Ulrich Trottenberg

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

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I

SummaryThis paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poissons 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.

SUPRENUM: System essentials and grid applications

Abstract 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.

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

We present some results concerning parallel multigrid (MG) algorithms for 3D problems. For certain-practically important-anisotropic cases, plane relaxation is needed for smoothing if standard MG (coarsening etc.) is used. Parallel versions of these sophisticated MG algorithms are described, and their time complexity and multiprocessor efficiency is studied.

Parallel multigrid methods: implementation on SUPRENUM-like architectures and applications

Multigrid (MG) methods for partial differential equations (and for other important mathematical models in scientific computing) have turned out to be optimal on sequential computers. Clearly, one wants to apply them also on vector and parallel computers in order to exploit both, the high MG-efficiency (compared to classical methods) and the full computational power of modern supercomputers. For this purpose, parallel MG methods are needed. It turns out that certain well-known standard MG methods (with RB and zebra-type relaxation, as described in [25]) already contain a sufficiently high degree of parallelism.

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II

SummaryIn a recent paper [11], two of the authors investigated a fast reduction method for solving difference equations which approximate certain boundary value problems for Poissons equation. In this second paper, we prove the numerical stability of the reduction method, and also report on further developments of the method. For the general case, the provided bounds for the numerical errors behave roughly like the condition numberO(n2) of the linear system; for more realistic model problems estimates of order less thanO(n) are obtained (n−1=h=mesh width). The number of operations required for the reduction method isO(n2), for the usual five-point difference formula, as well as for the common nine-point formula with discretization error of orderh4.

Parallel Processing on Distributed Memory Multiprocessors

In the past few years distributed memory multiprocessors have been among the most exciting computer developments. These machines offer very high peak performance compared to low prices. Meanwhile several systems are commercially available. Although there are many arguments in favor of these architectures (good price/performance ratio, scalability, low entry level) many users still hesitate to use distributed memory multiprocessors, mainly because of a lack of programming comfort.

On the multigrid acceleration approach in computational fluid dynamics

In this short note, two multigrid approaches for the treatment of computational fluid dynamics problems are distinguished: the “optimal approach”, where the specific model is to be treated entirely by multigrid and all multigrid components are to be defined optimally tailored - versus the “acceleration approach”, where one only tries to introduce some standard multigrid components into classical methods or into codes that are already available. For some examples, in particular the anisotropic convection-diffusion model operator and the (incompressible) Navier-Stokes equations, the gain that can be achieved by the acceleration approach is discussed.

Lösung linearer gewöhnlicher Randwertaufgaben 4. Ordnung mit Hilfe von Aufspaltungen

SummaryWe develop a method for splitting certain linear two-point boundary value problems of the fourth order into problems of the second order which can be solved successively. Both, the given differential equation and the boundary conditions, are involved in the splitting.The method is constructive and can be realized numerically. The resulting discrete second order problems are solved by thereduction method [4], which is distinguished by its special numerical stability.

GENESIS application software

Abstract In its second phase GENESIS was essentially a software project. However, already from its very beginning, the GENESIS project put a strong emphasis on parallel application software. More than ten research institutions and companies were involved in GENESIS on the application software side. The development of PARMACS, the portable message passing layer in GENESIS, allowed full portability across a substantial number of different parallel systems. This portability was demonstrated successfully with the GENESIS application codes.