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Conditional Stability Estimates and Regularization with Applications to Cauchy Problems for the Helmholtz Equation

In this article we consider the problem of reconstructing solutions x† of ill-posed problems Ax = y where A is a linear operator between Hilbert spaces X and Y. We assume that instead of exact data some noisy data y δ ∈ Y with ‖y − y δ‖ ≤ δ are given and that x† possesses a certain solution smoothness that we describe by x† ∈ M with some source set M ⊂ X. We discuss the following questions: (i) Which best possible accuracy can be obtained for identifying x† from noisy data y δ ∈ Y under the assumptions ‖y − y δ‖ ≤ δ and x† ∈ M? (ii) How to regularize such that the best possible accuracy can be guaranteed? We apply our results to Cauchy problems for the Helmholtz equation and show that, depending on different smoothness situations, the best possible accuracy may be of Hölder type, of logarithmic type, or of some other type. In addition, we study regularization methods that provide the best possible accuracy. In case of appropriate a posteriori parameter choice, the best possible order of accuracy can be obtained without using any smoothness information for x†.