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Numerical solution of the two-dimensional Calderon problem for domains close to a disk

Vladimir A. Sharafutdinov, Konstantin V. Storozhuk

TL;DR

This work addresses the 2D Calderón problem by focusing on simply connected domains near the unit disk and reformulating the inverse boundary-value problem in terms of a planar, flat-metric immersion. The authors derive a nonlinear condition $\mathcal{H}_\gamma\gamma=\gamma$ for the boundary parametrization $\gamma$ using the DtN-map through the Hilbert transform $\mathcal{H}_\gamma=D^{-1}\Lambda_\gamma$, alongside the constraint $|\frac{d\gamma}{ds}|=1$, and solve a finite-dimensional Fourier system to recover the boundary curve. A uniqueness result for the boundary traces of holomorphic functions ensures the reconstruction corresponds to the intended domain up to rigid motions. Numerical experiments on ellipses, dents, and asymmetric boundaries illustrate that small boundary deviations dramatically affect the DtN-map and demonstrate the method’s effectiveness for domains close to the disk, while also underscoring sensitivity to initialization and forward-problem accuracy. Overall, the paper provides a practical, Fourier-based pathway to numerically solving the Calderón problem in the near-disk regime and clarifies the intrinsic sensitivity of boundary measurements to geometry.

Abstract

For a compact Riemannian surface $(M,g)$ with non-empty boundary $Γ$, the Dirichlet-to-Neumann operator (DtN-map) $Λ_g:C^\infty(Γ)\to C^\infty(Γ)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_Γ$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_Γ=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $Λ_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}=\{(x,y)\mid x^2+y^2\le1\}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain.

Numerical solution of the two-dimensional Calderon problem for domains close to a disk

TL;DR

This work addresses the 2D Calderón problem by focusing on simply connected domains near the unit disk and reformulating the inverse boundary-value problem in terms of a planar, flat-metric immersion. The authors derive a nonlinear condition for the boundary parametrization using the DtN-map through the Hilbert transform , alongside the constraint , and solve a finite-dimensional Fourier system to recover the boundary curve. A uniqueness result for the boundary traces of holomorphic functions ensures the reconstruction corresponds to the intended domain up to rigid motions. Numerical experiments on ellipses, dents, and asymmetric boundaries illustrate that small boundary deviations dramatically affect the DtN-map and demonstrate the method’s effectiveness for domains close to the disk, while also underscoring sensitivity to initialization and forward-problem accuracy. Overall, the paper provides a practical, Fourier-based pathway to numerically solving the Calderón problem in the near-disk regime and clarifies the intrinsic sensitivity of boundary measurements to geometry.

Abstract

For a compact Riemannian surface with non-empty boundary , the Dirichlet-to-Neumann operator (DtN-map) is defined by , where is the unit outer normal vector to the boundary and is the solution to the Dirichlet problem . The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that is determined by uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces close to , where is the unit disk and is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain.
Paper Structure (6 sections, 5 theorems, 66 equations, 5 figures)

This paper contains 6 sections, 5 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Flat metrics
  3. The Calderón problem for a simply connected planar domain
  4. The uniqueness theorem
  5. Numerical solution of the Calderón problem
  6. Numerical examples

Key Result

Theorem 1.1

Let $(M_j,g_j)\ (j=1,2)$ be two compact Riemannian surfaces with boundaries $\Gamma_j$ and let $\varphi:(\Gamma_1,ds_{g_1})\rightarrow(\Gamma_2,ds_{g_2})$ be an isometry preserving the DtN-map, i.e., such that the following diagram is commutative: Then $\varphi$ extends to a diffeomorphism $\psi:M_1\rightarrow M_2$ such that $\psi|_{\Gamma_1}=\varphi$ and $\psi^*g_2=\rho g_1$ for some function $0

Figures (5)

  • Figure 1: self-intersecting solutions of \ref{['5.1']}--\ref{['5.2']}
  • Figure 2: Example 2: the original curve, two reconstructed curves for $N=5$ and for $N=20$, the velocity vector for $N=20$
  • Figure 3: Example 3: disk with a dent, the recovered and reconstructed curves, two "wrong solutions"
  • Figure 4: Example 3: two initial approximations, curves obtained by three iterations
  • Figure 5: Example 4: the original curve, the reconstructed curve for $N=3$, and for $N=10$

Theorems & Definitions (7)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof