计算反演问题中的优化与正则化方法及其应用

出版时间：2010-5  出版社：高等教育出版社  作者：王彦飞，（俄）亚哥拉，杨长春　主编  页数：350  
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前言

This volume contains the papers presented by invited speakers of the first inter-national workshop "Optimization and Regularization for Computational Inverse Problems and Applications". The workshop was organized under the auspices of the Chinese Academy of Sciences in the Institute of Geology and Geophysics, located in Beijing, the capital of China, and held during July 21-25, 2008, just before the opening of the Olympic Games. The workshop was sponsored by the National Natural Science Foundation of China, China-Russia Cooperative Re-search Project RFBR-07-01-92103-NFSC and the National "973" Key Basic Re-search Developments Program of China. The main goal of the workshop was to teach about 60 young Chinese participants （mostly geophysicists） how to solveinverse and ill-posed problems using optimization procedures. Eminent specialists from China, Russia （partially sponsored by the Russian Foundation of Basic Research）, USA and Austria were invited to present their lectures. Some of them could not participate personally but all invited speakers found a possibility to write papers especially for this publication.The book covers many directions in the modern theory of inverse and ill-posed problems the variational approach, iterative methods, using a prioriin formation for constructing regularizing algorithms, etc. But the most important for the papers is to show how these methods can be applied to effectively solving of practical problems in geophysics, astrophysics, vibrational spectroscopy, and image processing. This issue should encourage specialists in the inverse problems field not only to investigate mathematical methods and propose new approaches but also to apply them to processing of real experimental data. I would like to wish all of them great successes！

内容概要

Optimization and Regularization for Computational Inverse Problems and Applications focuses on advances in inversion theory and recent developments with practical applications, particularly emphasizing the combination of optimization and regularization for solving inverse problems. This book covers both the methods, including standard regularization theory, Fejer processes for linear and nonlinear problems, the balancing principle, extrapolated regularization, nonstandard regularization, nonlinear gradient method, the nonmonotone gradient method, subspace method and Lie group method; and the practical applications, such as the reconstruction problem for inverse scattering, molecular spectra data processing, quantitative remote sensing inversion, seismic inversion using the Lie group method, and the gravitational lensing problem.     Scientists, researchers and engineers, as well as graduate students engaged in applied mathematics, engineering, geophysics, medical science, image processing, remote sensing and atmospheric science will benefit from this book.

作者简介

编者：王彦飞 （俄国）亚哥拉（Anatoly G.Yagola） 杨长春Dr. Yanfei Wang is a Professor at the Institute of Geology and Geophysics, Chinese Academy of Sciences, China.Dr. Sc. Anatoly G. Yagola is a Professor and Assistant Dean of the Physical Faculty, Lomonosov Moscow State University, Russia.Dr. Changchun Yang is a Professor and Vice Director of the Institute of Geology and Geophysics, Chinese Academy of Sciences, China.

书籍目录

Part I Introduction1  Inverse Problems, Optimization and Regularization: A  Multi-Disciplinary Subject  Yanfei Wang and Changchun Yang  1.1  Introduction  1.2  Examples about mathematical inverse problems  1.3  Examples in applied science and engineering  1.4  Basic theory  1.5  Scientific computing  1.6  Conclusion  ReferertcesPart II Regularization Theory and Recent Developments2  Ill-Posed Problems and Methods for Their Numerical Solution  Anatoly G. Yagola  2.1  Well-posed and ill-posed problems  2.2  Definition of the regularizing algorithm  2.3  Ill-posed problems on compact sets  2.4  Ill-posed problems with sourcewise represented solutions  2.5  Variational approach for constructing regularizing algorithms   2.6  Nonlinear ill-posed problems  2.7  Iterative and other methods  References3  Inverse Problems with A Priori Information  Vladimir V. Vasin  3.1  Introduction  3.2  Formulation of the problem with a priori information  3.3  The main classes of mappings of the Fejer type and their properties  3.4  Convergence theorems of the method of successive approximations for the pseudo-contractive operators  3.5  Examples of operators of the Fejer type  3.6  Fejer processes for nonlinear equations  3.7  Applied problems with a priori information and methods for solution    3.7.1  Atomic structure characterization    3.7.2  Radiolocation of the ionosphere    3.7.3  Image reconstruction    3.7.4  Thermal sounding of the atmosphere    3.7.5  Testing a wellbore/reservoir  3.8  Conclusions  References4  Regularization of Naturally Linearized Parameter  Identification Problems and the Application of the Balancing  Principle  Hui Cao and Sergei Pereverzyev  4.1  Introduction  4.2  Discretized Tikhonov regularization and estimation of accuracy    4.2.1  Generalized source condition    4.2.2  Discretized Tikhonov regularization    4.2.3  Operator monotone index functions    4.2.4  Estimation of the accuracy  4.3  Parameter identification in elliptic equation    4.3.1  Natural linearization    4.3.2  Data smoothing and noise level analysis    4.3.3  Estimation of the accuracy    4.3.4  Balancing principle    4.3.5  Numerical examples  4.4  Parameter identification in parabolic equation    4.4.1 Natural linearization for recovering b(x) = a(u(T, x))    4.4.2  Regularized identification of the diffusion coefficient a(u)    4.4.3  Extended balancing principle    4.4.4  Numerical examples  References5  Extrapolation Techniques of Tikhonov Regularization  Tingyan Xiao, Yuan Zhao and Guozhong Su  5.1  Introduction  5.2  Notations and preliminaries  5.3  Extrapolated regularization based on vector-valued function approximation    5.3.1  Extrapolated scheme based on Lagrange interpolation    5.3.2  Extrapolated scheme based on Hermitian interpolation    5.3.3  Extrapolation scheme based on rational interpolation  5.4  Extrapolated regularization based on improvement of regularizing qualification  5.5  The choice of parameters in the extrapolated regularizing approximation  5.6  Numerical experiments  5.7  Conclusion  References6  Modified Regularization Scheme with Application in Reconstructing Neumann-Dirichlet Mapping  Pingli Xie and Jin Cheng  6.1  Introduction  6.2  Regularization method  6.3  Computational aspect  6.4  Numerical simulation results for the modified regularization  6.5  The Neumann-Dirichlet mapping for elliptic equation of second order  6.6  The numerical results of the Neumann-Dirichlet mapping  6.7  Conclusion  ReferencesPart III Nonstandard Regularization and Advanced Optimization Theory and Methods7  Gradient Methods for Large Scale Convex Quadratic Functions  Yaxiang Yuan  7.1  Introduction  7.2  A generalized convergence result  7.3  Short BB steps  7.4  Numerical results  7.5  Discussion and conclusion  References8  Convergence Analysis of Nonlinear Conjugate Gradient  Methods  Yuhong Dai  8.1  Introduction  8.2  Some preliminaries  8.3  A sufficient and necessary condition on βk    8.3.1  Proposition of the condition    8.3.2  Sufficiency of (8.3.5)    8.3.3  Necessity of (8.3.5)  8.4  Applications of the condition (8.3.5)    8.4.1  Property (#)    8.4.2  Applications to some known conjugate gradient methods    8.4.3  Application to a new conjugate gradient method  8.5  Discussion  References9  Full Space and Subspace Methods for Large Scale Image  Restoration  Yanfei Wang, Shiqian Ma and Qinghua Ma  9.1  Introduction  9.2  Image restoration without regularization  9.3  Image restoration with regularization  9.4  Optimization methods for solving the smoothing regularized functional    9.4.1  Minimization of the convex quadratic programming problem with projection    9.4.2  Limited memory BFGS method with projection    9.4.3  Subspace trust region methods  9.5  Matrix-Vector Multiplication (MVM)    9.5.1  MVM: FFT-based method    9.5.2  MVM with sparse matrix  9.6  Numerical experiments  9.7  Conclusions  ReferencesPart IV Numerical Inversion in Geoscience and Quantitative Remote Sensing10  Some Reconstruction Methods for Inverse Scattering Problems  Jijun Liu and Haibing Wang  10.1 Introduction  10.2 Iterative methods and decomposition methods    10.2.1 Iterative methods    10.2.2 Decomposition methods    10.2.3 Hybrid method  10.3 Singular source methods    10.3.1 Probe method    10.3.2 Singular sources method    10.3.3 Linear sampling method    10.3.4 Factorization method    10.3.5 Range test method    10.3.6 No response test method  10.4 Numerical schemes  References11 Inverse Problems of Molecular Spectra Data Processing  Gulnara Kuramshina  11.1 Introduction  11.2 Inverse vibrational problem  11.3 The mathematical formulation of the inverse vibrational problem  11.4 Regularizing algorithms for solving the inverse vibrational problem  11.5 Model of scaled molecular force field  11.6 General inverse problem of structural chemistry  11.7 Intermolecular potential  11.8 Examples of calculations    11.8.1 Calculation of methane intermolecular potential    11.8.2 Prediction of vibrational spectrum of fullerene C240  References12  Numerical Inversion Methods in Geoscience and Quantitative  Remote Sensing  Yanfei Wang and Xiaowen Li  12.1 Introduction  12.2 Examples of quantitative remote sensing inverse problems: land    surface parameter retrieval problem  12.3 Formulation of the forward and inverse problem  12.4 What causes ill-posedness  12.5 Tikhonov variational regularization    12.5.1 Choices of the scale operator D    12.5.2 Regularization parameter selection methods  12.6 Solution methods    12.6.1 Gradient-type methods    12.6.2 Newton-type methods  12.7 Numerical examples  12.8 Conclusions  References13  Pseudo-Differential Operator and Inverse Scattering of  Multidimensional Wave Equation  Hong Liu, Li He  13.1 Introduction  13.2 Notations of operators and symbols  13.3 Description in symbol domain  13.4 Lie algebra integral expressions  13.5 Wave equation on the ray coordinates  13.6 Symbol expression of one-way wave operator equations  13.7 Lie algebra expression of travel time  13.8 Lie algebra integral expression of prediction operator  13.9 Spectral factorization expressions of reflection data  13.10 Conclusions  References14 Tikhonov Regularization for Gravitational Lensing Research.  Boris Artamonov, Ekaterina Koptelova, Elena Shimanovskaya and  Anatoly G. Yagola  14.1 Introduction  14.2 Regularized deconvolution of images with point sources and smooth background    14.2.1 Formulation of the problem    14.2.2 Tikhonov regularization approach    14.2.3 A priori information  14.3 Application of the Tikhonov regularization approach to quasar profile reconstruction    14.3.1 Brief introduction to microlensing    14.3.2 Formulation of the problem    14.3.3 Implementation of the Tikhonov regularization approach    14.3.4 Numerical results of the Q2237 profile reconstruction  14.4 Conclusions  ReferencesIndex

章节摘录

插图：Indeed, if we introduce a very limited set of parameters （and thus create a very "rigid" model）, we are likely to fail achieving a good fit between experimental and calculated data. On the other hand, if a model is very flexible（that is, contains too many adjustable parameters）, we are likely to find a wide variety of solutions that all satisfy the experiment. Even if we employ the concept of a regularized solution, there must exist some kind of optimal parameter set that would correspond to the available experimental data. As for the force fielddetermination, it is a common knowledge that （except for a limited set of smallor very symmetrical molecules） we never have enough data to restore a complete force field. The ED data usually provides only a small additional data on force field , so as a rule we are in the situation when there exists a wide range of force fields compatible with spectroscopic experiment. Among the ways to reduce theambiguity of the force fields, we could mention the following:1. Introducing model assumptions based on general ideas of molecular structure  （e.g. valence force field, etc.）: these will result in neglecting some force constants, fixing the others, and/or introducing model potentials that would be allowed to generate force matrix depending on a small number of parameters.2. Transferring some force field parameters from similar fragments in related molecules and assuming they are not likely to be significantly changed in a different environment.

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