Thomas Schuster (auth.)3540712267, 9783540712268
Inverse problems arise whenever one tries to calculate a required quantity from given measurements of a second quantity that is associated to the first one. Besides medical imaging and non-destructive testing, inverse problems also play an increasing role in other disciplines such as industrial and financial mathematics. Hence, there is a need for stable and efficient solvers. The book is concerned with the method of approximate inverse which is a regularization technique for stably solving inverse problems in various settings such as L2-spaces, Hilbert spaces or spaces of distributions. The performance and functionality of the method is demonstrated on several examples from medical imaging and non-destructive testing such as computerized tomography, Doppler tomography, SONAR, X-ray diffractometry and thermoacoustic computerized tomography. The book addresses graduate students and researchers interested in the numerical analysis of inverse problems and regularization techniques or in efficient solvers for the applications mentioned above.
Table of contents :
Front Matter….Pages I-XIII
Front Matter….Pages 1-4
Ill-posed problems and regularization methods….Pages 5-9
Approximate inverse in L 2 -spaces….Pages 11-24
Approximate inverse in Hilbert spaces….Pages 25-38
Approximate inverse in distribution spaces….Pages 39-47
Conclusion and perspectives….Pages 49-49
Front Matter….Pages 51-54
A semi-discrete setup for Doppler tomography….Pages 55-61
Solving the semi-discrete problem….Pages 63-79
Convergence and stability….Pages 81-87
Approaches for defect correction….Pages 89-103
Conclusion and perspectives….Pages 105-106
Front Matter….Pages 107-110
The spherical mean operator….Pages 111-121
Design of a mollifier….Pages 123-131
Computation of reconstruction kernels….Pages 133-137
Numerical experiments….Pages 139-144
Conclusion and perspectives….Pages 145-145
Front Matter….Pages 147-149
Approximate inverse and X-ray diffractometry….Pages 151-164
A filtered backprojection algorithm….Pages 165-179
Computation of reconstruction kernels in 3D computerized tomography….Pages 181-185
Conclusion and perspectives….Pages 187-187
Back Matter….Pages 189-202
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