Optimal Shape Design by Partial Spectral Data

Abstract

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann--Poincaré operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann--Poincaré operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss--Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.