Convexification of Restricted Dirichlet-to-Neumann Map

Abstract

By our definition, "restricted Dirichlet-to-Neumann map" (DN) 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 the restricted DN data are non-overdetermined in the $n-$D case with $n\ge 2$. We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the 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: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography.

References in corpus (2)

• A Convergent Adaptive Finite Element Method for Electrical Impedance Tomography
• Numerical Solution of a Coefficient Inverse Problem with Multi-Frequency Experimental Raw Data by a Globally Convergent Algorithm

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• A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements
• Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
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