逆问题数学理论导论

出版时间：1999-10  出版社：世界图书出版公司  作者：A.Kirsch  页数：282  
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内容概要

Following Keller [119] we call two problems inverse to each other if the formulation of each of them requires full or partial knowledge of the other. By this definition, it is obviously arbitrary which of the two problems we call the direct and which we call the inverse problem. But usually, one of the problems has been studied earlier and, perhaps, in more detail. This one is usually called the direct problem, whereas the other is the inverse problem. However, there is often another, more important difference between these two problems. Hadamard (see [91]) introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem iU-posed. It turns out that many interesting and important inverse problems in science lead to ill-posed problems,, while the corresponding direct problems are well-posed. Often, existence and uniqueness can be forced by enlarging or reducing the solution space (the space of "models"). For restoring stability, however, one has to change the topology of the spaces,which is in many cases impossible because of the presence of measurement errors. At first glance, it seems to be impossible to compute the solution of a problem numerically if the solution of the problem does not depend continuously on the data, i.e., for the case of ill-posed problems. Under additional a priori information about the solution, such as smoothness and bounds on the derivatives, however, it is possible to restore stability and construct efficient numerical algorithms.

书籍目录

Preface Introduction and Basic Concepts 　1.1 Examples of Inverse Problems 　1.2 I11-Posed Problems 　1.3 The Worst-Case Error 　1.4 Problems 2 Regularization Theory for Equations of the First Kind 　2.1 A General Regularization Theory 　2.2 Tikhonov Regularization 　2.3 Landweber Iteration 　2.4 A Numerical Example 　2.5 The Discrepancy Principle of Morozov 　2.6 Landweber‘s Iteration Method with Stopping Rule 　2.7 The Conjugate Gradient Method 　2.8 Problems 3 Regularization by Discretization 　3.1 Projection Methods 　3.2 Galerkin Methods 　　3.2.1 The Least Squares Method 　　3.2.2 The Dual Least Squares Method　　3.2.3  The Bubnov-Galerkin Method for　　Coercive Operators　3.3  Application to Symm's Integral Equation of the First Kind　3.4  Collocation Methods　　3.4.1  Minimum Norm Colloction　　3.4.2 Collocation of Symm's Equation　3.5 "Numerical Experiments for Symm's Equation　3.6  The Backus-Gilbert Method　3.7  Problems4 Inverse Eigenvalue,Problems　4.1  Introduction　4.2  Construction of a Fundamental System　4.3  Asymptotics of the Eigenvalues and Eigenfunctions　4.4  Some Hyperbolic Problems　4.5  The Inverse Problem　4.6  A Parameter Identification Problem　4.7  Numerical Reconstrucion Techniques　4.8  Problems5 An Inverse Scattering Problem　5.1  Introduction　5.2  The Direct Scattering problem　5.3  Properties of the Far Field Patterns　5.4  Uniqueness of the Inverse Problem　5.5  Numerical Methods　　5.5.1  A Simplified Newton Method　　5.5.2  A Modified Gradient Method　　5.5.3  The Dual Space Method　5.6  Problems　　A Basic Facts from Functional Analysis　　　A.1  Normed Spaces and Hilbert Spaces　　　A.2  Orthonormal Systems　　　A.3  Linear Bounded and Compact Operators　　　A.4  Sobolev Spaces of Periodi  Functions　　　A.5  Spectral Theory for Compact  Operators in Hilbert Spaces　　　A.6  The Frechet Derivative　　B Proofs of the Results of Section 2.7ReferencesIndex

用户评论 (总计9条)

•   一般关于反问题的经典教材，这本书更侧重于计算方面！纯理论的可以看Isakov的Inverse problem for PDE
•   事一本很经典很前沿的数学专著
•   就喜欢Springer~
  内容偏重理论
•   这本书不错，值得一看，也值得收藏，同时建议购买了这本书的朋友，在反问题这方面，还有一本现在很容易能买到的书：王彦飞反演问题的数值解法与应用，也不错，而且还是精装本的，很好的！哦，对了，一定要感谢一下当当网的客服，速度快，而且态度好，呵呵，支持！！！
•   书中极品
•   很不错的一本书，就是理论性太强了，一般工科的人读起来比较晦涩
•   我喜欢，仔细地正在读。。。
•   听说这本书很好，所以我买了学习学习。速度真的很快，一天就到了，感谢当当
•   我是7.8折，价格便宜。运输有点慢，质量还不错。