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The linearization approach to the Calderón problem revisited: reconstruction via the Born approximation

Carlos Castro, Fabricio Macià, Cristóbal Meroño, Daniel Sánchez-Mendoza

TL;DR

This work revisits the linearization of the Calderón problem by formalizing the Born approximation and proving its existence beyond prior radial-only results. It establishes a rigorous two-step factorization into a linear, unstable stage and a nonlinear, stable map, with a detailed connection to Schrödinger data via $q(\gamma)=\frac{\Delta\sqrt{\gamma}}{\sqrt{\gamma}}$. In the unit disk, the authors link the Born approximation to a complex moment problem, derive explicit matrix-element formulas for the Fréchet derivative, and develop a robust numerical framework to reconstruct the Born approximation from boundary data. Numerical experiments on the disk demonstrate the method’s ability to locate perturbations, expose the role of priors (via $\sigma_{\kappa}$), and assess stability under noise. Overall, the paper provides rigorous existence and uniqueness results for the Born approximation in radial geometries, and presents a practical algorithmic path to approximate conductivities from boundary measurements using a moment-problem formulation.

Abstract

Linearization techniques are widely used in the analysis and numerical solution of the Calderón inverse problem, even if their theoretical basis is not fully understood. In this article, we study the effectiveness of linearization for reconstructing a conductivity from its Dirichlet-to-Neumann (DtN) map, combining rigorous analysis with numerical experiments. In particular, we prove that any DtN map arising from a radial conductivity in the unit ball of $\mathbb{R}^d$ admits an exact representation as a linearized DtN map for a uniquely determined integrable function, the Born approximation. We linearize on a family of background conductivities that includes the constant case, giving a rigorous foundation for linearization-based methods in this framework. We also characterize the Born approximation as a solution of a generalized moment problem. Since this moment problem is formally well-defined even for non-radial conductivities, we use it to develop a numerical algorithm to reconstruct the Born approximation of a general conductivity on the unit disk. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations. Finally, we show how it can be used as the starting point of an algorithm for reconstructing a conductivity from its DtN map.

The linearization approach to the Calderón problem revisited: reconstruction via the Born approximation

TL;DR

This work revisits the linearization of the Calderón problem by formalizing the Born approximation and proving its existence beyond prior radial-only results. It establishes a rigorous two-step factorization into a linear, unstable stage and a nonlinear, stable map, with a detailed connection to Schrödinger data via . In the unit disk, the authors link the Born approximation to a complex moment problem, derive explicit matrix-element formulas for the Fréchet derivative, and develop a robust numerical framework to reconstruct the Born approximation from boundary data. Numerical experiments on the disk demonstrate the method’s ability to locate perturbations, expose the role of priors (via ), and assess stability under noise. Overall, the paper provides rigorous existence and uniqueness results for the Born approximation in radial geometries, and presents a practical algorithmic path to approximate conductivities from boundary measurements using a moment-problem formulation.

Abstract

Linearization techniques are widely used in the analysis and numerical solution of the Calderón inverse problem, even if their theoretical basis is not fully understood. In this article, we study the effectiveness of linearization for reconstructing a conductivity from its Dirichlet-to-Neumann (DtN) map, combining rigorous analysis with numerical experiments. In particular, we prove that any DtN map arising from a radial conductivity in the unit ball of admits an exact representation as a linearized DtN map for a uniquely determined integrable function, the Born approximation. We linearize on a family of background conductivities that includes the constant case, giving a rigorous foundation for linearization-based methods in this framework. We also characterize the Born approximation as a solution of a generalized moment problem. Since this moment problem is formally well-defined even for non-radial conductivities, we use it to develop a numerical algorithm to reconstruct the Born approximation of a general conductivity on the unit disk. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations. Finally, we show how it can be used as the starting point of an algorithm for reconstructing a conductivity from its DtN map.
Paper Structure (17 sections, 16 theorems, 127 equations, 9 figures, 1 table)

This paper contains 17 sections, 16 theorems, 127 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting and results
  3. Relation to other works in the literature
  4. Structure of the article
  5. The Born approximation
  6. Linearized Calderón problem
  7. The Born approximation in the Calderón problem
  8. Uniqueness of the Born approximation
  9. Existence of the Born approximation in the radial case
  10. The Born approximation of a conductivity on the unit Disk
  11. The Born approximation and the complex moment problem when k=0
  12. Connection with moment problems in the general case
  13. Uniqueness of the complex moment problems
  14. Numerical results
  15. Direct problem method
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that $\Omega:=\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$, that $\gamma\in W^{2,\infty}(\mathbb{B}^d)$ is a radial conductivity, and that $\kappa\in(-\infty,\lambda_{\nu_d,1}^2)$. Then there exists a unique distribution $\gamma^{\mathrm{B}}_{\sigma_{\kappa,d}}\in W^{2,\infty}(\mathbb{B} In addition, $\gamma^{\mathrm{B}}_{\sigma_{\kappa,d}}$ is an integrable function that satisfies:

Figures (9)

  • Figure 1: Localization of positive circular bump. Iterative scheme run $N=2$ times. $(N_r,N_\theta)=(50,50)$, $(I,L)=(50,24)$.
  • Figure 2: Localization of negative circular bump. Iterative scheme run $N=1$ times. $(N_r,N_\theta)=(50,50)$, $(I,L)=(50,24)$.
  • Figure 3: Resolution near the boundary. Iterative scheme run $N=1$ times. $(N_r,N_\theta)=(50,100)$, $(I,L)=(50,49)$.
  • Figure 4: Resolution near the origin. Iterative scheme run $N=1$ times. $(N_r,N_\theta)=(50,100)$, $(I,L)=(50,49)$.
  • Figure 5: Linearization at $\kappa=4$. $(N_r,N_\theta)=(50,50)$, $(I,L)=(50,24)$
  • ...and 4 more figures

Theorems & Definitions (35)

  • Theorem 1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 25 more