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Long time ago I came across with a paper that covered early theoretical work (first half of 20th century) in the field of inverse problems. I remember there was a reference to a paper which proved that inverse problem can be solved uniquely for a spherical domain with infinitely many sources and very precise receivers distributed continuously over its surface.

Unfortunately, I cannot find neither of these papers. Does anyone know this work or anything similar?

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    $\begingroup$ Is it this? $\endgroup$ – J. M. May 15 '13 at 15:11
  • $\begingroup$ Guys, thanks for your help. Wolfgang got it right below. $\endgroup$ – Indalo May 17 '13 at 8:31
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I think you're looking for the paper by Calderon on electrical impedance tomography:

@InCollection{Cal80, author = {A. Calder{\'o}n}, title = {On an inverse boundary value problem}, booktitle = {Seminar on Numerical Analysis and its Applications to Continuum Physics}, pages = {65--73}, publisher = {Soc. Brasileira de Matematica, Rio de Janeiro}, year = 1980, editor = {W. H. Meyer and M. A. Raupp} }

There is a long sequence of papers that came later (by Uhlmann, Paivarinta and others) that showed unique solvability for larger and larger classes of parameters.

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  • $\begingroup$ Thank you. Looking through these works seems like a handful of people are still trying to address inverse problems from this perspective. All the rest went to numerical world and basically ended up with Tikhonov regularization. The Soviet school used to be very strong, but, as anything else there, faded away... $\endgroup$ – Indalo May 17 '13 at 7:49
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    $\begingroup$ No, they're still strong. But they're all in the US and Western Europe now :-) $\endgroup$ – Wolfgang Bangerth May 18 '13 at 2:06
  • $\begingroup$ But more seriously, I think the big questions have now been answered. The question was always: can you identify an $L^\infty$ coefficient from boundary data. It took a while till people could really answer this for this large space and they worked up to it, but it is now a closed case. $\endgroup$ – Wolfgang Bangerth May 18 '13 at 2:07
  • $\begingroup$ The theoretical result is known in two dimensions, but the proof does not work in higher dimensions. Partial data results and identification on manifolds are still under active investigation. $\endgroup$ – Tommi Nov 10 '17 at 8:10

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