Table Of ContentTHE SPLITTING
EXTRAPOUTION
METHOD
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and Kunio Tanabe
Vol. 1 International Conference on Scientific Computation
eds. T. Chan and Z.-C. Shi
Vol. 2 Network Optimization Problems — Algorithms, Applications and
Complexity
eds. D.-Z. Du and P. M. Pandalos
Vol. 3 Combinatorial Group Testing and Its Applications
by D.-Z. Du and F. K. Hwang
Vol. 4 Computation of Differential Equations and Dynamical Systems
eds. K. Feng and Z.-C. Shi
Vol. 5 Numerical Mathematics
eds. Z.-C. Shi and T. Ushijima
Vol. 6 Machine Proofs in Geometry
by S.-C. Chou, X.-S. Gao and J.-Z. Zhang
Vol. 7 The Splitting Extrapolation Method
by C. B. Liem, T. Lu and T. M. Shih
Series on
Applied Mathematics
A
Volume 1
THE SPLITTING
EXTRAPOLATION
METHOD
A New Technique in
Numerical Solution of
Multidimensional Problems
C. B. Liem
The Hong Kong I'olylcc/inic Univerai.ty
T. Lii
Chengdu Inal.il.utv of Computer Application a
T. M. Shih
The I long Kong Polytechnic University
World Scientific
SSiinngaaappoorree •• NNeeww J Jeerrsseeyy • LLo ndon • Hong Kong
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World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Tao, Lu, 1940-
Splitting extrapolation method / by Lii Tao, T.M. Shih, C.B. Liem.
p. cm. - (Series on applied mathematics: vol. 7)
Includes bibliographical references and index.
ISBN 9810222173
1. Splitting extrapolation method. I. Shih, T. M., 1937-
II. Liem, C. B., 1934- . III. Title. IV. Series.
QA297.5.T36 1995
519.4-dc20 95-18449
CIP
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Preface
Large scale computation in science and engineering is one of the most
challenging areas in computational mathematics. Despite the significant
progress made in digital computers and computational methods over the
last two decades, to date the solution of problems with complicated domains
and high dimension remains difficult. To solve a high dimensional problem,
the computational work and computer storage requirement increase expo
nentially with respect to the dimension. This phenomenon, known as "the
dimensional effect", is the major obstacle in the application of computa
tional methods to the solution of high dimensional problems.
In order to overcome this obstacle, a computational method would, ide
ally, fulfil the following requirements:
(1) it should be a parallel computational method which can be imple
mented in multiprocessor computers requiring minimal communica
tion among processors;
(2) it should possess a high order of accuracy with a posteriori error
estimation and adaptive grid refinement so that it is able to evaluate
the computational results in real time;
(3) the complexity of the algorithm should be independent or almost
independent of the dimension of the problems.
This volume describes the splitting extrapolation method that contains all
the above-mentioned characteristics.
The idea of splitting extrapolation was first developed by two Chinese
mathematicians Qun Lin and Tao Lii in 1983. The authors of this book
conducted a detailed analysis of splitting extrapolation and its applications
in 1990, and identified it to be an extension of the classical Richardson
extrapolation, that is, the multivariate Richardson extrapolation.
In recent years, promising progress has been made in parallel computa
tional methods, multilevel methods and sparse grid combination techniques.
The combination of these methods with splitting extrapolation has become
an effective way of tackling "the curse of dimensionality". A comprehensive
review of these methods can be found in U. Rude (1991, 1993).
v
vi Preface
A lot of the recent research on splitting extrapolation methods has been
published in Chinese journals but has not drawn much attention from over
seas. I hope that with this publication, it will help to promote interactions
among interested readers outside China.
Beijing, December 1994
Zhong-ci Shi
Member of Academia Sinica
Acknowledgements
First and foremost, we would like to acknowledge Professors Zhong-ci
Shi and Qun Lin for their valuable suggestions and unfailing encourage
ments. We also express our sincere gratitude to Professor Zhong-ci Shi for
writing the preface of this volume. We owe many thanks to our students at
the Hong Kong Polytechnic University, Har-ming Chu, Yu-kin Yiu, Kin-fai
Wong, Tung-kao Hung and Wai-lun Kwok, for performing many valuable
numerical experiments, and Mr. Yi-xun Wang for his help in editing the
manuscript. The authors would also like to thank World Scientific Publish
ing Co Pte Ltd and Ms E.H. Chionh for their friendly cooperation.
Finally, we thank the Hong Kong Polytechnic University Research Fund
(340/284 and 340/181) and the National Science Foundation of China
(19471075 and 863-306-05-01), without whose subsidies this work would
not have been possible.
Hong Kong, December 1994
Tao Lii
Institute of Chengdu Computer Applications
Academia Sinica
Tsi-min Shih, Chin-bo Liem
Department of Applied Mathematics
The Hong Kong Polytechnic University
vn
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Introduction
Most mathematical models of scientific and engineering problems are
described in continuous forms, such as integrals, integral equations, differ
ential equations and integro-differential equations. In order to numerically
evaluate these models, it is necessary to choose a discrete parameter h and
a discretization method, so as to convert the continuous forms into alge
braic equations, and then to obtain the numerical solution uh. Therefore,
the accuracy of uh depends on h.
If an exact solution is regarded as the limit of numerical solutions, con
sider the dependence between the numerical solution and discrete parame
ters, one can derive naturally an extrapolation or an extrapolation to the
limit. In essence, one can choose several different discrete parameters and
evaluate the relevant approximate solutions. These solutions are then ex
trapolated to obtain another approximate solution with higher order of
accuracy.
In fact, the idea of extrapolation was used in the ancient time for ac
celerating the convergence. The Chinese mathematicians Liu Hui (A.D.
263) and Zhu Chongzhi (429-500) used the idea to evaluate the value of 7r,
and Huygens (1654), a French mathematician, proposed an extrapolation
4 1
formula -Th - -T2h to evaluate TT.
o o
However, it was Richardson (1910) who first studied the extrapolation
as common and effective algorithms, and established polynomial extrap
olations. This was followed by the ^-extrapolation by Wynn (1956), the
rational extrapolation by Bulirsch and Stoer (1964), and the general ex
trapolation by C. Brezinski (1980).
In practice, extrapolation methods have become effective techniques for
accelerating the convergence in various branches of numerical mathemat
ics, including the Romberg algorithm (1955) in numerical integration, the
Stetter Theorem (1965) and the Gragg-Bulirsch-Stoer algorithms (1964)
in ordinary differential equations, the extrapolation in the finite difference
method by G. Marchuk and V. Shaidurov (1979), the extrapolation in the
finite element method by Q. Lin and T. Lii (1983), and by R. Rannacher
(1988).
IX
X Introduction
The above-mentioned extrapolations, though varying in forms, are all
based on the extrapolation with respect to a single discrete parameter. Nat
urally, the extrapolation of such a kind will cause difficulties in solving high
dimensional problems because of the dimensional effect. Take the evalua
tion of an s-multiple integral by Richardson's extrapolation as an example,
in order to obtain an algebraic precision of degree (2m -f 1), function val
ues at 2m5 points are needed. This leads to an exponential increase in the
computational complexity as the dimension increases.
The dimension effect becomes more serious in solving partial differ
ential equations: for an s-dimensional problem by means of one step of
Richardson's extrapolation, apart from solving an algebraic equation with
Ns unknowns, one has to solve an auxiliary problem with (2N)S unknowns.
However, problems arising from science and engineering, such as the explo
ration and exploitation of oil and gas, weather forecasting, pollution control,
are often described by high dimensional mathematical models. It was on
the grounds of such demands that Q. Lin and T. Lii (1983) proposed the
Splitting Extrapolation Method (SEM).
The Richardson extrapolation is based on a single-variable asymptotic
expansion of the error, while the SEM is based on a multivariate asymptotic
expansion of the error. This simple change greatly improves the efficiency
of the extrapolation. First, the computation complexity is almost optimal.
For example, to evaluate an s-multiple integral by the SEM, in order to
reach the algebraic precision of degree (2m +1), only the function values at
( m m) Pom^s are needed. Second, SEM converts a large scale problem
into several mutually independent subproblems which can be computed in
parallel by a multiprocessor computer with a few communication among
the processors. Here, the degree of parallelism depends on the numbers of
independent discrete parameters, which do not need to coincide with the
dimension of the problem, and can be set according to the scale of the
problem and the available number of processors. Third, SEM provides a
posteriori error estimations and self-adaptive algorithms.
The latest developments of extrapolation methods are related to the
multilevel methods. Multilevel methods (such as multigrid methods) re
quire the solution of several discrete equations with different grid sizes.
These results can be used in extrapolation methods to obtain economically
a higher order of accuracy. There are two branches of this idea: A. Brandt
(1983) and W. Hackbusch (1984) combined the Richardson extrapolation
and the multigrid methods (called the r-extrapolation), and A. Schiiller
and Q. Lin (1985) combined the SEM and multigrid methods. Further,
based on finite element multilevel subspace splitting, C. Zenger (1990) pro
posed a sparse-grid method for solving multidimensional problems, and
