Regularizing inverse problems
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| Опубліковано в:: | ProQuest Dissertations and Theses (2014) |
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ProQuest Dissertations & Theses
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| Онлайн доступ: | Citation/Abstract Full Text - PDF |
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| Короткий огляд: | An inverse problem reconstructs the unknown internal parameters of a subject based on collected data derived synthetically or from real measurements. Inverse problems often lack the well-posedness defined by J. Hadamard; in other words, solutions of inverse problems, namely the reconstructions of the parameters, may not exist, may not be unique or may be unstable. Regularization is a technique that deals with such situations. The well-known Tikhonov regularization method translates the original inverse problem to optimization problems of minimizing the norm of the data misfit plus a weighted regularization functional that incorporates the a priori information we may have about the original problem. The choices of the regularization functional r(q) include ||q||2L 2||q||2L 1|q|BV and |q| TV. However, each of these has its limitations. In this work, we develop a novel Hs seminorm regularization method and present numerical results for model problems. This method relies on the evaluation of the seminorms of an intermediary Hilbert space, namelyHs space, that stays between L2 and H1. The Hs seminorm regularization is designed to minimize the undesirable aspects of the existing L2 and H1 regularization functionals. The Hs seminorm regularization also allows discontinuities and stabilizes the perturbations. We study theHs seminorm regularization method both theoretically and numerically. We consider the theoretical analysis of this new regularization method based on a model problem. We show that a stable solution can be achieved with some conditions. In addition, we prove the convergence and guarantee a convergence rate provided additional conditions for the model problem when the considered domain is 1D. Numerically, we produce an approximated discretization of theHs seminorm regularization that can be applied to 1D, 2D or 3D examples. We also provide reconstructions of both continuous and discontinuous parameters from synthetic data and a comparison of these solutions to the ones based on existing L2 and H1 regularization methods. Furthermore, we also apply theHs seminorm regularization method to a fluorescence optical tomography problem. In summary, we study and implement theHs seminorm regularization method for inverse problems, which can provide a stable solution to the model problem. The numerical results indicate the robustness of the new method and suggests that theHs seminorm regularization method produces the closest approximation of the exact solution than the L2 norm and H 1 seminorm regularization methods for the model problem. |
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| ISBN: | 9781321587531 |
| Джерело: | ProQuest Dissertations & Theses Global |