Niall Donlon | Searcharxiv

arXiv2025

A uniqueness result in the inverse problem for the anisotropic Schrödinger type equation from local measurements

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $σ$ and $q$ of the equation $-\mbox{div}(σ\nabla u)+qu = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion $Σ$ of the boundary $\partial Ω$ of a domain $Ω\subset \mathbb{R}^n$, with $n\geq 3$. We assume that $σ$ and $q$ are \textit{a-priori} known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of $Ω$ with curved interfaces. We prove that $σ$ and $q$ can be uniquely determined in $Ω$ from the knowledge of the local map.

math.AP↗

arXiv2022

Stability and reconstruction of a special type of anisotropic conductivity in magneto-acoustic tomography with magnetic induction

We consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain $Ω\subset\mathbb{R}^3$ by means of the hybrid inverse problem of magneto-acoustic tomography with magnetic induction (MAT-MI). The class of anisotropic conductivities considered here is of type $σ(\cdot)=A(\cdot,γ(\cdot))$ in $Ω$, where $[λ^{-1}, λ]\ni t\mapsto A(\cdot, t)$ is a one-parameter family of matrix-valued functions which are \textit{a-priori} known to be $C^{1,β}$, allowing us to stably reconstruct $γ$ in $Ω$ in terms of an internal functional $F(σ)$. Our results also extend previous results in MAT-MI where $σ(\cdot) = γ(\cdot) D(\cdot)$, with $D$ an \textit{a-priori} known matrix-valued function on $Ω$ to a more general anisotropic structure which depends non-linearly on the scalar function $γ$ to be reconstructed.

math.AP↗