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Other titles in the Lecture Notes in Mathematics series:
Lecture Notes in Mathematics #1906: The Method of Approximate Inverse: Theory and Applicationsby Thomas Schuster
Synopses & Reviews
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.
This book is concerned with the method of approximate inverse which is a regularization technique for stably solving inverse problems in various settings. It demonstrates the performance and functionality of the method on several examples from medical imaging and non-destructive testing, such as computerized tomography, Doppler tomography, SONAR, X-ray diffractometry and thermoacoustic computerized tomography.
About the Author
1990 - 1995 Study of Mathematics at Saarland University Saarbrücken (Germany) 1996 - 2004 Scientific assistant at Saarland University Saarbrücken (Germany) 1999 PhD at Saarland University Saarbrücken (Germany) 2002 - 2003 Research stay at Tufts University Medford, MA (USA) 2004 Habilitation at Saarland University Saarbrücken (Germany) 2004 - 2006 Assistant Professor at Saarland University Saarbrücken (Germany) 2007 - today Associate Professor at the Helmut Schmidt University Hamburg (Germany)
Table of Contents
Part I: Inverse and Semi-discrete Problems: Ill-posed Problems and Regularization Methods.- Approximate Inverse in L2-spaces.- Approximate Inverse in Hilbert Spaces.- Approximate Inverse in Distribution Spaces.- Conclusion and Perspectives.- Part II: Application to 3D Doppler Tomography.- A Semi-discrete Setup for Doppler Tomography.- Solving the Semi-discrete Problem.- Convergence and Stability.- Approaches for Defect correction.- Conclusion and Perspectives.- Part III: Application to the Spherical mean operator.- The Spherical Mean Operator.- Design of a Mollifier.- Computation of Reconstruction Kernels.- Numerical Experiments.- Conclusion and Perspectives.- Part IV: Further Applications.- Approximate Inverse and X-ray Diffractometry.- A Filtered Backprojection Algorithm for Thermoacoustic Computerized Tomography (TCT).- Computation of Reconstruction Kernels in 3D Computerized Tomography.- Conclusion and Perspectives.- References.- Index.
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