Monotonicity-based regularization of inverse medium scattering for shape reconstruction

TL;DR

This paper introduces a monotonicity-based regularization method for inverse medium scattering problems, combining shape reconstruction with linearization to accurately recover scatterer shapes from noisy far field data.

Contribution

It proposes a novel regularization scheme that stabilizes shape reconstruction using monotonicity principles and linearization, with proven convergence and numerical validation.

Findings

Successfully recovers scatterer shape with noise-free data

Converges to the exact shape as noise decreases

Numerical examples confirm theoretical results

Abstract

We consider the scattering of time-harmonic plane waves by a compactly supported inhomogeneous scattering obstacle governed by the Helmholtz equation. Given far field observations of the scattered fields corresponding to plane wave incident fields for all possible incident and observation directions we study the inverse problem to recover the support of the scatterer. We propose a qualitative monotonicity-based regularization scheme which combines monotonicity-based shape reconstruction with one-step linearization to reconstruct a discrete approximation of the shape of the scatterer from noisy far field data. The purpose of the one-step linearization is to stabilize the monotonicity approach to shape reconstruction. We show that the monotonicity-based regularization scheme recovers the correct shape of the scatterer for noise-free data. Furthermore, we establish that the solution of the monotonicity-based regularization converges to the exact solution as the noise level tends to zero. We present numerical examples to illustrate our theoretical findings.