Iterative Incomplete Factorization Methods

200 Pages·1992·13.711 MB·English

by  V P IlК№in

Series on Soviet and East European Mathematics - Vol. 4
ITERATIVE
INCOMPLETE
FACTORIZATION
METHODS
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Series on Soviet and East European Mathematics - Vol. 4
ITERATIVE
INCOMPLETE
FACTORIZATION
METHODS
V P II in
Russian Academy of Sciences
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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ITERATIVE INCOMPLETE FACTORIZATION METHODS
Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof may not be reproduced in any form
orby any means, electronic or mechanical, including photocopying, recording or any
information storage and retrieval system now known or to be invented, without
written permission from the Publisher.
ISBN 981-02-0996-7
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
To N. I. Buleev,
the founder of incomplete factorization methods
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Preface
Fortunately, the author in his younger years witnessed the beginning of a new
trend in computational algebra - methods of incomplete factorization. It goes
back to the end of the 50-s at the Institute of Physics and Energy, Depart­
ment of Mathematics headed by G.I. Marchuk, in the town of Obninsk near
Moscow, which used to be one of the secret towns, where the first atomic power
station in the world was constructed. Prof. N. I. Buleev, head of the Laboratory
of Numerical Methods in Hydrodynamics and the author of one of the original
turbulent theories was keen on calculations and he created his own algorithm
for iterative computations of two- and three-dimensional flows, which were suc­
cessfully used on URAL-1 computer, with the capacity of 100 operations per
second.
The method was described in the "close" research reports and was first
to appear in the book "Methods of Calculation of Nuclear Reactors" by
G.I.Marchuk in 1958. Though the new class of algorithms was practically ef­
fective and presented generalizations, it did not draw much attention. First, it
was due to the absence of an adequate theory and forced purely experimental
grounds. The second reason was due to the preference of then popular methods
of optimal relaxation and alternating directions which had attractive theoreti­
cal investigations. Now, when decades have passed, the incomplete factorization
methods are the most widely spread and efficient techniques for solving sparse
grid systems of equations of higher dimensions. Although there are still many
questions to be solved, the theory is simple enough to be lectured to students.
For those researchers interested in the subject, there is much to do in the area
of optimization and invention of new algorithms.
I would like to aknowledge my gratitude to Professors O.Axellson and
R. B. Beauwens for the many fruitful discussions. A lot of experimental compu­
tations were carried out by E. A. Itskovich, N. E. Kozorezova and L. K. Kositsina.
I would also like to thank E. L. Makovskaya and A. Yu. Shadrin for preparing
the English version of this book.
Novosibirsk, March 1992 V. P. Il'in
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Contents
Introduction 1
1. Some Elements from Linear Alegbra 7
1.1. Short Extracts from Matrix Theory 7
1.2. Algebraic Peculiarities of Grid Systems of Equations 18
1.3. Common Properties and Comparative Analysis of Iterative
Algorithms 35
2. Buleev Methods and 'Grid' Principles of Incomplete
Factorization 46
2.1. The Explicit Buleev Method for Two-Dimensional
Five-Point Equations (EBM-2) 46
2.2. The Explicit Buleev Method for Three-Dimensional
Seven-Point Equations (EBM-3) 52
2.3. Examples of Other Factorizations for Two-Dimensional
Problems 56
2.4. Second-Order Factorization Methods 62
2.5. Implicit 'Grid' Methods of Incomplete Factorization 68
2.6. Methods of /i-Factorization 77
2.6.1. Tee's Method 78
2.6.2. Ginkin's Method 81
3. Matrix Analysis of Incomplete Factorization Methods 86
3.1. Algorithms with Incomplete Triangualar Factorization 86
3.2. Implicit Incomplete Factorization Methods for Block
Tri-Diagonal Systems 98
3.2.1. Preliminaries 98
3.2.2. Overimplicit Seidel Method 102
3.2.3. Implicit Algorithms for Symmetric Matrices 105
3.3. Application of Incomplete Factorization to Domain
Decomposition Methods 113
3.3.1. Domain Decomposition into Two Subdomains 114
3.3.2. Domain Decomposition into Several Subdomains ... 118
3.3.3. 'Diagonal' Decomposition with Application
of the Explicit Buleev Method 123
3.3.4. Reduction Methods without Backward Step 125
X Contents
4. Problems of Theoretical Background 128
4.1. Correctness and Stability of Incomplete Factorization 128
4.2. Estimates of Convergence Rate of Iterations 133
4.3. On Application of Incomplete Factorization to Solution
of Parabolic Equations 140
4.4. Solution of Diffusion Convection Equations 145
4.4.1. Equations with Constant Coefficients 146
4.4.2. Equations with Separable Variables 152
4.5. On the Symmetric Alternating Direction Methods 155
5. Examples of Numerical Experiments 159
5.1. Description of Experiment Conditions 160
5.2. Explicit Algorithms 163
5.3. Implicit Algorithms 172
Confinement. Some Conclusions and Unsolved Problems 177
References 181
Subject Index 189