Reconstruction of Inhomogeneous Conductivities via the Concept of Generalized Polarization Tensors
Reconstruction of Inhomogeneous Conductivities via the Concept of Generalized Polarization Tensors
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DOI：
10.1016/j.anihpc.2013.07.008
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发表时间：
2012-11
期刊：
影响因子：
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通讯作者：
H. Ammari;Youjun Deng;Hyeonbae Kang;Hyundae Lee
中科院分区：
文献类型：
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作者：
H. Ammari;Youjun Deng;Hyeonbae Kang;Hyundae Lee
本文将广义极化张量（GPTs）的概念推广到非均匀电导率夹杂，该概念以前是为具有均匀电导率的夹杂定义的。我们开始给出两个略有不同但等价的定义的GPT的非均匀夹杂。然后，我们表明，在均匀的情况下，GPTs的远场扩展的电压在导电性夹杂物的存在下，是基本的积木。将GPT与纽曼-狄利克雷（NtD）图联系起来，可以看出，对GPT的充分了解可以独特地确定电导率分布。此外，我们显示的GPT的重要性质，如对称性和积极性，并推导出满足其调和和的界限。我们还计算的GPT相对于电导率分布的变化的灵敏度，并提出了一种算法，用于重建电导率分布从他们的GPT。这为求解高度非线性、不适定的电导率反问题提供了一种新的策略。我们证明了所提出的算法的可行性进行灵敏度分析，并给出了一些数值例子。
This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.