Localized sensitivity analysis at high-curvature boundary points of reconstructing inclusions in transmission problems

TL;DR

This paper analyzes how high-curvature boundary points of inclusions in electrostatics and acoustic scattering can be reconstructed more accurately, using novel mathematical tools involving GPTs and microlocal analysis.

Contribution

It introduces a new local resolution analysis framework that links boundary curvature to reconstruction stability, advancing wave super-resolution imaging techniques.

Findings

High-curvature points are reconstructed more stably.

Develops a novel mathematical scheme using GPTs and SCs.

Establishes local resolution effects via sensitivity analysis.

Abstract

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarisation tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via the microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one steps depend on the global geometry, and one particular step depends on the mean curvature at a given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave super-resolution imaging.