Piotrowska-Kurczewski, Iwona
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Piotrowska-Kurczewski, Iwona
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Piotrowska-Kurczewski, Iwona
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Kurcezewski, Iwona
Piotrowska, Iwona
Piotrowska-Kurczewski, I.
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Item-typ:Veröffentlichung, Regularization of ill-posed inverse problems with tolerances and sparsity in the parameter spaceWe consider the solution of ill-posed inverse problems using regularization with tolerances. In particular, we are interested in the reconstruction of solutions that lie within or close to an area outlined by a tolerance measure. To approximate the true solution of the problem in a stable way, we propose a Tikhonov functional with a tolerance function in the regularization term. The tolerances allow us to neglect errors in the penalty term up to a certain threshold. Our theoretical analysis proves that the proposed method complies with all the requirements of variational regularization methods. In addition, we establish convergence rates for the convergence of minimizers to the true solution. Moreover, we are interested in obtaining sparse solutions. For this purpose, we extend the proposed approach with the idea of elastic net regularization by introducing an additional penalty term that promotes the sparsity of the solution. We establish theoretical results for this elastic net approach and give a convergence rate analysis for the minimizers. To confirm our analytical findings, we illustrate the effect of tolerances in the computed regularized solutions on some numerical examples.Dissertation295 842 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Tikhonov functionals incorporating tolerances in discrepancy term for inverse problems(2023-01-13); ; ; This thesis contributes to the field of inverse problems and their regularization through Tikhonov-type regularization schemes. Tikhonov-type regularization schemes use a discrepancy term on the operator evaluation and a regularization term on the parameter. One can change the penalty term to incorporate different a-priori information about the true parameter. For example, most research focuses on the regularization term, such as applying sparsity constraints instead of $L_2$-penalty. This work takes a different approach and adds an $\varepsilon$-insensity to the discrepancy term. This insensitivity does add another regularization and accounts for confidence bands. We can obtain these bands through multiple measurements, for example. Besides the mathematical framework, this work explores possible numerical solvers for the altered Tikhonov functional. Depending on the chosen norm of the discrepancy term and the type of penalty term, the altered Tikhonov functional may not be differentiable. In this case, a non-smooth solver is necessary. This thesis compares existing non-smooth solvers with a newly introduced subgradient method with adaptive step size. Finally, we apply the theory to a parameter identification problem. The example is from micro-milling and the resulting surface. First, an existing cutting process model is taken and expanded to account for wear on the cutting tool during the process. Then we use the model for the parameter identification.Dissertation247 232