Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Niall Donlon, Romina Gaburro

TL;DR

This work addresses the inverse problem of simultaneously recovering an anisotropic conductivity $\sigma$ and a nonnegative potential $q$ in $\Omega$ from a local Neumann-to-Dirichlet map on a curved boundary portion $\Sigma$, under the assumption that $\sigma$ and $q$ are piecewise constant on a known partition with curved interfaces. The authors extend the prior $q=0$ results to the case $q\ge 0$ by leveraging the local data, the metric relation $g=(\det\sigma)^{1/(n-2)}\sigma^{-1}$, and the asymptotics of the Neumann kernel near boundary poles, together with singular solutions and a unique continuation framework to propagate the information across layers. They prove that $\sigma$ and $q$ are uniquely determined in $\Omega$ from the local ND map, and in the layered setting show that the interfaces can also be recovered (Nested Domain Theorem). The results have implications for diffusion-type imaging problems such as diffuse optical tomography, where local boundary measurements and piecewise-constant anisotropic coefficients are natural models. Overall, the paper advances uniqueness theory for anisotropic inverse problems with local data and layered media.

Abstract

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.

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

TL;DR

This work addresses the inverse problem of simultaneously recovering an anisotropic conductivity and a nonnegative potential in from a local Neumann-to-Dirichlet map on a curved boundary portion , under the assumption that and are piecewise constant on a known partition with curved interfaces. The authors extend the prior results to the case by leveraging the local data, the metric relation , and the asymptotics of the Neumann kernel near boundary poles, together with singular solutions and a unique continuation framework to propagate the information across layers. They prove that and are uniquely determined in from the local ND map, and in the layered setting show that the interfaces can also be recovered (Nested Domain Theorem). The results have implications for diffusion-type imaging problems such as diffuse optical tomography, where local boundary measurements and piecewise-constant anisotropic coefficients are natural models. Overall, the paper advances uniqueness theory for anisotropic inverse problems with local data and layered media.

Abstract

We consider the inverse boundary value problem of the simultaneous determination of the coefficients and of the equation from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion of the boundary of a domain , with . We assume that and are \textit{a-priori} known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of with curved interfaces. We prove that and can be uniquely determined in from the knowledge of the local map.

Paper Structure

This paper contains 10 sections, 7 theorems, 151 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. The Neumann-to-Dirichlet map
  4. Unique determination of $\sigma$ and $q$ in $\Omega$.
  5. Unique determination of $\sigma$, $q$ and the layers' interfaces
  6. Proof of Main Result (Theorem \ref{['Main theorem of uniqueness']})
  7. Unique determination of $\sigma$ in $\mathcal{U}$
  8. Unique determination of $\sigma$ and $q$ on $D_1$
  9. Unique determination of $\sigma$ and $q$ in $\Omega$
  10. Appendix

Key Result

Theorem 2.1

Let $\sigma^{(i)}$ and $q^{(i)}$, for $i=1,2$, be two conductivities and potentials satisfying respectively, where $\{D_j\}_{j=1}^N$ satisfies assumptions $(i)-(iii)$, $\sigma^{(i)}_j \in Sym_n$ is a constant matrix satisfying ellipticity condition 0, $q^{(i)}_j\geq 0$ is constant, for $j=1,\dots , N$ and there is $J_i\in\{1,\dots N\}$ such that $q^{(i)}_{J_i}>0$, for $i=1,2$. If then

Figures (4)

  • Figure 1: Schematic figure representing a chain of domains belonging to the partition of $\Omega$, $\{D_j\}_{j=1}^K$, connecting $D_1$ with $D_{K}$.mmmmmmmmmmmmmmmmmmmmmmmmmmmm
  • Figure 2: The family of nested domains $\{\Omega\}_{j=0}^{N-1}$ and the layers $\{ D\}_{j=1}^{N}$.
  • Figure 3: Schematic figure displaying the non-physical domain $D_0$, subdomain $(D_0)_r$ and the augmented domain $\Omega_0$.
  • Figure 4: Schematic figure representing $\Omega_1^{(1)}$, $\Omega_1^{(2)}$, the connected component $E_1$ of $\Omega \backslash \overline{(\Omega_1^{(1)} \cup \Omega_1^{(2)})}$, such that $\Sigma \subset \partial E_1$, the subdomain $\mathcal{E}_1 \subset E_1$, the open portion $\Sigma_2\subset(\partial \Omega^{(1)}_1 \backslash \overline{\Omega^{(2)}_1}) \cap \partial E_1$ and the neighborhood $\mathcal{U}_2$ of $y_2\in\Sigma_2$.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['Neumann kernel theorem']}
  • Lemma 3.2
  • ...and 11 more