This volume deals with problems of modern effective algorithms for the numerical solution of the most frequently occurring elliptic partial differential equations. From the point of view of implementation, attention is paid to algorithms for both classical sequential and parallel computer systems. The first two chapters are devoted to fast algorithms for solving the Poisson and biharmonic equation. In the third chapter, parallel algorithms for model parallel computer systems of the SIMD and MIMD types are described. The implementation aspects of parallel algorithms for solving model elliptic boundary value problems are outlined for systems with matrix, pipeline and multiprocessor parallel computer architectures. A modern and popular multigrid computational principle which offers a good opportunity for a parallel realization is described in the next chapter. More parallel variants based in this idea are presented, whereby methods and assignments strategies for hypercube systems are treated in more detail. The last chapter presents VLSI designs for solving special tridiagonal linear systems of equations arising from finite-difference approximations of elliptic problems. For researchers interested in the development and application of fast algorithms for solving elliptic partial differential equations using advanced computer systems.

The use of parallel programming and architectures is essential for simulating and solving problems in modern computational practice. There has been rapid progress in microprocessor architecture, interconnection technology and software devel- ment, which are in?uencing directly the rapid growth of parallel and distributed computing. However, in order to make these bene?ts usable in practice, this dev- opment must be accompanied by progress in the design, analysis and application aspects of parallel algorithms. In particular, new approaches from parallel num- ics are important for solving complex computational problems on parallel and/or distributed systems. The contributions to this book are focused on topics most concerned in the trends of today's parallel computing. These range from parallel algorithmics, progr- ming, tools, network computing to future parallel computing. Particular attention is paid to parallel numerics: linear algebra, differential equations, numerical integ- tion, number theory and their applications in computer simulations, which together form the kernel of the monograph. We expect that the book will be of interest to scientists working on parallel computing, doctoral students, teachers, engineers and mathematicians dealing with numerical applications and computer simulations of natural phenomena.

Most of the papers in this volume were presented at the NATO Advanced Research Workshop High Performance Computing: Technology and Application, held in Cetraro, Italy from 24 to 26 of June, 1996. The main purpose of the Workshop was to discuss some key scientific and technological developments in high performance computing, identify significant trends and defme desirable research objectives. The volume structure corresponds, in general, to the outline of the workshop technical agenda: general concepts and emerging systems, software technology, algorithms and applications. One of the Workshop innovations was an effort to extend slightly the scope of the meeting from scientific/engineering computing to enterprise-wide computing. The papers on performance and scalability of database servers, and Oracle DBMS reflect this attempt We hope that after reading this collection of papers the readers will have a good idea about some important research and technological issues in high performance computing. We wish to give our thanks to the NATO Scientific and Environmental Affairs Division for being the principal sponsor for the Workshop. Also we are pleased to acknowledge other institutions and companies that supported the Workshop: European Union: European Commission DGIII-Industry, CNR: National Research Council of Italy, University of Calabria, Alenia Spazio, Centro Italiano Ricerche Aerospaziali, ENEA: Italian National Agency for New Technology, Energy and the Environment, Fujitsu, Hewlett Packard-Convex, Hitachi, NEC, Oracle, and Silicon Graphics-Cray Research. Editors January 1997 vii LIST OF CONTRIBUTORS Ecole Nonnale Sucentsrieure de Lyon, 69364 Abarbanel. Robert

'Et moi, ..., si j'avait su comment. One service mathematics has ren- en revenir, je n'y serais point alle'. dered the human race. It has put common sense back where it be- Jules Verne longs, on the topmost shelf next to the dusty canister labelIed 'discard- The series is divergent; therefore we ed nonsense'. may be able to do something with Eric T. Bell it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathema- tical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics ...'. All ar- guably true. Alld all statements obtainable this way form part of the raison d 'etre of this serics.

The use of parallel programming and architectures is essential for simulating and solving problems in modern computational practice. There has been rapid progress in microprocessor architecture, interconnection technology and software devel- ment, which are in?uencing directly the rapid growth of parallel and distributed computing. However, in order to make these bene?ts usable in practice, this dev- opment must be accompanied by progress in the design, analysis and application aspects of parallel algorithms. In particular, new approaches from parallel num- ics are important for solving complex computational problems on parallel and/or distributed systems. The contributions to this book are focused on topics most concerned in the trends of today's parallel computing. These range from parallel algorithmics, progr- ming, tools, network computing to future parallel computing. Particular attention is paid to parallel numerics: linear algebra, differential equations, numerical integ- tion, number theory and their applications in computer simulations, which together form the kernel of the monograph. We expect that the book will be of interest to scientists working on parallel computing, doctoral students, teachers, engineers and mathematicians dealing with numerical applications and computer simulations of natural phenomena.