Shape reconstructions by using plasmon resonances

¹ School of Mathematics, Hunan University, Changsha 410082, P.R. China
² Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, P.R. China

^* Corresponding author: hongyliu@cityu.edu.hk (Hongyu Liu); zhenggh2012@hnu.edu.cn (Guang-Hui Zheng)

Received: 23 November 2021
Accepted: 14 February 2022

Abstract

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.