Exponential instability in the fractional Calderón problem

In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this result, we follow the strategy introduced by Mandache in (2001 Inverse Proble...

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Bibliographic Details
Published in:Inverse problems 2018-04, Vol.34 (4), p.45003
Main Authors: Rüland, Angkana, Salo, Mikko
Format: Article
Language:English
Subjects:
exponential instability
fractional Calderón problem
Runge approximation
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Summary:In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this result, we follow the strategy introduced by Mandache in (2001 Inverse Problems 17 1435) for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in Rüland and Salo (2017 arXiv:1708.06294). Finally, in one dimension, we show a close relation between the fractional Calderón problem and the truncated Hilbert transform.
ISSN:0266-5611
1361-6420
DOI:10.1088/1361-6420/aaac5a