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Abstract
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Tikhonov regularization, which is one of the classical regularization methods, has been well Known to be efficient in solving (non)-linear ill-posed problems. This scheme usually involves a single penalty term from which the stabilization of reconstructed solutions is expected. At the same time, the reconstructed solutions are mostly constructed upon a single scale. Here, in the first part of the talk, we examine some recent results on multi-parameter Tikhonov regularization where multiple penalty terms are incorporated in building the Tikhonov functional. These multiple penalty terms are implemented to different a priori information or structures of the exact solution.
In the second part of the talk, we bring forth the idea of multiscale approximation and present an iterative Tikhonov regularization method where compactly supported radial basis functions of varying radii are employed to accommodate different scales which are owned by the exact solution. Several numerical examples are shown to illustrate the appropriateness of the proposed methods.
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