Frontiers And Prospects of Contemporary Applied Mathematics (Series in Contemporary Applied Mathematics Cam)

276 Pages·2006·6.11 MB·English

by  Tatsien Li

Series in Contemporary Applied Mathematics
CAM 6
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Tatsien Li • P ingwen Zhang
frontiers and Prospects of
Contemporary Applied Mathematics
Series in Contemporary Applied mathematics C AM
Honorary Editor: Chao-Hao Gu (Fudan University)
Editors: P. G. Ciarlet (City University of Hong Kong),
Tatsien Li (Fudan University)
1. Mathematical Finance Theory and Practice
(Eds. Yong Jiongmin, Rama Cont)
2. New Advances in Computational Fluid Dynamics
Theory, Methods and Applications
(Eds. F. Dubois, Wu Huamo)
3. Actuarial Science Theory and Practice
(Eds. Hanji Shang, Alain Tosseti)
4. Mathematical Problems in Environmental Science and
Engineering
(Eds. Alexandre Ern, Liu Weiping)
5. Ginzburg-Landau Vortices
(Eds. Haim Brezis, Tatsien Li)
6. Frontiers and Prospects of Contemporary Applied
Mathematics
(Eds. Tatsien Li, Pingwen Zhang)
Series in Contemporary Applied Mathematics CAM 6
Frontiers and Prospects of
editors
Tatsien Li
Fudan University, China
Pingwen Zhang
Peking University, China
Higher Education Press
\[p World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANCHAI • HONGKONG • TAIPEI • CHENNAi
Tatsien Li Pingwen Zhang
Department of Mathematics School of Mathematical Sciences
Fudan University Peking University
Shanghai, 200433 Beijing, 100871
China China
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ISBN 7-04-018575-X
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Copyright © 2005 by
Higher Education Press
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World Scientific Publishing Co Pte Ltd
5 Toh Tuch Link, Singapore 596224
All rights reserved. No part of this book may be reproduced or transmitted in
any form or by any means, electronic or mechanical, including photocopying,
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ISBN 7-04-018575-X
Printed in P. R. China
Contents
Preface
Invited Talks
Jin Cheng, Mourad Choulli, Xin Yang: An Iterative BEM for the
Inverse Problem of Detecting Corrosion in a Pipe 1
Weinan E, Pinghing Ming: Analysis of the Local Quasicontinuum
Method 18
Houde Han: The Artificial Boundary Method Numerical Solutions
of Partial Differential Equations on Unbounded Domains 33
Kerstin Hesse, Ian H. Sloan: Optimal order integration on
the sphere 59
Jialin Hong: A Survey of Multi-symplectic Runge-Kutta Type
Methods for Hamiltonian Partial Differential Equations 71
Ming Jiang, Yi Li, Ge Wang: Inverse Problems in
Bioluminescence Tomography 114
Fangting Li, Ying Lu, Chao Tang, Qi Ouyang: Global Dynamic
Properties of Protein Networks 149
Wei Li, Yunqing Huang: A Modified Adaptive Algebraic
Multigrid Algorithm for Elliptic Obstacle Problems 160
Zeyao Mo: Parallel Algorithms and Implementation
Techniques for Terascale Numerical Simulations of
Typical Applications 179
Yaguang Wang: Long Time Behaviour of Solutions to Linear
Thermoelastic Systems with Second Sound 191
Contributed Talks
Hongxuan Huang, Changjun Wang: Distance Geometry Problem
and Algorithm Based on Barycentric Coordinates 208
Jijun Liu: On IU-Posedness and Inversion Scheme for 2-D
Backward Heat Conduction 227
Yirang Yuan, Ning Du, Yuji Han: Careful Numerical
Simulation and Analysis of Migration-Accumulation 242
Rongxian Yue: Error Analysis on Scrambled Quasi-Monte Carlo
Quadrature Rules Using Sobol Points 254
Preface
During the period of the 8th Annual Conference of the China Society
for Industrial and Applied Mathematics (CSIAM) held on August 24 -
30, 2004 in Xiangtan, Hunan Province, China, the Symposium on Fron
tiers and Prospects of Contemporary Applied Mathematics was held.
About 300 representatives from over 100 domestic universities, scien
tific research institutions and enterprises and from abroad attended the
conference. At the symposium some Chinese and foreign scholars and
experts were invited to give plenary lectures. They introduced current
progress and expressed their prospects on some important topics of the
industrial and applied mathematics. Besides, at the section meetings
many participants gave academic reports. Considering that these ple
nary lectures have high academic values due to, their representative and
perspective, we collected them in a volume for publication. Meanwhile a
small part of the academic reports provided in the sections was also se
lected for this volume. We hope that the publication of this book would
effectively help readers understand the current situation of the industrial
and applied mathematics and the hot issues in this area. Also we hope
the publication of this book would be helpful in pushing the industrial
and applied mathematics forward.
We would like to take this opportunity to express our heartfelt thanks
to all of the speakers at the symposium for their great support, especially
to those speakers who wrote papers for this book. We would also like
to show our sincere thanks and respect to the National Natural Science
Foundation of China, the Mathematical Center of Ministry of Education
of China and Xiangtan University for their financial help and support;
and to Higher Education Press and World Scientific Publishing Company
for their hard work and efforts in publishing this book.
Li Tatsien
October 2005
1
An Iterative BEM for the Inverse Problem
of Detecting Corrosion in a Pipe*
Jin Cheng
School of Mathematical Sciences, Fudan University,
Shanghai 200433, China.
E-mail: [email protected]
Mourad Choulli
Departement de Mathematiques Universite de Metz He
du Saulcy, 57045 Metz cedex, France.
E-mail: [email protected]
Xin Yang
Department of Computer Science and Engineering, Penn
State University, University Park, PA 16802, USA.
E-mail: [email protected]
Abstract
In this paper, we consider an inverse problem of determining
the corrosion occurring in an inaccessible interior part of a pipe
from the measurements on the outer boundary. The problem is
modelled by the Laplace equation with an unknown term 7 in the
boundary condition on the inner boundary. Based on the Maz'ya
iterative algorithm, a regularized BEM method is proposed for
obtaining approximate solutions for this inverse problem. The
numerical results show that our method can be easily realized
and is quite effective.
1 Introduction
Detecting the corrosion inside a pipe is one of the most important top
ics in engineering, especially in the safety administration of the nuclear
power station. There are several ways to do this. In this paper, we will
discuss the mathematical theory and numerical algorithm for a method of
detecting the corrosion by electrical fields. More exactly, we consider an
*The authors are partly supported by NNSF of China (No. 10271032 and No.
10431030) and Shuguang Project of Shanghai Municipal Education Commission
(N.E03004).
2 Jin Cheng, Mourad Choulli, Xin Yang
inverse problem of determining the corrosion occurring in an inaccessible
interior part of a pipe from the measurements on the outer boundary.
Our goal is to determine information about the corrosion that possibly
occurs on an interior surface of the pipe, which is an 'inaccessible' part,
and we collect electrostatic data on the part of the exterior surface of
the pipe, which is an 'accessible' part.
In the case that the thickness of the pipe is sufficiently small when
compared with the radius of the pipe and the Cauchy data are given on
the whole outer boundary, this inverse problem can be treated by the
Thin Plate Approximation,method (TPA). The algorithm and numerical
analysis can be found in [7]. But this algorithm works only under the
assumption that the thickness is small enough when compared with the
radius of the pipe. The case, in which the Cauchy data are given on
part of the outer boundary and the smallness assumption is abandoned,
has not been studied and it is obvious that it is of great importance for
practice problems.
The main difficulty for this inverse problem is the ill-posedness of the
inverse problem. The measured data are given only on part of the outer
boundary and we want to determine an unknown function in the inner
boundary. Because of the ill-posedness, the errors in measured data will
be enlarged in the numerical treatment if we do not treat it suitably.
In this paper, based on the Maz'ya iterative method, we propose a new
BEM algorithm for this inverse problem. It can be easily realized. The
numerical results show the efficiency of this method.
This paper is organized as follows:
1. Formulation of the inverse problem,
2. The iterative boundary element method,
3. Numerical examples,
4. Conclusions.
2 Formulation of the inverse problem
Suppose a domain fi = {x | n < \x\ < r2} C R2 (see Figure 2.1) and the
boundaries Ti = {x\ \x\ = ri}and T2 = {x\ \x\ = r2}-