Simultaneous Estimation of Noise Level and Solution Smoothness for Ill-Posed Problems
Final Report Abstract
Inverse problems are often ill-posed, specifically not continuously invertible, meaning that small errors in the measured data may yield reconstructed causes far off the true one. In order to prevent this and to stabilize the recovery, many algorithms have been developed and many theoretical results have been established that not only guarantee the stability of the reconstruction process but also yield bounds on the reconstruction errors and rules for the choice of the regularization parameters that allow to tune between stability and reconstruction accuracy. Unfortunately, the theory requires assumptions on the unknown solution and on the size of the noise level that are often not realistic in practice. The main objective of this project was to develop a method to extract both the noise level and the smoothness parameter in a source condition. This goal has been reached, and several interesting results have been discovered along the way. A new doctrine has been developed for understanding regularization of inverse problems. Namely, one may understand regularization as approximating the unknown in a suffciently smooth space. This extends to some extent to the approximation of the measured data, which in turn can be used to reveal the hidden parameters. Assuming the classical source condition in the setting of linear inverse problems in Hilbert spaces, we have shown that, calculating reconstructions for many different regularization parameters in classical Tikhonov regularization, the residuals follow a distinct pattern that can be used to extract both the measurement noise and the solution smoothness. The new doctrine also lead to new parameter choice rules, to showing that many other parameter choice rules are equivalent, and to a new and in the eyes of the author much simpler derivation of the classical convergence theory for linear ill-posed problems in Hilbert spaces. The project made significant contribution to theory and computational reality of inverse problems. In a sub project on the Hausdorff moment problem it was shown how difficult and sometimes seemingly contradicting theory and numerics can be for ill-posed systems.
Publications
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The Kurdyka–Łojasiewicz Inequality as Regularity Condition. Springer Proceedings in Mathematics & Statistics, 257-274. Springer Singapore.
Gerth, Daniel & Kindermann, Stefan
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A new interpretation of (Tikhonov) regularization. Inverse Problems, 37(6), 064002.
Gerth, Daniel
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Estimating Solution Smoothness and Data Noise with Tikhonov Regularization. Numerical Functional Analysis and Optimization, 43(1), 88-115.
Gerth, Daniel & Ramlau, Ronny
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THE HAUSDORFF MOMENT PROBLEM IN THE LIGHT OF ILL-POSEDNESS OF TYPE I. Eurasian Journal of Mathematical and Computer Applications, 9(2), 57-87.
Gerth, Daniel; Hofmann, Bernd; Hofmann, Christopher & Kindermann, Stefan
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A note on numerical singular values of compositions with non-compact operators. ETNA - Electronic Transactions on Numerical Analysis, 57, 57-66.
Gerth, Daniel
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A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM. Eurasian Journal of Mathematical and Computer Applications, 10(1), 40-50.
Gerth, D. & Hofmann; B.
