Convexification of Restricted Dirichlet-to-Neumann Map
This work provides a novel numerical framework for non-overdetermined inverse problems with restricted boundary data, addressing a key bottleneck in computational inverse problems.
The paper introduces a globally convergent numerical method for coefficient inverse problems using restricted Dirichlet-to-Neumann data, applicable to elliptic, parabolic, and hyperbolic PDEs. The method achieves Hölder stability and uniqueness by truncating a Fourier-like series, with applications in land mine detection, crosswell imaging, and electrical impedance tomography.
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 $n \geq 2$. 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.