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The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

Juan A. Barceló, Carlos Castro, Fabricio Macià, Cristóbal J. Meroño

TL;DR

This work analyzes the Born approximation for the three-dimensional Calderón problem in the radial setting, where the DtN map diagonalizes in spherical harmonics and the linearized problem reduces to a Hausdorff moment problem. It provides explicit formulas linking the conductivity Born data $\gamma_{\mathrm{exp}}$ to Schrödinger Born data $q_{\mathrm{exp}}$, including a scattering limit as the radius grows and a linearization around $\gamma=1$ via $\widehat{(\gamma_{\mathrm{exp}}-1)}(\xi)=-2\widehat{q_{\mathrm{exp}}}(\xi)/|\xi|^2$. The paper develops a numerical pipeline to compute the Born approximation from the DtN spectrum, demonstrates well-definedness even for discontinuous conductivities, and shows depth-dependent accuracy and a clear convergence trend under a smallness assumption through an iterative refinement scheme. It also draws parallels with the inverse Schrödinger problem, illustrating recovery of singularities and a scattering limit, and discusses stability challenges tied to the underlying Hausdorff moment problem. Overall, the work provides both theoretical insight and practical algorithms for radial EIT-like reconstructions and offers a structured approach to assess the impact of domain size on Born-data quality.

Abstract

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $B_R \subset \mathbb{R}^3 $ of radius $R$, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $L^{\infty}$ conductivities, and we present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $\partial B_R$. We then investigate how increasing the radius $R$ affects the quality of the Born approximation, and the existence of a scattering limit as $R\to \infty$. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $-Δ+q$, and strong recovery of singularity results are observed in this case.

The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

TL;DR

This work analyzes the Born approximation for the three-dimensional Calderón problem in the radial setting, where the DtN map diagonalizes in spherical harmonics and the linearized problem reduces to a Hausdorff moment problem. It provides explicit formulas linking the conductivity Born data to Schrödinger Born data , including a scattering limit as the radius grows and a linearization around via . The paper develops a numerical pipeline to compute the Born approximation from the DtN spectrum, demonstrates well-definedness even for discontinuous conductivities, and shows depth-dependent accuracy and a clear convergence trend under a smallness assumption through an iterative refinement scheme. It also draws parallels with the inverse Schrödinger problem, illustrating recovery of singularities and a scattering limit, and discusses stability challenges tied to the underlying Hausdorff moment problem. Overall, the work provides both theoretical insight and practical algorithms for radial EIT-like reconstructions and offers a structured approach to assess the impact of domain size on Born-data quality.

Abstract

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball of radius , in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for conductivities, and we present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary . We then investigate how increasing the radius affects the quality of the Born approximation, and the existence of a scattering limit as . Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator , and strong recovery of singularity results are observed in this case.
Paper Structure (13 sections, 5 theorems, 52 equations, 15 figures)

This paper contains 13 sections, 5 theorems, 52 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Two closely related inverse problems
  3. The Born approximation in the inverse Schrödinger boundary value problem
  4. The scattering limit
  5. The formula for exp
  6. Connection with the Hausdorff moment problem
  7. Numerical computation of the Born approximation
  8. Computation of the DtN map
  9. Implementation details in the Born approximation
  10. Numerical experiments: The conductivity case
  11. Numerical experiments: The potential case
  12. Iterative algorithm
  13. Support of the Born approximation

Key Result

Lemma 2.1

Let $d\ge 2$, $R\geq 1$, and $q = q_0(|\cdot|)$, where $q_0 \in L^\infty(\mathbb{R}_+,\mathbb{R})$ is supported in $(0,1]$. Assume that $q \in \mathcal{Q}_d^R$ for all $R\ge 1$. Then the eigenvalues $(\lambda_k[q])_{k\in\mathbb{N}_0}$ of $\Lambda[q]$ in the unit ball determine the eigenvalues $(\lam

Figures (15)

  • Figure 1: Experiment 1: A step conductivity and its Born approximation (left), and a comparison of their Fourier transforms (right). Lower simulations correspond to a conductivity closer to the reference conductivity $\gamma=1$.
  • Figure 2: Experiment 2: Three different conductivities that coincide in the interval $(1/3,1)$ (left) and their Born approximations (right). We observe that they also coincide in this same interval $(1/3,1)$.
  • Figure 3: Experiment 3: Scattering limit. A conductivity $\gamma$, its Born approximation $\gamma_{\mathrm{exp}}$ given by \ref{['id:gae_formula']} with $R =1$, and the scattering limit $\gamma_{\mathrm{exp}}(\centerdot,\infty)$ given by formula \ref{['id:gae_scattering']} (left).
  • Figure 4: Experiment 4: Born approximations (left) and their Fourier transform (right) for smooth conductivities with different sizes.
  • Figure 5: Experiment 5: Born approximation of increasingly larger step potentials (left) and its Fourier transforms (right).
  • ...and 10 more figures

Theorems & Definitions (8)

  • Lemma 2.1
  • proof
  • Proposition A.1
  • Lemma A.2
  • proof
  • Proposition A.3
  • Proposition A.4
  • proof