Calder\'on cavities inverse problem as a shape-from-moments problem

TL;DR

This paper presents a non-iterative method to recover the shape and location of cavities in a 2D domain using the Dirichlet-to-Neumann map, by transforming the inverse problem into a shape-from-moments problem.

Contribution

It introduces a novel approach combining generalized Pólia-Szegö tensors with shape-from-moments techniques for cavity reconstruction in inverse conductivity problems.

Findings

Method effectively reconstructs cavity shapes from DtN data.

Numerical results demonstrate efficiency for arbitrary geometries.

The approach simplifies the inverse problem into a shape-from-moments framework.

Abstract

In this paper, we address a particular case of Calder\'on's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non iterative and uses two main ingredients. First, we show how to compute the so-called generalized P\'olia-Szeg\"o tensors (GPST) of the cavities from the DtN of the cavities. Secondly, we show that the obtained shape from GPST inverse problem can be transformed into a shape from moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries.

Equations

$$-\nabla \cdot (\sigma \nabla u^f)$$

$$u^f$$

$$-\Delta u^f$$

$$u^f$$

$${\int }_{{\gamma }_k}{\partial }_n{u}^f\mathrm{d}\sigma =0,k=1,\dots ,N.$$

$$G(x)=-\frac{1}{2\pi }\log |x|$$

$${\mathcal{H}}^{\frac{1}{2}}(\gamma ):=H^{\frac{1}{2}}({\gamma }_1)\times H^{\frac{1}{2}}({\gamma }_2)\times \cdots \times H^{\frac{1}{2}}({\gamma }_N),$$

$${\mathcal{H}}^{-\frac{1}{2}}(\gamma ):=H^{-\frac{1}{2}}({\gamma }_1)\times H^{-\frac{1}{2}}({\gamma }_2)\times \cdots \times H^{-\frac{1}{2}}({\gamma }_N),$$

$$\Vert q{\Vert }_{\frac{1}{2},\gamma }={\left(\Vert q_1{\Vert }_{\frac{1}{2},{\gamma }_1}^2+\cdots +\Vert q_N{\Vert }_{\frac{1}{2},{\gamma }_N}^2\right)}^{\frac{1}{2}},\forall q=(q_1,\dots ,q_N)\in {\mathcal{H}}^{\frac{1}{2}}(\gamma ),$$

$$\Vert \hat{q}{\Vert }_{-\frac{1}{2},\gamma }={\left(\Vert {\hat{q}}_1{\Vert }_{-\frac{1}{2},{\gamma }_1}^2+\cdots +\Vert {\hat{q}}_N{\Vert }_{-\frac{1}{2},{\gamma }_N}^2\right)}^{\frac{1}{2}},\forall \hat{q}=({\hat{q}}_1,\dots ,{\hat{q}}_N)\in {\mathcal{H}}^{-\frac{1}{2}}(\gamma ).$$

$${\mathcal{S}}_{\gamma }\hat{q}(x)={\int }_{\gamma }G(x-y)\hat{q}(y)\mathrm{d}{\sigma }_y=\sum _{k=1}^N{\mathcal{S}}_{{\gamma }_k}{\hat{q}}_k(x),\forall x\in {\mathbb{R}}^2\setminus \gamma .$$

$${\mathcal{S}}_{\gamma }\hat{q}(x)=-\frac{1}{2\pi }⟨\hat{q},1{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }\log |x|+O(|x{|}^{-1}),$$

$$⟨\hat{q},p{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }:=\sum _{k=1}^N⟨{\hat{q}}_k,p_k{⟩}_{-\frac{1}{2},\frac{1}{2},{\gamma }_k},$$

$$\hat{\mathcal{H}}(\gamma ):=\hat{H}({\gamma }_1)\times \hat{H}({\gamma }_2)\times \dots \times \hat{H}({\gamma }_N),$$

$$\hat{H}({\gamma }_k):=\{{\hat{q}}_k\in H^{-\frac{1}{2}}({\gamma }_k):⟨{\hat{q}}_k,1{⟩}_{-\frac{1}{2},\frac{1}{2},{\gamma }_k}=0\}.$$

$$[{\mathcal{S}}_{\gamma }\hat{q}{]}_{{\gamma }_k}=0,{\left[{\partial }_n({\mathcal{S}}_{\gamma }\hat{q})\right]}_{{\gamma }_k}=-{\hat{q}}_k,\forall \hat{q}=({\hat{q}}_1,\dots ,{\hat{q}}_N)\in {\mathcal{H}}^{-\frac{1}{2}}(\gamma ).$$

$$q=(q_1,\dots ,q_N),q_i=\sum _{j=1}^N{\mathsf{S}}_{ij}{\hat{q}}_j,$$

$${\mathsf{S}}_{ij}{\hat{q}}_j={\mathrm{Tr}}_{{\gamma }_i}({\mathcal{S}}_{{\gamma }_j}{\hat{q}}_j),\forall i,j\in \{1,\dots ,N\}.$$

$${\mathsf{S}}_{\gamma }\hat{q}(x)={\int }_{\gamma }G(x-y)\hat{q}(y)\mathrm{d}{\sigma }_y,\forall x\in \gamma .$$

$$⟨\hat{q},{\mathsf{S}}_{\gamma }\hat{q}{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }={\int }_{{\mathbb{R}}^2}{\left|\nabla ({\mathcal{S}}_{\gamma }\hat{q})\right|}^2\mathrm{d}x<+\infty ,\forall \hat{q}\in \hat{\mathcal{H}}(\gamma ).$$

$${\hat{\mathsf{e}}}^k:=({\hat{\mathsf{e}}}_1^k,\dots ,{\hat{\mathsf{e}}}_N^k)\in {\mathcal{H}}^{-\frac{1}{2}}(\gamma )$$

$$⟨{\hat{\mathsf{e}}}_{\ell }^k,1{⟩}_{-\frac{1}{2},\frac{1}{2},{\gamma }_{\ell }}={\delta }_{k,\ell },\ell =1,\dots ,N$$

$$\mathcal{H}(\gamma ):=\{q\in {\mathcal{H}}^{\frac{1}{2}}(\gamma ):⟨{\hat{\mathsf{e}}}^k,q{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }=0,k=1,\dots ,N\}.$$

$$⟨{\hat{\mathsf{e}}}^i,q{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }=⟨\hat{q},{\mathsf{e}}^i{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }=\sum _{j=1}^N⟨{\hat{q}}_j,{\mathsf{e}}_j^i{⟩}_{-\frac{1}{2},\frac{1}{2},{\gamma }_j}=\sum _{j=1}^N{\mathsf{e}}_j^i⟨{\hat{q}}_j,1{⟩}_{-\frac{1}{2},\frac{1}{2},{\gamma }_j},$$

$${\mathsf{S}}_{\gamma }:\hat{q}\in \hat{\mathcal{H}}(\gamma )⟼q\in \mathcal{H}(\gamma ),$$

$$q:={\mathsf{S}}_{\gamma }\hat{q}\in \mathcal{H}(\gamma ).$$

$$⟨\hat{p},\hat{q}{⟩}_{-\frac{1}{2},\gamma }=⟨p,q{⟩}_{\frac{1}{2},\gamma }=⟨\hat{p},q{⟩}_{-\frac{1}{2},\frac{1}{2},\gamma }.$$

$$\Vert \hat{q}{\Vert }_{-\frac{1}{2},\gamma }^2=\Vert q{\Vert }_{\frac{1}{2},\gamma }^2={\int }_{{\mathbb{R}}^2}|\nabla ({\mathcal{S}}_{\gamma }\hat{q}){|}^2\mathrm{d}x,\forall \hat{q}\in \hat{\mathcal{H}}(\gamma ).$$

$$\forall \hat{q}\in {\mathcal{H}}^{-\frac{1}{2}}(\gamma ):{\hat{\Pi }}_{\gamma }\hat{q}$$

$$\forall q\in {\mathcal{H}}^{\frac{1}{2}}(\gamma ):({\Pi }_{\gamma }q{)}_k$$

$$W_0^1({\mathbb{R}}^2)=\{u\in {\mathcal{D}}^{\prime }({\mathbb{R}}^2):\rho u\in L^2({\mathbb{R}}^2),\nabla u\in (L^2({\mathbb{R}}^2){)}^2\}/\mathbb{R},$$

$$\rho (x):={\left(\sqrt{1+|x{|}^2}\log (2+|x{|}^2)\right)}^{-1},x\in {\mathbb{R}}^2,$$

Full text

††copyright: ©2009: Brown University

1. Introduction

Let be given a simply connected open bounded set $\Omega$ in $\mathbb{R}^{2}$ with Lipschitz boundary $\Gamma$. Let $\sigma$ be a positive function in $L^{\infty}(\Omega)$ and consider the elliptic boundary value problem:

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In a recent work [46], the authors investigated this problem in the particular case of piecewise conductivity with infinitely high contrast (see for instance Friedman and Vogelius [21] who considered this problem in the case of small inclusions). Combining an integral formulation of the problem with tools from complex analysis, they proposed an explicit reconstruction formula for the geometry of the unknown cavity. However, due to the crucial use of the Riemann mapping theorem, the proposed approach was limited to the case of a single cavity. The aim of this paper is to investigate the case of a multiply connected cavity. More precisely, we suppose that $\Omega$ contains a multiply connected domain $\omega=\cup_{k=1}^{N}\omega_{k}$, where the open sets $\omega_{k}$, for $k=1,\dots,N$ are non intersecting simply connected domains with $C^{1,1}$ boundaries $\gamma_{k}$