Schuster T. The method of approximate inverse: theory and applications (Вerlin, 2007) - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаSchuster T. The method of approximate inverse: theory and applications. - Berlin: Springer, 2007. - 198 p. - (Lecture notes in mathematics; 1906). - ISSN 0075-8434; ISBN 978-3-540-71226-8
 

Место хранения: 013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents
 
Part I.  Inverse and Semi-discrete Problems

1.  III-posed problems and regularization methods ............... 5
2.  Approximate inverse in L2-spaces ........................... 11
    2.1.  The idea of approximate inverse ...................... 11
    2.2.  A first example: The Radon transform ................. 17
3.  Approximate inverse in Hilbert spaces ...................... 25
    3.1.  Semi-discrete operator equations ..................... 25
    3.2.  Convergence and stability ............................ 32
4.  Approximate inverse in distribution spaces ................. 39
    4.1.  Mollifier and reconstruction kernels in dual
          spaces of smooth functions ........................... 40
    4.2.  Dealing with semi-discrete equations ................. 44
5.  Conclusion and perspectives ................................ 49

Part II.  Application to 3D Doppler Tomography

6.  A semi-discrete setup for Doppler tomography ............... 55
7.  Solving the semi-discrete problem .......................... 63
    7.1.  Definition of the operators Пp,q,r and Ed ............. 63
    7.2.  Computation of reconstruction kernels for Dj ......... 70
    7.3.  The method of approximate inverse for
          Ωр,q,rD ............................................... 76
8.  Convergence and stability .................................. 81
9.  Approaches for defect correction ........................... 89
    9.1.  Potentials as solutions of elliptic boundary
          value problems ....................................... 90
    9.2.  The Neumann problem .................................. 93
          9.2.1.  A boundary element method for the Neumann
                  problem ...................................... 93
          9.2.2.  The computation of the Newton potentials ..... 94
          9.2.3.  Numerical results ........................... 100
    9.3.  The Dirichlet problem ............................... 101
10. Conclusion and perspectives ............................... 105

Part III. Application to the spherical mean operator

11. The spherical mean operator ............................... 111
    11.1. Spherical means in SONAR and SAR .................... 111
    11.2. Properties of the spherical mean operator ........... 113
    11.3. Approximate inverse for M ........................... 118
12. Design of a mollifier ..................................... 123
13. Computation of reconstruction kernels ..................... 133
14. Numerical experiments ..................................... 139
15. Conclusion and perspectives ............................... 145

Part IV.  Further Applications

16. Approximate inverse and X-ray diffractometry .............. 151
    16.1. X-ray diffractometry ................................ 151
    16.2. Approximate inverse for the Laplace transform ....... 153
    16.3. A solution scheme for the X-ray diffractometry
          problem ............................................. 161
17. A filtered backprojection algorithm for thermoacoustic
    computerized tomography (TCT) ............................. 165
    17.1. Thermoacoustic computerized tomography (TCT) ........ 165
    17.2. An inversion method for the spherical geometry ...... 168
    17.3. Numerical results ................................... 175
18. Computation of reconstruction kernels in 3D computerized
    tomography ................................................ 181
19. Conclusion and perspectives ............................... 187

References .................................................... 189
Index ......................................................... 197


 
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