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- Iterative regularization methods for nonlinear ill-posed problems.
In some cases successful discretization method can be viewed as regularization method and then other regularization is not needed. Namely, if the discretization method converges in case of exact data, then in case of noisy data we can choose the discretization step in dependence of the noise level of the data in such a way, that the solution of the discretized problem converges to the solution of continuous problem if noise level tends to zero.
Such phenomenon is called self-regularization. We consider self-regularization of ill-posed problems in Hilbert and Banach spaces by the following projection methods: least squares method, least error method, collocation method. In these methods the regularization parameter is the dimension of the projected equation.
Regularization of Ill-Posed Problems by Iteration Methods
We choose this dimension by the discrepancy principle or by the monotone error rule and give convergence results. Since many practical problems are related to shape recovery, chapter 6 describes applications of the level set methods and their adaptation for solving inverse problems.
Finally, chapter 7 presents two applications: reconstruction of transducer pressure fields from Schlieren tomography and a parameter estimation problem from nonlinear magnetics. The book is intended as a text book for graduate students and for specialists working in the field of inverse problems.