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

276 Pages·2006·6.11 MB·English
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Series in Contemporary Applied Mathematics CAM 6 liiiiiiiiiiimiiMlIlIH 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 ^ t t M t f c^ Wftff&^MW. ^Frontiers and Prospects of Contemporary Applied Mathematics / ^^kM (Li, Tatsien), ft^fX (Zhang, Pingwen) if. —JfcJiC: mmm^&mt, 2005.12 ISBN 7-04-018575-X i.Mf... n.©$. ..©&... m.mffi&-ffi% —XM^X IV.29-53 ^ l a w s ^ it CIP mm?f (2005) m 13795s ^ Copyright © 2005 by Higher Education Press 4 Dewai Dajie, Beijing 100011, P. R. China, and 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, recording or by any information storage and retrieval system, without permis sion in writing from the Publisher. 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}-

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