On Sobolev Instability Of The Interior Problem Of Tomography

Keywords

Interior problem; Prior knowledge; Singular value decomposition; Sobolev estimates; Tomography

Abstract

As is known, solving the interior problem with prior data specified on a finite collection of intervals Ii is equivalent to analytic continuation of a function from Ii to an open set J. In the paper we prove that this analytic continuation can be obtained with the help of a simple explicit formula, which involves summation of a series. Our second result is that the operator of analytic continuation is not stable for any pair of Sobolev spaces regardless of how close the set J is to Ii. Our main tool is the singular value decomposition of the operator He-1 that arises when the interior problem is reduced to a problem of inverting the Hilbert transform from incomplete data. The asymptotics of the singular values and singular functions of He-1, the latter being valid uniformly on compact subsets of the interior of Ii, was obtained in [5]. Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval J. Our last result is the convergence of the asymptotic approximation of the singular functions in the L2(Ii) sense. We also present a preliminary numerical experiment, which illustrates how to use our results for reducing the instability of the analytic continuation by optimizing the position of the intervals with prior knowledge.

Publication Date

6-15-2016

Publication Title

Journal of Mathematical Analysis and Applications

Volume

438

Issue

2

Number of Pages

962-990

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jmaa.2015.12.062

Socpus ID

84961218465 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84961218465

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