Abstract
In this paper, we identify the Hilbert uniqueness method for a boundary control problem with the calculation of the pseudo inverse. Because of its ill-posedness, we approximate it by a regularized Hilbert uniqueness method, which we prove to be identical with Tikhonov regularization. By this equivalence, we can find sufficient conditions for convergence and convergence rates, which require approximation rates in Müntz spaces. We show that these conditions are fulfilled by an a priori bound in Sobolev norms on the exact solution.
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Kindermann, S. Convergence Rates of the Hilbert Uniqueness Method via Tikhonov Regularization. Journal of Optimization Theory and Applications 103, 657–673 (1999). https://doi.org/10.1023/A:1021792225922
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DOI: https://doi.org/10.1023/A:1021792225922