The Method of Approximate Inverse: Theory and Applications

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)

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.

From the reviews:

"The powerful method of the approximate inverse is a good bunch of regularization techniques, and this monograph presents a comprehensive outline of this method. Application to 3D Doppler tomography and the spherical mean operator is then studied in details, and further results on X-ray diffractometry, thermoacoustic computerized tomography and reconstruction kernels in 3D are attached. The book is naturally recommended for computer tomographers and graduate students heading toward computer tomography, but it contains many beneficial results for researchers of Radon transforms too." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

"The book under review which deals with a particular class of regularization methods, the so called method of approximate inverse, is the result of continuous study of the author for more than a decade, by himself for his habilitation thesis and also in collaborations with many experts in the field, including A. K. Louis (his own teacher), A. Rieder and many others. ... No doubt, the book is a good addition to the literature on regularization of ill-posed inverse problems." (M. Thamban Nair, Zentralblatt MATH, Vol. 1171, 2009)

From the reviews:"The powerful method of the approximate inverse is a good bunch of regularization techniques, and this monograph presents a comprehensive outline of this method. Application to 3D Doppler tomography and the spherical mean operator is then studied in details, and further results on X-ray diffractometry, thermoacoustic computerized tomography and reconstruction kernels in 3D are attached. The book is naturally recommended for computer tomographers and graduate students heading toward computer tomography, but it contains many beneficial results for researchers of Radon transforms too." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 74, 2008)“The book under review which deals with a particular class of regularization methods, the so called method of approximate inverse, is the result of continuous study of the author for more than a decade, by himself for his habilitation thesis and also in collaborations with many experts in the field, including A. K. Louis (his own teacher), A. Rieder and many others. … No doubt, the book is a good addition to the literature on regularization of ill-posed inverse problems.” (M. Thamban Nair, Zentralblatt MATH, Vol. 1171, 2009)