Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

TL;DR

This paper presents a novel, globally convergent numerical method for phase retrieval in inverse scattering, combining WKB-based phase reconstruction, Fourier dimension reduction, and Carleman convexification, effectively handling high noise levels.

Contribution

It introduces a new framework that integrates phase retrieval, dimension reduction, and convexification to solve the challenging phaseless inverse scattering problem.

Findings

Successfully reconstructs dielectric constants from noisy data

Demonstrates robustness and accuracy in numerical simulations

Effectively recovers shape and contrast of scatterers

Abstract

This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and severe ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.