作者：(德)佛罗莱恩·弗吕豪夫(Florian
页数：148页
出版社：哈尔滨工业大学出版社
出版日期：2022
ISBN：9787560398549
电子书格式：pdf/epub/txt
内容简介
在本书中, 作者调查研究了解决某些反问题的方法, 根据阿达玛 (Hadamard) 的观点, 如果存在唯一的解并且该解持续依赖于数据, 那么这个问题就很突出了。如果该解的某一个性质不满足这个问题, 那么这种情况称为不适定的。逆问题通常是不适定的。在许多应用中, 可能不需要去详细地解反间题, 但在本书中作者研究了解决这种二进制反问题的三种不同的方法。本书共分为五章, 具体内容包括线性正则化、非线性吉洪诺夫 (Tikhonov) 正则化、金兹堡·朗道 (Ginzburg-Landau) 正则化等相关理论。
目录
1 Introduction
I Regularization Methods
2 Analysis of regularization methods
2.1 Linear regularization
2.2 Nonlinear Tikhonov regularization
2.3 BV functions, sets of finite perimeter and their relation to level sets
2.4 Level set regularization
2.4.1 Analysis of Level Set Regularization
2.4.2 Towards an Analysis of Level Set Regularization Techniques
2.4.3 Minimizing Concept
2.4.4 Convergence Analysis ffinctiohs
2.5 Ginzburg-Landau regularization
2.5.1 Ginzburg-Landan functional
2.5.2 Regularization with a Ginzburg-Landau functional
3 Numerical implementation of regularizatiou methods
3.1 Continuous regularizatinn and a connection to iterative regularization
3.1.1 Continuous regularization
3.1.2 Convex analysis
3.1.3 Connection between continuous and iterative regu]arization
3.2 Implementation of the level set regularization
3.2.1 Numerical solution
3.2.2 Iterative Regularization and the Relation to Dynamic Level Set Methods
3.3 Implementation of the Ginzhurg-Landan regularization method
3.4 Numerical examples
3.4.1 The inverse conductivity problem
3.4.2 Implementation
3.4.3 Results and Discussion
II A parabolic-elliptic problem
4 The direct problem
4.1 Analysis of the parabolic-elliptic problem
4.2 Numerical realization of the parabolic-elliptic problem
4.2.1 Reformulation of the direct problem
4.2.2 Convergence analysis of the reformulated problem
4.2.3 Implementation of the reformulated problem
4.2.4 Numerical examples
5 The inverse problem
5.1 The Factorization Method
5.1.1 The Factorization of A0 – A1
5.1.2 Range characterization
5.1.3 Characterization of the inclusion
5.2 Implementation of the inverse problem
5.2.1 Implementation by regularization
5.2.2 Implementation applying the Picard criterion
5.2.3 Numerical examples
A Additional results:constraint ill-posed operator equation
A.1 A modified level set regularization method
A.2 Ginzburg-Landau Regularization
A.3 Implementation of the constraint operator equation
A.4 Examples of the numerical implementation
B Additional results:Image processing
I Regularization Methods
2 Analysis of regularization methods
2.1 Linear regularization
2.2 Nonlinear Tikhonov regularization
2.3 BV functions, sets of finite perimeter and their relation to level sets
2.4 Level set regularization
2.4.1 Analysis of Level Set Regularization
2.4.2 Towards an Analysis of Level Set Regularization Techniques
2.4.3 Minimizing Concept
2.4.4 Convergence Analysis ffinctiohs
2.5 Ginzburg-Landau regularization
2.5.1 Ginzburg-Landan functional
2.5.2 Regularization with a Ginzburg-Landau functional
3 Numerical implementation of regularizatiou methods
3.1 Continuous regularizatinn and a connection to iterative regularization
3.1.1 Continuous regularization
3.1.2 Convex analysis
3.1.3 Connection between continuous and iterative regu]arization
3.2 Implementation of the level set regularization
3.2.1 Numerical solution
3.2.2 Iterative Regularization and the Relation to Dynamic Level Set Methods
3.3 Implementation of the Ginzhurg-Landan regularization method
3.4 Numerical examples
3.4.1 The inverse conductivity problem
3.4.2 Implementation
3.4.3 Results and Discussion
II A parabolic-elliptic problem
4 The direct problem
4.1 Analysis of the parabolic-elliptic problem
4.2 Numerical realization of the parabolic-elliptic problem
4.2.1 Reformulation of the direct problem
4.2.2 Convergence analysis of the reformulated problem
4.2.3 Implementation of the reformulated problem
4.2.4 Numerical examples
5 The inverse problem
5.1 The Factorization Method
5.1.1 The Factorization of A0 – A1
5.1.2 Range characterization
5.1.3 Characterization of the inclusion
5.2 Implementation of the inverse problem
5.2.1 Implementation by regularization
5.2.2 Implementation applying the Picard criterion
5.2.3 Numerical examples
A Additional results:constraint ill-posed operator equation
A.1 A modified level set regularization method
A.2 Ginzburg-Landau Regularization
A.3 Implementation of the constraint operator equation
A.4 Examples of the numerical implementation
B Additional results:Image processing
