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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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# Convexification of restricted Dirichlet-to-Neumann map

Michael V. Klibanov
• Corresponding author
• Department of Mathematics and Statistics, University of North Carolina atCharlotte, Charlotte, NC 28223, USA
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Published Online: 2017-08-05 | DOI: https://doi.org/10.1515/jiip-2017-0067

## Abstract

By our definition, “restricted Dirichlet-to-Neumann (DN) map” means that the Dirichlet and Neumann boundary data for a coefficient inverse problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with restricted DN data are non-overdetermined in the n-dimensional case, with $n\ge 2$. We develop, in a unified way, a general and radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of second order, such as, e.g., elliptic, parabolic and hyperbolic ones. Namely, using Carleman weight functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well-acceptable one in the numerical analysis, that is, we truncate a certain Fourier-like series with respect to some functions depending only on the position of the point source. At least three applications are imaging of land mines, crosswell imaging and electrical impedance tomography.

MSC 2010: 35R30

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Accepted: 2017-07-06

Published Online: 2017-08-05

Published in Print: 2017-10-01

Funding Source: Army Research Laboratory

Award identifier / Grant number: W911NF-15-1-0233

Funding Source: Army Research Office

Award identifier / Grant number: W911NF-15-1-0233

Funding Source: Office of Naval Research

Award identifier / Grant number: N00014-15-1-2330

This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-15-1-0233 and by the Office of Naval Research grant N00014-15-1-2330.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 5, Pages 669–685, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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© 2017 Walter de Gruyter GmbH, Berlin/Boston.

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