Convexification of restricted Dirichlet-to-Neumann map

• Published in: Journal of inverse and ill-posed problems Vol. 25; no. 5; pp. 669 - 685
• Main Author: Klibanov, Michael V.
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.10.2017; Walter de Gruyter GmbH
• Subjects: 35R30; convexification; Dirichlet problem; Electrical impedance; Fourier series; global strict convexity, Carleman weight functions; Inverse problems; Land mines; Mathematical analysis; Medical imaging; Numerical analysis; Numerical methods; Restricted Dirichlet-to-Neumann data; Uniqueness
• ISSN: 0928-0219; 1569-3945
• DOI: 10.1515/jiip-2017-0067

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 -dimensional case, with . 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.