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arXiv · 1603.05133

Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods

Thorsten Hohage·Frederic Weidling
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Abstract

We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of ill-posedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. If the forward operator can be written in terms of the functional calculus of a Laplace-Beltrami operator, variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.

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Thorsten Hohage, Frederic Weidling. 2016-03-16. Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods. https://doi.org/10.1137/16m1067445

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