Optimal Shape Design by Partial Spectral Data

TL;DR

This paper introduces a method to design shapes using spectral data like polarization tensors and Fredholm eigenvalues, with applications in nanoparticle plasmonics and imaging.

Contribution

It proposes a novel approach to recover Neumann-Poincaré eigenvalues from polarization tensors and develops a regularized optimization method for shape reconstruction.

Findings

Numerical results validate the effectiveness of the proposed methods.

The approach reveals properties of Fredholm eigenvalues and eigenfunctions.

Applications include plasmon resonance design and anomaly imaging.

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-Poincar\'e operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann-Poincar\'e 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.