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Numerical Methods for Inverse Problems [Kõva köide]

  • Formaat: Hardback, 240 pages, kõrgus x laius x paksus: 241x163x18 mm, kaal: 499 g
  • Ilmumisaeg: 08-Apr-2016
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • ISBN-10: 1848218184
  • ISBN-13: 9781848218185
Teised raamatud teemal:
Numerical Methods for Inverse ProblemsSuurem pilt
  • Formaat: Hardback, 240 pages, kõrgus x laius x paksus: 241x163x18 mm, kaal: 499 g
  • Ilmumisaeg: 08-Apr-2016
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • ISBN-10: 1848218184
  • ISBN-13: 9781848218185
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781848218185.html
Märksõnad:
This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system.

The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications.

This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.

Arvustused

"The book is very carefully written, in a reader-friendly style. It can be considered as an introductory textbook for the theory of ill-posed problems and their numerical solution." (Mathematical Reviews/MathSciNet 11/05/2017)

Preface ix
Part 1 Introduction and Examples
1(28)
Chapter 1 Overview of Inverse Problems
3(6)
1.1 Direct and inverse problems
3(1)
1.2 Well-posed and ill-posed problems
4(5)
Chapter 2 Examples of Inverse Problems
9(20)
2.1 Inverse problems in heat transfer
10(3)
2.2 Inverse problems in hydrogeology
13(3)
2.3 Inverse problems in seismic exploration
16(5)
2.4 Medical imaging
21(4)
2.5 Other examples
25(4)
Part 2 Linear Inverse Problems
29(74)
Chapter 3 Integral Operators and Integral Equations
31(14)
3.1 Definition and first properties
31(5)
3.2 Discretization of integral equations
36(6)
3.2.1 Discretization by quadrature--collocation
36(3)
3.2.2 Discretization by the Galerkin method
39(3)
3.3 Exercises
42(3)
Chapter 4 Linear Least Squares Problems -- Singular Value Decomposition
45(26)
4.1 Mathematical properties of least squares problems
45(7)
4.1.1 Finite dimensional case
50(2)
4.2 Singular value decomposition for matrices
52(5)
4.3 Singular value expansion for compact operators
57(3)
4.4 Applications of the SVD to least squares problems
60(5)
4.4.1 The matrix case
60(3)
4.4.2 The operator case
63(2)
4.5 Exercises
65(6)
Chapter 5 Regularization of Linear Inverse Problems
71(32)
5.1 Tikhonov's method
72(11)
5.1.1 Presentation
72(1)
5.1.2 Convergence
73(8)
5.1.3 The L-curve
81(2)
5.2 Applications of the SVE
83(5)
5.2.1 SVE and Tikhonov's method
84(1)
5.2.2 Regularization by truncated SVE
85(3)
5.3 Choice of the regularization parameter
88(6)
5.3.1 Morozov's discrepancy principle
88(3)
5.3.2 The L-curve
91(1)
5.3.3 Numerical methods
92(2)
5.4 Iterative methods
94(4)
5.5 Exercises
98(5)
Part 3 Nonlinear Inverse Problems
103(64)
Chapter 6 Nonlinear Inverse Problems - Generalities
105(22)
6.1 The three fundamental spaces
106(5)
6.2 Least squares formulation
111(5)
6.2.1 Difficulties of inverse problems
114(1)
6.2.2 Optimization, parametrization, discretization
114(2)
6.3 Methods for computing the gradient -- the adjoint state method
116(7)
6.3.1 The finite difference method
116(2)
6.3.2 Sensitivity functions
118(1)
6.3.3 The adjoint state method
119(1)
6.3.4 Computation of the adjoint state by the Lagrangian
120(3)
6.3.5 The inner product test
123(1)
6.4 Parametrization and general organization
123(2)
6.5 Exercises
125(2)
Chapter 7 Some Parameter Estimation Examples
127(28)
7.1 Elliptic equation in one dimension
127(2)
7.1.1 Computation of the gradient
128(1)
7.2 Stationary diffusion: elliptic equation in two dimensions
129(8)
7.2.1 Computation of the gradient: application of the general method
132(2)
7.2.2 Computation of the gradient by the Lagrangian
134(1)
7.2.3 The inner product test
135(1)
7.2.4 Multiscale parametrization
135(1)
7.2.5 Example
136(1)
7.3 Ordinary differential equations
137(10)
7.3.1 An application example
144(3)
7.4 Transient diffusion: heat equation
147(5)
7.5 Exercises
152(3)
Chapter 8 Further Information
155(12)
8.1 Regularization in other norms
155(2)
8.1.1 Sobolev semi-norms
155(2)
8.1.2 Bounded variation regularization norm
157(1)
8.2 Statistical approach: Bayesian inversion
157(6)
8.2.1 Least squares and statistics
158(2)
8.2.2 Bayesian inversion
160(3)
8.3 Other topics
163(4)
8.3.1 Theoretical aspects: identifiability
163(1)
8.3.2 Algorithmic differentiation
163(1)
8.3.3 Iterative methods and large-scale problems
164(1)
8.3.4 Software
164(3)
Appendices
167(38)
Appendix 1
169(14)
Appendix 2
183(10)
Appendix 3
193(12)
Bibliography 205(8)
Index 213
Michel Kern is a research scientist in the Serena group at the Inria Research Center in Paris, France