应用反问题中的计算方法

章节摘录

版权页：   插图：   The harmonic Bz algorithm for the reconstruction of conductivity σ applies△B,the Laplacian operation of magnetic flux,as input data,rather than Bz.Of course,this algorithm amplifies the noise in the measurenlent data Bz obviously.Thus the performance of the harmonic Bz algorithm could deteriorate ifthe noise in the measurement of Bz is not so small,which is the practical caseof MREIT.To deal with this noise problem,solne algorithms were developedin order to weaken the ill—posedness of the harmonic Bz algorithm,such asthe gradient Bz decomposition algorithm and variational gradient Bz algo—rithm,which need the first order derivative of Bz only.Although the noiseamplification problem is weakened in some senses in these two schemes,thefirstorder derivative of Bz is still needed.Due to these difficulties in treatingwith the derivative of the measurement data Bz,it is prefereable to establish animage reconstruction algorithm using the magnetic flux density data directly,rather than its derivative.Recently,we proposed an integral equation method,where the conductivity σ was reconstructed from the Bz data directly,see.The integral version of the Biot-Savart law was used to describe the relationbetween Bz and σ,see(14.2.7)in the next,section.The validity of the integralequation method was shown by some numerical simulations in. In this chapter.we will review MREIT from mathematical models,imagingreconstruction algorithms to its numerical simulations. In section 14.2,themathematical models of the forward problem and inverse problem in MREITare described in details.Since the imaging reconstruction algorithms are thekey to the practical applications of MREIT technique,two specific algorithms：the harmonic Bz algorithm and the integral equation method are discussed inSection 14.3 and Section 14.4,respectively.In the last section,we give somenumerical simulations to show the validity of the above image reconstructionalgorithms.

内容概要

作者：（俄罗斯）亚哥拉（Anatoly G. Yagola） 王彦飞 杨长春

书籍目录

Preface Editor's Preface Ⅰ Introduction 1 S.I.Kabanikhin Inverse Problems of Mathematical Physics 1.1 Introduction 1.2 Examples of Inverse and Ill—posed Problems 1.3 Well—posed and Ill—posed Problems 1.4 The Tikhonov Theorem 1.5 The Ivanov Theorem：Quasi—solution 1.6 The Lavrentiev,s Method 1.7 The Tikhonov Regularization Method References Ⅱ Recent Advances in RegulariZation Theory and Methods 2 D.V.Lukyanenko and A.G.Yagola Using Parallel Computing for Solving Multidimensional Ill—posed Problems 2.1 Introduction 2.2 Using Parallel Computing 2.2.1 Main idea of parallel computing 2.2.2 Parallel computing limitations 2.3 Parallelization of Multidimensional Ill—posed Problem 2.3.1 Formulation of the problem and method of sohltion 2.3.2 Finite—difference approximation of the functional and its gradient 2.3 3 Parallelization of tile minimization problem 2.4 Some Examples of Calculations 2.5 Conclusions References 3 M.T.Nair RegulariZation of Fredholm Integral Equations of the First Kind using NystrSm Approximation 3.1 Introduction 3.2 Nystroin Method for Regularized Equations 3.2.1 NystrSm approximation of integral operators 3.2.2 Approximation of regularized equation 3.2.3 Solvability of approximate regularized equation 3.2.4 Method of numerical solution 3.3 Error Estimates 3.3.1 Some preparatory results 3.3.2 Error estimate with respect to 3.3.3 Error estimate with respect to 3.3.4 A modified method 3.4 Conclusion References 4 T.Y.Xiao.H.Zhang and L.L.Hao Regularization of Numerical DifFerentiation：Methods and Applications 4.1 Introduction 4.2 Regularizing Schemes 4.2.1 Basic settings 4.2.2 Regularized difference method(RDM) 4.2.3 Smoother-Based regularization(SBR) 4.2.4 Mollifier regularization method(MRM) 4.2.5 Tikhonov's variational regulariZation(TiVR) 4.2.6 Lavrentiev regularization method(LRM) 4.2.7 Discrete regularization method(DRM) 4.2.8 Semi-Discrete Tikhonov regularization(SDTR) 4.2.9 Total variation regularization(TVR) 4.3 Numerical Comparisons 4.4 Applied Examples 4.4.1 Simple applied problems 4.4.2 The inverse heat conduct problems(IHCP) 4.4.3 The parameter estimation in new product diffusion model 4.4.4 Parameter identification of sturm-liouville operator 4.4.5 The numerieal inversion of Abel transform 4.4.6 The linear viscoelastic stress analysis 4.5 Discussion and Conclusion References …… Ⅲ Optimal Inverse Design and Optimization Methods Ⅳ Recent Advances in Inverse Scattering Ⅴ Inverse Vibration，Data Processing and Imaging Ⅵ Numerical Inversion in Geosciences Index

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《应用反问题中的计算方法(英文版)》可作为高等学校环境科学、环境工程和其他相关专业的本科生教学用书及研究生与博士生的参考书。

作者简介

《应用反问题中的计算方法(英文版)》主要阐明了环境伦理学的基本理论及其在中国的发展，西方环境伦理学派的发展简况，环境道德与环境教育的关系，可持续发展和环境立法中的环境伦理，环境伦理的实践（实践部分主要包括环境伦理与管理政策、人口、科学技术等）、环境意识中的平等观等方面的内容。本着简明扼要的宗旨，《应用反问题中的计算方法(英文版)》重点阐述环境伦理学的基本知识和基本理论。

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