We study the shape reconstruction of a dielectric inclusion from the faraway measurement of the associated electric field. This is an inverse problem of practical importance in biomedical imaging and is known to be notoriously ill-posed. By incorporating Drude's model of the dielectric parameter, we propose a novel reconstruction scheme by using the plasmon resonance with a significantly enhanced resonant field. We conduct a delicate sensitivity analysis to establish a sharp relationship between the sensitivity of the reconstruction and the plasmon resonance. It is shown that when plasmon resonance occurs, the sensitivity functional blows up and hence ensures a more robust and effective construction. Then we combine the Tikhonov regularization with the Laplace approximation to solve the inverse problem, which is an organic hybridization of the deterministic and stochastic methods and can quickly calculate the minimizer while capture the uncertainty of the solution. We conduct extensive numerical experiments to illustrate the promising features of the proposed reconstruction scheme.

翻译：我们研究了从远距离测量相关电场中重建电离层的形状。 这是一个在生物医学成像中具有实际重要性的反面问题,众所周知,这个问题是臭名昭著的弊病。我们通过采用德鲁德的电离参数模型,提出一个新的重建计划,用显著增强的共振场来使用普拉斯蒙共振;我们进行了微妙的敏感性分析,以便在重建的敏感度和普拉斯蒙共振之间建立一种尖锐的关系。我们发现,当出现普拉斯蒙共振反应时,灵敏功能就会爆炸,从而保证更稳健有效的构造。然后,我们将蒂赫诺诺夫的正规化与拉普特近距离结合起来,解决反向问题,这是确定性和随机性方法的有机混合,可以快速计算最小化,同时捕捉解决方案的不确定性。我们进行了广泛的数字实验,以说明拟议重建计划的有希望的特征。