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Direct reconstruction of anisotropic self-adjoint inclusions in the Calderón problem

Henrik Garde, David Johansson, Thanasis Zacharopoulos

TL;DR

The paper advances direct reconstruction in the Calderón problem for anisotropic conductivities by extending the monotonicity method to general self-adjoint cases and to extreme (insulating or conducting) inclusions. It develops outer and inner reconstruction strategies, introduces linearized variants for computational practicality, and proves exact recovery of the outer shape of inclusions from partial boundary data via monotone test-operators. A key technical contribution is the extension of localized potentials to anisotropic and even non-self-adjoint coefficients, together with a careful limit analysis that connects finite perturbations to extreme models. These results illuminate non-uniqueness phenomena in the anisotropic Calderón problem and broaden the applicability of monotonicity-based inclusion detection beyond isotropic conductivities, with potential impact on noninvasive imaging where interior properties are sought from boundary measurements.

Abstract

We extend the monotonicity method for direct exact reconstruction of inclusions in the partial data Calderón problem, to the case of general anisotropic conductivities in any spatial dimension $d\geq 2$. From a local Neumann-to-Dirichlet map, we give reconstruction methods of inclusions based on unknown anisotropic self-adjoint perturbations to a known anisotropic conductivity coefficient. This additionally provides new insights into the non-uniqueness issues of the anisotropic Calderón problem. The main assumption is a definiteness condition for the perturbations near the outer inclusion boundaries. Beyond this condition, they are $L^\infty$-perturbations that may be indefinite away from the outer inclusion boundaries, and with no boundary regularity requirement for the inclusions. Alternatively, we allow extreme parts that are perfectly insulating or perfectly conducting, in which case we require Lipschitz regularity of the outer inclusion boundaries.

Direct reconstruction of anisotropic self-adjoint inclusions in the Calderón problem

TL;DR

The paper advances direct reconstruction in the Calderón problem for anisotropic conductivities by extending the monotonicity method to general self-adjoint cases and to extreme (insulating or conducting) inclusions. It develops outer and inner reconstruction strategies, introduces linearized variants for computational practicality, and proves exact recovery of the outer shape of inclusions from partial boundary data via monotone test-operators. A key technical contribution is the extension of localized potentials to anisotropic and even non-self-adjoint coefficients, together with a careful limit analysis that connects finite perturbations to extreme models. These results illuminate non-uniqueness phenomena in the anisotropic Calderón problem and broaden the applicability of monotonicity-based inclusion detection beyond isotropic conductivities, with potential impact on noninvasive imaging where interior properties are sought from boundary measurements.

Abstract

We extend the monotonicity method for direct exact reconstruction of inclusions in the partial data Calderón problem, to the case of general anisotropic conductivities in any spatial dimension . From a local Neumann-to-Dirichlet map, we give reconstruction methods of inclusions based on unknown anisotropic self-adjoint perturbations to a known anisotropic conductivity coefficient. This additionally provides new insights into the non-uniqueness issues of the anisotropic Calderón problem. The main assumption is a definiteness condition for the perturbations near the outer inclusion boundaries. Beyond this condition, they are -perturbations that may be indefinite away from the outer inclusion boundaries, and with no boundary regularity requirement for the inclusions. Alternatively, we allow extreme parts that are perfectly insulating or perfectly conducting, in which case we require Lipschitz regularity of the outer inclusion boundaries.

Paper Structure

This paper contains 19 sections, 23 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Outlining the main results
  3. Previous results for the monotonicity method
  4. Remarks on notation
  5. The forward problem
  6. Outer reconstruction of general anisotropic inclusions
  7. Inner reconstruction of uniformly definite anisotropic inclusions
  8. Operator inequalities for Neumann-to-Dirichlet maps
  9. Localized potentials for anisotropic coefficients
  10. Proof of Theorem \ref{['thm:main']}
  11. Proof of Theorem \ref{['thm:mainlinear']}
  12. Proof of Theorem \ref{['thm:mainlinearinnerpos']}
  13. Proof of Theorem \ref{['thm:mainlinearinnerneg']}
  14. The forward problem with extreme coefficients
  15. Reconstruction of general anisotropic and extreme inclusions
  16. ...and 4 more sections

Key Result

Theorem 3.6

2 Under Assumption assump:recon(i), for any measurable $C\subseteq \overline{\Omega}$ we have Under all of Assumption assump:recon, for any $C\in\mathcal{A}$ we have

Theorems & Definitions (52)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.5: Remarks on Assumption \ref{['assump:recon']}
  • Theorem 3.6
  • proof
  • Theorem 3.7
  • proof
  • Remark 3.8
  • Theorem 4.2
  • ...and 42 more