Klaus Stüben

A review of algebraic multigrid

Abstract Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical one-level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrix-based approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes, both in 2D and 3D. Since AMG does not make use of any geometric information, it is a “plug-in” solver which can even be applied to problems without any geometric background, provided that the underlying matrix has certain properties.

Multigrid methods on parallel computers—a survey of recent developments

Abstract Multigrid methods have been established as being among the most efficient techniques for solving complex elliptic equations. We sketch the multigrid idea, emphasizing that a multigrid solution is generally obtainable in a time directly proportional to the number of unknown variables on serial computers. Despite this, even the most powerful serial computers are not adequate for solving the very large systems generated, for instance, by discretization of fluid flow in three dimensions. A breakthrough can be achieved here only by highly parallel supercomputers. On the other hand, parallel computers are having a profound impact on computational science. Recently, highly parallel machines have taken the lead as the fastest supercomputers, a trend that is likely to accelerate in the future. We describe some of these new computers, and issues involved in using them. We describe standard parallel multigrid algorithms and discuss the question of how to implement them efficiently on parallel machines. The natural approach is to use grid partitioning. One intrinsic feature of a parallel machine is the need to perform interprocessor communication. It is important to ensure that time spent on such communication is maintained at a small fraction of computation time. We analyze standard parallel multigrid algorithms in two and three dimensions from this point of view, indicating that high performance efficiencies are attainable under suitable conditions on moderately parallel machines. We also demonstrate that such performance is not attainable for multigrid on massively parallel computers, as indicated by an example of poor efficiency on 65,536 processors. The fundamental difficulty is the inability to keep 65,536 processors busy when operating on very coarse grids. This example indicates that the straightforward parallelization of multigrid (and other) algorithms may not always be optimal. However, parallel machines open the possibility of finding really new approaches to solving standard problems. In particular, we present an intrinsically parallel variant of standard multigrid. This “PSMG” (parallel superconvergent multigrid) method allows all processors to be used at all times. even when processing on the coarsest grid levels. The sequential version of this method is not a sensible algorithm

Parallel algebraic multigrid based on subdomain blocking

Abstract 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.

Parallel multigrid solution of the Navier-Stokes equations on general 2D domains

Abstract Multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency. Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).

Parallelization and vectorization aspects of the solution of tridiagonal linear systems

Abstract We are concerned with the parallel solution of large tridiagonal systems on message-based MIMD computers with vector processors. A subset of equations is assigned to eacg process. The algorithms we present consist of ‘local’, fully parallel parts (based on a modified cyclic reduction), and a ‘global’ part. The latter requires the solution of an ‘interface system’ with couplings between all processes; only here is communication required. We present several strategies and compare them theoretically as well as by concrete tests on the iPSC2-VX.

NON-STANDARD MULTIGRID TECHNIQUES USING CHECKERED RELAXATION AND INTERMEDIATE GRIDS

Publisher Summary This chapter examines the nonstandard multigrid (MG) techniques using checkered relaxation and intermediate grids. The MG techniques offer sensational perspectives in the numerical treatment of partial differential equations. There are close connections between the MG and the total reduction ideas suggesting certain combinations leading to the so called MGR methods. The quantitative results refer to model problems in the unit square with Dirichlet or Neumann boundary conditions, respectively. With respect to both theoretical rate of convergence, and the computational effort, MG methods turn out to be considerably superior to standard MG methods. MGR-CH-1 and MGR-CH-2 turn out to be the most efficient methods. It is observed that as only five-point operators are used in these methods, they can in principle be applied to very general problems. It is found that checkered relaxation techniques yield considerable improvements also if no intermediate grids are used explicitly.

L i SS —an environment for the parallel multigrid solution of partial differential equations on general 2D domains

LiSS is a very general package for the parallel multigrid solution of nonlinear partial differential equations. This paper gives the reader an impression of what LiSS can do. It includes the description of two important current perspectives, the implementation of adaptive structures and the generalization to 3D. An overview of numerical and parallel results is given.

Europort-1: Porting industrial codes to parallel architectures

Europort is a European initiative (within the ESPRIT III programme) the goal of which is to demonstrate the benefit of parallel computer technology for industry by porting a vast range of real industrial applications to parallel platforms. This porting action will serve as an exemplar for industry to increase the awareness of parallel high performance computing (HPC) within industry at large and demonstrate the cost-effectiveness of parallel systems. Europort as a whole is subdivided into two projects, Europort-1 and Europort-2. This paper gives an overview of the Europort-1 project, namely, those activities dealing with fluid dynamics and structural mechanics. Europort-1 is partially funded by the European Commission under contract EP 8421.

Algebraic Multigrid for Selected PDE Systems

In this paper, strategies for solving systems of partial differential equations by algebraic multigrid are discussed. In particular, a general framework for so-called point-based strategies is introduced. For a demonstration, we have investigated several industrial applications from semiconductor process and device simulation. It is shown that this framework allows to construct robust and fast algebraic multigrid approaches even for cases, where iterative solvers of the type commonly used in such applications exhibit bad convergence or even fail.

Industrial parallel computing with real codes

Abstract The significant amount of work required to move large commercial application programs to parallel architectures in a portable way has stalled the take up of parallel technology in industry, thereby preventing the extra competitiveness of this technology from being fully utilized. Therefore, the European Commission has decided to promote European industry through the Europort initiative (within the ESPRIT III programme) by partially funding the porting of 38 industrially relevant codes to parallel computers. The goal is to demonstrate the benefit and cost-effectiveness of such systems and foster their industrial use. This paper presents current trends and first official benchmark results in the area of fluid dynamics and structural mechanics (Europort-1). These clearly show that it is not necessary to develop new codes in order to be able to gain essential benefit from the power of parallel architectures. In spite of serious constraints which come along with the porting of commercial codes in an industrial environment, all parallel codes will outperform their sequential counterpart at a considerably better price-performance ratio.