We study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data yd are given satisfying ||y - yd|| ≤ d with known noise level d. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximation provides order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extend recent results for infinitely smoothing operators.
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M. Thamban Nair, Ulrich Tautenhahn, E. Schock, Morozov's Discrepancy Principle under General Source Conditions. Z. Anal. Anwend. 22 (2003), no. 1, pp. 199–214DOI 10.4171/ZAA/1140