Algorithms for Elliptic Problems: Efficient Sequential and Parallel Solvers

309 Pages·1993·13.266 MB·English

by  Marián Vajteršic (auth.)

ALGORITHMS FOR
ELLIPTIC PROBLEMS
Efticient Sequential
and Parallel Solvers
Mathematics and Its Applications (East European Series)
Managing Editor:
M. HAZEWlNKEL
Centl'e /01' Mathematics and Computel> Science, Amsterdam, The Ndhe1'lands
Editorial Board:
A. BIALYNICKI-BIRULA, Institute 0/ Mathematics, W'jl'saw Ullivu'sity, Poland
H. KURKE, Humboldt University, Bel'iin, Germany
J. KURZWEIL, Mathematics Institute, Academy 0/ Sciences, Pmgue, Czechoslovakia
L. LElNDLER, Bolyai Institute, Szeged, Hungar'y
L. LOV Asz , Bolyai Institute, Szeged, Iltmgary
D. S. MITRINOVIC, Unive7'sity 0/ Belegmde, }'ugoslavia
S. ROLEWICZ, Polish Academy 0/ Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy 0/ Sciences, Sofia, Bulgaria
1. T. TODOROV, Bulgarian Academy 0/ Sciences, Sofia, Bulgar'ia
H. TRIEBEL, University 0/ Jena, Germany
Volume 58
ALGORITHMS FOR
ELLIPTIC PROBLEMS
Efficient Sequential
and Parallel Solvers
by
Mariau Vajtersic
Institute 0/ Contml Theory amI Robotics,
Slovak Academy 0/ Sciences,
Bratislava, Czechoslollakia
..
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Libr-ar-y of Congr-ess Cataloging-in-Publication Data
Vajtersic, Marian.
[Moderne algoritrny na ricsenie niektorych eliptickych parcialnych
diferencialnych rovnic. English]
Aigorithrns for elliptic problems. Efficient scquenli,tl and parallcl
solvers / by Marian Vajtersic.
p. cm. -- t..lathernatics and its applications : 58
ISBN 978-90-481-4190-6 ISBN 978-94-017-0701-5 (eBook)
DOI 10.1007/978-94-017-0701-5
1. Differential cquations. Elliptic--Nurnerical solutiolls.
I. Title. II. Series: Mathernatics ancl its applications (1\:111\\'('1'
Academic Pu blishcrs) : v ..5 8.
QA377.V:l513 1992
515' ,,353--dc20 92-2.S I:lr;
ISBN 978-90-481-4190-6
Original title: Modcrne algori tmy na riescnie niektorych eliptickS'ch parci alnych
diferencialnych rovnic
© 1993 by Marian Vajtersic
Originally published by Kluwer Academic Publishers in 1993
Translation © Jozef Draveckj
All rights re~er\'ed.
No part of the material prot ected by this copyrigh t notiel' llla)' be reprod uced or
utilized in any form or by any means, electronic 01' mechanical, including photocopy
ing, recordillg or by all,\' information ~torage and retrievel systelll, \\'ithollt writ.ten
permission from the cop.vrighl owner.
SERIES EDITOR'S PREFACE
'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.
This series, Mathematics and fts Applications, started in 1977. Now
that over one hUlldred volumes have appeared it seems opportune to
reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of mono
graphs and textbooks on increasingly specialized topics. However, the 'tree'
of knowledge of mathematics and related fields does not grow only by putting
forth new branches. It also happens, quite often in fact, that branches which
were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in vari
ous seien ces has changed drastically in recent years: measure theory is used
(non-trivially) in regional and theoretical economics; algebraic geometry in
t.eracts with physics; the Minkowsky lemma, coding theory and the struc
ture of water meet one another in packing and covering theory; quantum
fields, crystal defects and mat hematical programming profi t from homotopy
theory; Lie algebra.s are relevant to filtering; and prediction and eleetrical
engineering can use Stein spaees. And in addition 1.0 this tllere are such new
emerging subdiseiplines as 'experimental mathematics', 'CFD', 'eompletely
integrable systems', 'chaos, synergetics and largescale order', which are al
most impossible to fit into existing classifieation schemes. They draw upon
widely different sect.ions of mathematics."
By and large, all this still applies today. It is still true that at finit
sight mathematics seems rat her fragmented and that to find, see, and
exploit the deeper underlying interrelations more effort is needed and so
vi Series editor's preface
are books that can help mathematicians and scientists do so. Accordingly
MIA will continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement.
To the examples of interaction areas one should add string theOl'y where
Riemann surfaces, algebraic geometry, modular functions, kaots, quan
tum field theory, Kac-Moody algebras, monstrous moonshine (all<! more)
all co me together. And to the ex am pIes of things which can be usefully
applied let me add the topic 'finite geometry'; a cornbination of words
which sounds like it might not even exist, let alolle be applicable. And
yet it is being applied: to statistics via designs, to radar/sonar detec
tiOll arrays (via finite projective planes), and to bus connections of VLSI
chips (via difference sets). There seems to be no part of (so-called pure)
mathematics that is not in immediate danger of being applied. And, ac
cordingly, the appliecl mathematiciall neeels to be aware of much more.
Besicles analysis eUld llumerics, the traditional workhorses, he may need
all kinds of combinatorics, algebra, probability, alld so Oll.
In addition, the applied scientist needs to cope increasingly with the
nOlllinear world and the extra mathematica.l sophistication that this re
quires. For that is where the rewards are. Linear models are honest and
a bit sad and depressing: proportional efforts and results. It is the non
linear world that infinitesimal inputs may result in macroscopic outputs
(01' vice versa). To appreciate what I am hinting at: if electronics were
linear we would have no fun with transistors and computers; we would
have no TV; in fact .vou woulel not be reading these lines.
There is also no safety in ignoring such outlanclish things as nonstan
darel analysis, superspace and anticommuting integratioll, p-adic and ul
trametric space. All three have applications in both plectrica! engineering
and physics. Once, complex numbers were equally outlandish, hut they
frequently proved the shortest path between 'real' results. SimilarIy, the
first two topics named have already provieled a number of 'wormhole'
paths. There is no telling where aU this is leadiag- fortunately.
Thus the original scope of the series, which for various (sound) reasons
now comprises five subseries: white (Japan), yellow (China), red (USSR),
blue (Eastern Europe) and green (everything else), still applies. It has
been enlarged a bit to include books treating of the tools from one sub
discipline wh ich are used in others. Thus the se ries still aims at books
dealing with:
a central concept which plays an important role in several different
mathematical and/or scientific specialization areas;
new applications of the results anel ideas from one area of scientific
endeavour into a.nother;
Series editor's preface vii
infiuences which the results, problems and concepts of one field of
enquiry have, and have had, on the development of another.
The shortest path between t.wo trut.hs N ever lend books, for no one ever
in the real domain passes t.hrough the returns them; the only books I have
complex domain. in my library are books that other
folk have len\. me.
J. Hadamard
Anatole France
La physique ne nous donne pas seule
ment l'occasion de resoudre des prob The function of an expert is not to
lemes . .. elle nous fait pressen tir la so be more right thau other people,
lution. but to be wrong for 1Il0re sophis
ticated reasons.
H. PoillCare
David Butler
Bussum, 1992 Michiel Hazewinkel
Contents
Series editor's preface v
INTRODUCTION xiii
FAST METHODS FOR SOLVING THE POISSON EQUATION 1
1.1 Introduction 1
1.2 Direct methods 5
1.2.1 Introduction 5
1.2.2 Shintani's algorithm 7
1.2.3 Fundamental marching algorit.hms 10
1.2.4 Mejran's modificat.ion of t.he matrix decomposition algorithm 12
1.2.5 Lorenz's variant of the marching method ........ 14
1.3 Applications of direct methods to solving other boundary value problems 17
1.3.1 Neumann's boundary value problem ..... 17
1.3.2 Problem with periodic boundary value conditions 23
1.4 Iterative met.hods . . . . . . . . 26
1.4.1 Tntroduction 26
1.4.2 Fundamental iterative methods . . . 27
1.4.3 Two-step and optimalization met.hods 29
1.4.4 Relaxation multigrid met.hod . . . . 33
1.5 Solving the Poisson equation on non-rectangular domains 35
1.5.1 Introduction . . . . . . . . . . . 35
1.5.2 Method of embedding into a rectangle 37
1.5.3 Method of fictive unknowns 39
1.5.4 Method of decomposition 41
1.5.5 An algorithm for an octagonal domain 44
1.5.6 An algorithm for a disc 47
References . . . . . . . . . . . . . . . . 50
2 FAST SERIAL ALGORITHMS FOR SOLVING BIHARMONIC EQUATION 53
2.1 Introduction 53
2.2 Direct methods 56
2.2.1 Golub's algorithm 58
2.2.2 The Buzbee-Dorr algorithm 60
2.2.3 Bj~rstad's algorithm 61
2.3 Algorithms based on splitting 63
2.3.1 The splitting method 63
x Contents
2.3.2 A fast iterative process for the splitting method anel it.s algorithmie
and program realization . . . . .. ......... 68
2.3.3 An algorithm for one iteration using an elimination proceclure 73
2.4 Solving the eigenvalue problem for a biharmonic operat.or 78
References . . . . . . . . . . . . . . . . . . . . . . . 84
3 PARALLEL ALGORITHMS FOR SOLVING CERTAIN ELLIPTIC BOUND-
ARY VALUE PROBLEMS 87
3.1 Introcluction 87
3.2 Parallel methods for solving the Poisson equation on multiprocessors 90
3.2.1 Introduction . . . . . . . . . . 90
3.2.2 Parallel block matrix c1ecomposition . . 91
3.2.3 Parallel marching algorithms . . . . . 95
3.2.4 A parallel domain decomposition method 101
3.2.5 A parallel variant of the cyclie odd-even reduction sol ver 105
3.2.6 An iterative parallel algorithm . . . . . . . . . . . 108
3.3 Parallel algorithms for solving biharmonic equations on SIMD comput.ers 111
3.3.1 Introduction . . . . . . . . . . . . . . . 111
3.3.2 Parallel algorithms using matrix decomposition 114
3.3.3 Application of reduction algorithms 118
3.3.4 Elimination parallel algorithm 124
3.3.5 A block explicit. iterat.ive method 128
References . . . . . . . . . . . . . . 130
4 IMPLEMENTATION OF PARALLEL ALGORITHMS ON SPECIALIZED
COMPUTERS .................... . 134
4.1 lntroduct.ion 134
4.2 ImplementatiOJI of parallel algorit.hms on matrix processors 139
4.2.1 Introduction ................. . 139
4.2.2 Algorithms for the mat.rix processor ICL DAP 141
4.2.3 Multicolour iterat.ion schemes for t.he specialized FEM processor 147
4.3 Parallel algorithms for pipeline comput.ers .... . 149
4.3.1 Int.roduction . . . . . . . . .. .... . 149
4.3.2 Conjugate gradient algorithm for CDC STAR-IOD 152
4.3.3 Black-white iteration scheme on the CRAY-1 comput.er 157
4.4 Implementation of fast parallel algorithms for solving Poisson and biharmo
nie equations on multiprocessor computers 161
4.4.1 Introduction . . . . . . . . . . . . . . 161
4.4.2 Principal characteristics of the EGPA system 164
4.4.3 Formulation of algorithms for EGPA 166
4.4.4 The results of the implementation 170
4.5 Algorithms for massively parallel compu ters 173
4.5.1 IntroductiOJI . . . . . . . . . . . 173
4.5.2 An example for the Connect.ion Machine 174
4.5.3 Matrix multiplicat.ion on the MasPar comput.er 179
References . . . . . . . . . . . . . . . . . . . . 198
5 PARALLEL MULTIGRID ALGORITHMS 203
5.1 Introduction 203
5.2 Parallelizat.ion principles for mult.igrid algorit.hms 207
Contents xi
5.2.1 Introduction . . . . . . . . . . . . . . . . 207
5.2.2 Parallel implementation of a multigrid cyde 207
5.2.3 SIMD implementatiou of the multigrid algorithm 211
5.2.4 Algorithm for a parallel computation over all grids 214
5.2.5 Complexity of multigrid algorithms far some parallel computer to-
pologies . . . . . . . . . . . . . . . . . . . . . . . 218
5.3 Multigrid algorithms far parallel systems with hypercube structure 226
5.3.1 IntroductiOil . . . . . . . . . . . . . . . . . . 226
5.3.2 Hypercube and Gray code . . . . . . . . . . . . 228
5.3.3 Parallelization of multigrid algarithms by Gray codes 231
5.4 Experiments with multigrid algorithms on parallel computers 237
5.4.1 Introduction . . . . . . . . . . . . . . . . . 237
5.4.2 The multigrid parallel method far the CRAY X-MP . . 238
5.4.3 Algorithms for the DIRMU modular system 243
5.4.4 Multigrid method amenable for implemeutation on systems with
massive parallelism 246
References . . . . . . . . 249
ß VLSI ELLIPTIC SOLVERS 252
6.1 Introduction 252
6.2 VLSI algorithms for special band systems 253
6.2.1 Introduction . . . . . . . . . . 253
6.2.2 Recursive algorithm for band systems 253
6.2.3 VLSI cydic reduction 256
6.3 VLSI Poisson sol vers . . . . . . . . . . 258
6.3.1 Introduction . . . . . . . . . . . 258
6.3.2 Matrix decomposition VLSI algorithm 259
6.3.3 VLSI implementat.ion of an elimination Poisson sol ver 261
6.3.4 VLSI cydic odd-even designs . . 266
6.3.5 VLSI multigrid solver . . . . . 270
6.4 VLSI Heimholtz and biharmonic solvers 272
6.4.1 Introduction 272
6.4.2 A VLSI capacitance matrix solver for the Helmholtz equation 273
6.4.3 A VLSI biharmonic semidirect solver with multigrid Poisson blocks 277
6.4.4 An application of direct VLSI Poisson solvers 1.0 the biharmonic
problem 283
References 286
Subject index 288