order discrete‐difference formulation in primitive variables. Here smoothing is stabilized using a minimum residual control harness and velocity–pressure coupling is maintained by employing a special interpolation during the construction of the inter‐grid transfer operators. The second is an indirect approach that avoids the coupling problem altogether by using a fourth‐order discrete‐difference formulation in a single scalar‐field variable, primitive variables being recovered in post‐processing steps. In both approaches the discrete‐difference equations are for the steady‐state limit (infinite time step) with a fully implicit treatment of advection based on central differencing using uniform and non‐uniform unstructured meshes. They are solved by Picard iteration, the AMG solvers being used repeatedly for each linear approximation.Both classical AMG (C‐AMG) and smoothed‐aggregation AMG (SA‐AMG) are used. In the direct approach, the SA‐AMG solver (with inter‐grid transfer operators based on mixed‐order interpolation) provides an almost mesh‐independent convergence. In the indirect approach for uniform meshes, the C‐AMG solver (based on a Jacobi‐relaxed interpolation) provides solutions with an optimum scaling of the convergence rates. For non‐uniform meshes this convergence becomes mesh dependent but the overall solution cost increases relatively slowly with increasing mesh bandwidth. Copyright © 2006 John Wiley & Sons, Ltd.

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.

An Algebraic Multigrid Method for Linear Elasticity

We present an algebraic multigrid (AMG) method for the efficient solution of linear block-systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio $\nu$. We obtain rates $\rho<0.4$ for $\nu<0.4$. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.

Parallel algebraic multigrid based on subdomain blocking

The algebraic multigrid (AMG) approach provides a purely algebraic means to tackle the efficient solution of systems of equations posed on large unstructured grids, in 2D and 3D. While sequential AMG has been used for increasingly large problems (with several million unknowns), its application to even larger applications requires a parallel version. Since, in contrast to geometric multigrid, the hierarchy of coarser levels and the related operators develop dynamically during the setup phase of AMG, a direct parallelization is very complicated. Moreover, a “naive” parallelization would, in general, require unpredictable and highly complex communication patterns which seriously limit the achievable scalability, in particular of the costly setup phase. In this paper, we consider a classical AMG variant which has turned out be highly robust and efficient in solving large systems of equations corresponding to elliptic PDEs, discretized by finite differences or finite volumes. Based on a straightforward partitioning of variables (using one of the available algebraic partitioning tools such as Metis), a parallelization approach is proposed which minimizes the communication without sacrificing convergence in complex situations. Results will be presented for industrial CFD and oil-reservoir simulation applications on distributed memory machines, including PC-clusters.