We study the shape reconstruction of an 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 permittivity 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:我们研究从相关电场的 {faraway} 测量中包含相关电场的形状重建。 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。 通过纳入Drude的许可参数模型,我们提出了一个新颖的重建计划,用显著增强的共振场来使用plasmon共振; 我们进行了微妙的敏感性分析,以便在重建的敏感性和普lasmon共振之间建立一种尖锐的关系; 显示当出现共振时,敏感功能就会爆炸,从而保证更稳健有效的建设。 然后,我们将Tikhonov的正规化与Laplace近似点结合起来,以解决反向问题,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。