Convexification of Restricted Dirichlet-to-Neumann Map

TL;DR

This paper introduces a new numerical approach for solving coefficient inverse problems using restricted Dirichlet-to-Neumann data, applicable to various PDEs, with proven stability, uniqueness, and practical applications like land mine detection and electrical impedance tomography.

Contribution

It develops a unified, globally convergent numerical method for CIPs with restricted DN data across multiple PDE types, using Carleman weights and Fourier series truncation.

Findings

Proved Hölder stability and uniqueness for the method.

Constructed globally convergent numerical algorithms.

Demonstrated applications in land mine imaging, crosswell imaging, and electrical impedance tomography.

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 $n \geq 2$. 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. H\"{o}lder 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.