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A Stable Iterative Direct Sampling Method for Elliptic Inverse Problems with Partial Cauchy Data

Bangti Jin, Fengru Wang, Jun Zou

TL;DR

This work introduces a stable Iterative Direct Sampling Method (IDSM) for elliptic inverse problems with partial Cauchy data. By combining data completion, a heterogeneously regularized Dirichlet-to-Neumann map, and a stabilization-correction scheme, the method restores near-orthogonality of probing functions, suppresses noise-induced artifacts, and guarantees stability through a uniformly bounded resolver. The approach is validated across diverse models, including EIT, DOT, and cardiac electrophysiology, demonstrating robust localization of inhomogeneities under partial data and significant measurement noise. The results highlight practical efficiency (tens of PDE solves) and broad applicability to nonlinear and multi-physics settings, with clear guidance on algorithmic parameters and stopping indicators derived from the damping factor behavior.

Abstract

We develop a novel iterative direct sampling method (IDSM) for solving linear or nonlinear elliptic inverse problems with partial Cauchy data. It integrates three innovations: a data completion scheme to reconstruct missing boundary information, a heterogeneously regularized Dirichlet-to-Neumann map to enhance the near-orthogonality of probing functions, and a stabilization-correction strategy to ensure the numerical stability. The resulting method is remarkably robust with respect to measurement noise, is flexible with the measurement configuration, enjoys provable stability guarantee, and achieves enhanced resolution for recovering inhomogeneities. Numerical experiments in electrical impedance tomography, diffuse optical tomography, and cardiac electrophysiology show its effectiveness in accurately reconstructing the locations and geometries of inhomogeneities.

A Stable Iterative Direct Sampling Method for Elliptic Inverse Problems with Partial Cauchy Data

TL;DR

This work introduces a stable Iterative Direct Sampling Method (IDSM) for elliptic inverse problems with partial Cauchy data. By combining data completion, a heterogeneously regularized Dirichlet-to-Neumann map, and a stabilization-correction scheme, the method restores near-orthogonality of probing functions, suppresses noise-induced artifacts, and guarantees stability through a uniformly bounded resolver. The approach is validated across diverse models, including EIT, DOT, and cardiac electrophysiology, demonstrating robust localization of inhomogeneities under partial data and significant measurement noise. The results highlight practical efficiency (tens of PDE solves) and broad applicability to nonlinear and multi-physics settings, with clear guidance on algorithmic parameters and stopping indicators derived from the damping factor behavior.

Abstract

We develop a novel iterative direct sampling method (IDSM) for solving linear or nonlinear elliptic inverse problems with partial Cauchy data. It integrates three innovations: a data completion scheme to reconstruct missing boundary information, a heterogeneously regularized Dirichlet-to-Neumann map to enhance the near-orthogonality of probing functions, and a stabilization-correction strategy to ensure the numerical stability. The resulting method is remarkably robust with respect to measurement noise, is flexible with the measurement configuration, enjoys provable stability guarantee, and achieves enhanced resolution for recovering inhomogeneities. Numerical experiments in electrical impedance tomography, diffuse optical tomography, and cardiac electrophysiology show its effectiveness in accurately reconstructing the locations and geometries of inhomogeneities.

Paper Structure

This paper contains 16 sections, 4 theorems, 78 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The IDSM for full boundary data
  4. The IDSM for partial boundary measurement
  5. Data completion
  6. HR-DtN map
  7. Stabilization-correction scheme
  8. Algorithmic implementation
  9. Numerical experiments
  10. Example 1: EIT
  11. Example 2: DOT model
  12. Example 3: DOT model
  13. Example 4: CE model
  14. Example 5: Modulus nonlinearity model
  15. Damping factor
  16. ...and 1 more sections

Key Result

Lemma 1

\newlabellemma10 Let the elliptic operator $\mathcal{A} \in \mathcal{L}(H^{1}(\Omega), H^{1}(\Omega)')$ satisfy the following conditions: where $m, M > 0$ are constants. Then, there exists a unique solution $p \in L^{2}(\Gamma)$ of eqn12 such that

Figures (7)

  • Figure 1: The reconstruction of conductivity inclusion $u_c$ for Example \ref{['exam1']}. Columns 2–4: Estimates for 15% noise. Columns 5–7: Estimates for 30% noise. \newlabelfig1.10
  • Figure 2: The reconstructions for Example \ref{['exam1']}. Top row: Partial data (columns 2–4) versus full data (columns 5–7). Bottom row: Homogeneous DtN map (columns 2–4) and unstabilized scheme (columns 5–7). \newlabelfig1.20
  • Figure 3: The reconstructed conductivity ($u_c$, columns 2–4) and potential ($u_{p}$, columns 5–7) inclusions for Example \ref{['exam2']}. \newlabelfig20
  • Figure 4: The reconstructions for Example \ref{['exam3']} with integral indices $p=1$ (top) and $p=99$ (bottom). Columns 2–4: Conductivity inclusion $u_c$. Columns 5–7: Potential inclusion $u_{p}$. \newlabelfig30
  • Figure 5: The reconstruction of characteristic function $u$ for Example \ref{['exam4']}. Columns 2–4: Estimates for $\alpha_{d}=10^{-3}$ and $\alpha_{n}=2.0$. Columns 5–7: Estimates for $\alpha_{d}=0.05$ and $\alpha_{n}=10.0$. \newlabelfig40
  • ...and 2 more figures

Theorems & Definitions (8)

  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Theorem 3
  • Proof 3
  • Lemma 1
  • Proof 4