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A Computationally Efficient Finite Element Method for Shape Reconstruction of Inverse Conductivity Problems

Lefu Cai, Zhixin Liu, Minghui Song, Xianchao Wang

TL;DR

This work addresses the inverse conductivity problem in electrical impedance tomography by reformulating the finite-measurement reconstruction as a non-iterative finite-element method. A one-step linearization leads to a regularized Frobenius-norm residual, which is recast as a variational problem and solved via a standard Lax-Milgram FE framework, enabling direct shape reconstruction of the conductivity. The authors provide rigorous a priori error estimates for both reconstruction and discretization, derive an explicit regularization-parameter rule that accounts for data size and noise, and validate the approach through numerical experiments on diverse shapes. The resulting methodology reduces computational cost by avoiding iterative forward solves and offers quantitative guidance for parameter selection, making it practical for shape reconstruction from limited boundary data.

Abstract

The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such approaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. {\it A prior} error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

A Computationally Efficient Finite Element Method for Shape Reconstruction of Inverse Conductivity Problems

TL;DR

This work addresses the inverse conductivity problem in electrical impedance tomography by reformulating the finite-measurement reconstruction as a non-iterative finite-element method. A one-step linearization leads to a regularized Frobenius-norm residual, which is recast as a variational problem and solved via a standard Lax-Milgram FE framework, enabling direct shape reconstruction of the conductivity. The authors provide rigorous a priori error estimates for both reconstruction and discretization, derive an explicit regularization-parameter rule that accounts for data size and noise, and validate the approach through numerical experiments on diverse shapes. The resulting methodology reduces computational cost by avoiding iterative forward solves and offers quantitative guidance for parameter selection, making it practical for shape reconstruction from limited boundary data.

Abstract

The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such approaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. {\it A prior} error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

Paper Structure

This paper contains 8 sections, 8 theorems, 114 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Minimization problem and its variational formulation
  3. Error estimates
  4. Error of the reconstruction
  5. Error of the discretization
  6. Numerical experiments
  7. Numerical method
  8. Numerical examples

Key Result

Theorem 2.1

The minimization problem min is equivalent to the following variational formulation: find $\kappa_r\in L^2(\Omega)$, such that where the bilinear form is defined by and the linear form is given by Moreover, the variational problem variation2 admits a unique solution.

Figures (4)

  • Figure 1: Reconstruction of a smooth circular object under different regularization parameter $\alpha$, with noise level $\delta=5\%$.
  • Figure 2: Reconstruction of an arch-shaped object under different regularization parameter $\alpha$, with noise level $\delta=1\%$.
  • Figure 3: Reconstruction results obtained using the iterative method and the finite element method.
  • Figure 4: Reconstruction of a loop-shaped object and two circle-shape object.

Theorems & Definitions (22)

  • Theorem 2.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm2']}.
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Theorem 3.1
  • ...and 12 more