电阻抗成像的二阶形状最优化方法研究
On second order shape optimization methods for electrical impedance tomography
摘要：
本文分析了由边界测量恢复物体内部夹杂$ω$未知形状的二阶方法。这个逆问题-被称为电阻抗断层成像-有许多重要的实际应用，因此在过去的几年中，已经集中了很多注意力。然而，据我们所知，还没有工作考虑二阶的方法来解决这个问题。本文的目的是填补这一空白：我们调查的存在性二阶导数的状态$u$相对于扰动的界面$\partialω$的形状，然后我们选择一个费用函数，以恢复的几何$\partial ω$，并推导出表达式的导数需要实现相应的牛顿法。然后，我们调查的过程的稳定性，并解释为什么这个反问题是严重不适定的证明在全球最小的Hessian的紧性。
英文摘要：
This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $ω$ inside a body $Ω$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partialω$, then we choose a cost function in order to recover the geometry of $\partial ω$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.
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