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Comparison of direct regularization methods based on least squares for problems corrupted by noise
 Cepko, Tomáš ; Hnětynková, Iveta (advisor) ; Kučera, Václav (referee)
In this thesis we are going to deal with the inverse linear approximation problem Ax ≈ b, where our goal is to find the best approximation x of the unknown exact solution. We are going to especially focus on the so-called rank-deficient and ill-posed problems, which are very ill-conditioned and sensitive to possible random noise present in b. To solve these problems, we must use regularization methods, which suppress this sensitivity. The main goal of this thesis is to get a comprehensive overview of direct methods T-SVD, T- TLS and Tikhonov regularization, and analyse their close connection with classical least squares methods. One possible approach is to formulate these regularization methods as so-called filtering. In this way we implement them for numerical experiments. This thesis will also include a numerical comparision of these methods for selected problems from the Regularization Toolbox and in the application problem of image reconstruction. 1
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