These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians. The use of more advanced mathematical tools, such as functional analysis, is avoided. The course is intended to be accessible to a wide audience of users of computational methods. We restrict ourselves to finite volume and finite difference discretization. The basic principles are given. Smoothing methods and Fourier smoothing analysis are reviewed. The fundamental multigrid algorithm is studied. The smoothing and coarse grid approximation properties are discussed. Multigrid schedules and structured programming of multigrid algorithms are treated. Robustness and efficiency are considered.

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An Introduction to Multigrid Methods

BoomerAMG: a parallel algebraic multigrid solver and preconditioner

Multigrid methods: Fundamental algorithms, model problem analysis and applications

This paper describes the design of an interactive system for the semi-automatic transformation of FORTRAN 77 programs into parallel programs for the SUPERNUM machine. The system is characterized by a powerful analysis component, a catalog of MIMD and SIMD parallelization transformations, and a flexible dialog facility. It contains specific knowledge about the parallelization of an important class of numerical algorithms.

SUPERB: A tool for semi-automatic MIMD/SIMD parallelization☆

Programmable active memories (PAM) are a novel form of universal reconfigurable hardware coprocessor. Based on field-programmable gate array (FPGA) technology, a PAM is a virtual machine, controlled by a standard microprocessor, which can be dynamically and indefinitely reconfigured into a large number of application-specific circuits. PAM's offer a new mixture of hardware performance and software versatility. We review the important architectural features of PAM's, through the example of DECPeRLe-1, an experimental device built in 1992. PAM programming is presented, in contrast to classical gate-array and full custom circuit design. Our emphasis is on large, code-generated synchronous systems descriptions; no compromise is made with regard to the performance of the target circuits. We exhibit a dozen applications where PAM technology proves superior, both in performance and cost, to every other existing technology, including supercomputers, massively parallel machines, and conventional custom hardware. The fields covered include computer arithmetic, cryptography, error correction, image analysis, stereo vision, video compression, sound synthesis, neural networks, high-energy physics, thermodynamics, biology and astronomy. At comparable cost, the computing power virtually available in a PAM exceeds that of conventional processors by a factor 10 to 1000, depending on the specific application, in 1992. A technology shrink increases the performance gap between conventional processors and PAM's. By Noyce's law, we predict by how much the performance gap will widen with time.

Programmable active memories: reconfigurable systems come of age

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

Multigrid methods on parallel computers—a survey of recent developments

This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional toh ?2 (h=mesh width). The method is one oftotal reduction compared with the method ofblock-cyclic reduction (odd-even reduction) [2], which we describe as a method ofpartial reduction.--Due to the developed calculus, many generalizations are possible.--In a following part II of the paper, the algorithm and numerical results will be described in detail.

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I

The SUPRENUM idea, the project, and the system has generally been described and presented in several papers. There is also a great deal of more detailed technical papers describing SUPRENUM as a whole or certain elements of it. Here we want to give only a very general and rough survey on the essentials of the SUPRENUM system in order to enable the reader to categorize and understand the more specific SUPRENUM papers in this special issue. Most of the supercomputer applications today are based on grid or grid-like data structures. Grid applications play also an essential role in the SUPRENUM development: in the top-down design of the architecture, in the programming environment, in the parallelization concept of algorithms, and, of course, in the application software development itself. We therefore place some emphasis on this grid orientation in our presentation.

Ulrich Trottenberg

Papers

Multigrid methods on parallel computers—a survey of recent developments

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I

SUPRENUM: System essentials and grid applications

On fast poisson solvers and applications

A short note on standard parallel multigrid algorithms for 3D problems