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Shape reconstruction of a conductivity inclusion using the Faber polynomials

Doosung Choi, Junbeom Kim, Mikyoung Lim

TL;DR

This work addresses 2D shape reconstruction of conductivity inclusions from exterior measurements by leveraging Faber polynomial Polarization Tensors (FPTs). It builds a bridge between boundary integral GPTs, Faber polynomials, and the exterior conformal mapping to derive an exact inversion for extreme conductivities ($\lambda=\pm\tfrac12$) and to extract mapping coefficients for the inclusion boundary. An optimization framework is proposed that uses the extreme-conductivity inversion as a high-fidelity initial guess, with a GPT-based cost and a gradient-based shape update to recover general conductivity values. Numerical experiments validate the approach, showing that the extreme-conductivity-based reference shape yields accurate initializations and improves reconstruction when high-order GPTs are utilized. Overall, the paper provides a principled route from conformal geometry and boundary-integral descriptors to practical shape recovery for conductivity inclusions.

Abstract

We consider the shape reconstruction of a conductivity inclusion in two dimensions. We use the concept of Faber polynomials Polarization Tensors (FPTs) introduced in \cite{choi:2018:GME} to derive an exact shape recovery formula for an inclusion with the extreme conductivity. This shape can be a good initial guess in the shape recovery optimization for an inclusion with either small or large conductivity values. We illustrate and validate our results with numerical examples.

Shape reconstruction of a conductivity inclusion using the Faber polynomials

TL;DR

This work addresses 2D shape reconstruction of conductivity inclusions from exterior measurements by leveraging Faber polynomial Polarization Tensors (FPTs). It builds a bridge between boundary integral GPTs, Faber polynomials, and the exterior conformal mapping to derive an exact inversion for extreme conductivities () and to extract mapping coefficients for the inclusion boundary. An optimization framework is proposed that uses the extreme-conductivity inversion as a high-fidelity initial guess, with a GPT-based cost and a gradient-based shape update to recover general conductivity values. Numerical experiments validate the approach, showing that the extreme-conductivity-based reference shape yields accurate initializations and improves reconstruction when high-order GPTs are utilized. Overall, the paper provides a principled route from conformal geometry and boundary-integral descriptors to practical shape recovery for conductivity inclusions.

Abstract

We consider the shape reconstruction of a conductivity inclusion in two dimensions. We use the concept of Faber polynomials Polarization Tensors (FPTs) introduced in \cite{choi:2018:GME} to derive an exact shape recovery formula for an inclusion with the extreme conductivity. This shape can be a good initial guess in the shape recovery optimization for an inclusion with either small or large conductivity values. We illustrate and validate our results with numerical examples.

Paper Structure

This paper contains 9 sections, 4 theorems, 33 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Faber polynomial Polarization Tensors (FPTs)
  3. Generalized Polarization Tensors (GPTs)
  4. Faber polynomials and FPTs
  5. Exact inversion formula from the GPTs of extreme conductivities
  6. Optimization scheme to reconstruct shape from the GPTs
  7. Initial guess
  8. Recursive scheme
  9. Numerical results

Key Result

Lemma 2.1

Let $C$ be the Grunsky matrix with elements $\{c_{mk}\}$, and $\gamma^{\pm2\mathbb{N}}$ be diagonal matrices whose $(k,k)$-entry is $\gamma^{\pm2k}$. For each $m,k\in\mathbb{N}$, the FPTs satisfy Here, $\delta_{mk}$ is the Kronecker delta function.

Figures (6)

  • Figure 3.1: The initial guess for the cap-shaped domain. The black curve is the reconstructed curve and the gray curve is the actual shape. Starting from the left side to the right side, the conductivity is 10, 50, 100.
  • Figure 3.2: The initial guess with errors. The black curve is the reconstructed curve and the gray curve is the actual shape. Starting from the left side to the right side, the results are reconstructed shapes from the data with 0%, 10%, 20% error. Here, the conductivity $k= 50$.
  • Figure 4.1: The initial guess for sinusoidal perturbation of a disk. The black curve is the reconstructed curve and the gray curve is the actual shape. Starting from the left side to the right side, the conductivity is 3, 10, 50, and the first row shows the equivalent ellipse and the second row shows the reference shape.
  • Figure 4.2: The initial guess for the kite-shaped domain. The black curve is the reconstructed curve and the gray curve is the actual shape. Starting from the left side to the right side, the conductivity is 0.5, 0.1, 0.02, and the first row shows the equivalent ellipse and the second row shows the reference shape.
  • Figure 4.3: The reconstructed inclusion for the kite-shaped target with conductivity 10. The left figures are the initial guesses and the right figures are the reconstructed results. The first row is shape reconstruction starting from the equivalent ellipse and the second row is reconstruction approximating from FPTs.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 2.1.1
  • Definition 2.2.1
  • Lemma 2.1: choi:2018:GME
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1: ammari:2012:GPT