Iterative Direct Sampling Method for Elliptic Inverse Problems with Limited Cauchy Data

TL;DR

This work addresses nonlinear elliptic coefficient inverse problems from limited Cauchy data by introducing an iterative direct sampling method (IDSM). The IDSM combines a regularized Dirichlet-to-Neumann map $\Lambda_{\alpha}(\mathcal{A})$, an operator-based forward map $\mathcal{G}[u]$ that targets the inclusion $u$ directly, and low-rank updates to a resolver to progressively refine the imaging function $\eta$ with modest extra cost. An abstract framework and multiple numerical demonstrations (including EIT, diffuse optical tomography, and cardiac electrophysiology) establish that IDSM is robust to substantial data noise and capable of distinguishing inclusions of different physical types, outperforming the standard DSM in accuracy and stability. The method offers a scalable, direct imaging tool for linear and nonlinear elliptic inverse problems and suggests potential extensions to time-dependent PDEs and moving inhomogeneities.

Abstract

In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an iterative mechanism, enhancing its performance with a modest increase in computational effort but a clear improvement in its stability against data noise. The method is formulated in an abstract framework of operator equations and is applicable to a broad range of elliptic inverse problems. Numerical results on electrical impedance tomography, optical tomography and cardiac electrophysiology etc. demonstrate its effectiveness and robustness, especially with an improved accuracy for identifying the locations and geometric shapes of inhomogeneities in the presence of large noise, when compared with the standard DSM.

Figures (5)

• Figure 1: Visualization of $u_{k}$ for Example 1. The conductivity is $0.3$ inside the inclusions and $1.0$ outside. The first and second rows show the estimates for $\varepsilon = 10\%$, by the BFG and DFP corrections, respectively. The third and fourth rows depict the estimates for $\varepsilon = 30\%$, by the BFG and DFP corrections, respectively. \newlabelfig10
• Figure 2: Visualization of $u_{k}$ for Example 2 with a noise level $\varepsilon = 10\%$. The conductivity is $0.3$ inside the conductivity inclusions and $1.0$ outside, and the potential is $6$ inside the potential inclusions and $1.0$ outside. The first two rows show the estimates for conductivity inclusion, by the BFG and DFP corrections, respectively. The last two rows depict the estimates for the potential inclusion, by the BFG and DFP corrections, respectively.
• Figure 3: Visualization of $u_{k}$ for Example 3 with a noise level $\varepsilon = 10\%$. The potential is $6$ inside the inclusions and $0.0$ outside. The first and second rows show the estimates with $\alpha=\text{1e-3}$, using the BFG and DFP corrections, respectively. The third and fourth rows depict the estimates with $\alpha=1$, using the BFG and DFP corrections, respectively. \newlabelfig30
• Figure 4: Visualization of $u_{k}$ for Example 4 with a noise level $\varepsilon = 10\%$. The conductivity is $\text{1e-4}$ inside the inclusions and $1.0$ outside. The top row displays the outcomes using the BFG correction, while the bottom row shows the results using the DFP correction.
• Figure 5: Visualization of $u_{k}$ for Example 5 with a noise level $\varepsilon = 10\%$. The coefficient of $u$ is set to $40$ inside the white circle, $20$ inside the black circle, and $0$ outside both regions. The top row displays the outcomes using the BFG correction, while the bottom row shows the results using the DFP correction.

Theorems & Definitions (7)

• Example 2.1
• Example 2.2
• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Example 3.1: The IDSM for cardiac electrophysiology