A highly efficient iterative approach for inverse acoustic obstacle scattering problems in three dimensions

Jul 16, 2026

This paper addresses a three-dimensional inverse acoustic obstacle scattering problem using scattered-field or phased/phaseless far-field data. By formulating boundary integral equations on a homothetic surface, the work proposes an iterative reconstruction method for obstacles that eliminates singularity handling. The injectivity and dense-range properties of the Fréchet derivative are established to guarantee solvability of the linearized equivalent data equation. It is further shown that fields generated on the homothetic surface can approximate the exact scattered field arbitrarily, with numerical tests confirming efficiency and robustness.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages31

Reading Time1 h 18 min

Introduction

Problem setting and direct scattering formulation

Phased/phaseless data and non-uniqueness

Inverse problem variants and data types

Existing reconstruction methods overview

arXiv :2607 .02180v1 [math .NA] 2 Jul 2026 A HIGHLY EFFICIENT ITERATIVE APPROACH FOR INVERSE ACOUSTIC OBSTACLE SCATTERING PROBLEMS IN THREE DIMENSIONS ZHIYONG CHENG, HEPING DONG, AND LU ZHAO Abstract. This paper concerns a three-dimensional inverse acoustic obstacle scattering problem from scattered field or phased/phaseless far-field data. Based on the boundary integral defined on a homothetic surface, we propose a highly efficient iterative approach for obstacle reconstruction that completely avoids dealing with any singularity. Here, the injectivity and dense-range property of the Fr´echet derivative have been proved to ensure the solvability of the linearized equivalent data equation. We also prove that the scattered field generated by the homothetic surface can arbitrarily approximate the exact one. Numerical experiments are presented to verify the superiority and robustness of the proposed approach. 1. Introduction In this paper, we consider an inverse acoustic scattering problem of reconstructing a bounded obstacle from scattered field or phased/phaseless far-field data. This type of inverse problem isan important research topic with many practical applications such as geophysical exploration [5], biomedical imaging [1] and nondestructive testing [2] . We begin by presenting the mathematical formulation of the direct acoustic scattering problem in three dimensions. Assume that D ⊂ R3 is a bounded, simply connected domain with sufficiently smooth boundary Γ, and that the exterior region R3 \ D is filled with a homogeneous medium. We further assume that the incident wave can be either a plane wave uinc(x, d) = eiκx ·d or a point source uinc(x, z) = G(x, z) := ~~e~~4iκπ|~~ ~~x|xxxxxzzzzzz||~~ ~~ , where d is an incident direction, z ∈ R3 \ D is a location of the point source, and G is the Green function of Helmholtz equation in three dimensions. Let utot denote the total field, that means usc = utot − uinc. Then the direct scattering problem is to find the scattered field usc that satisfies the following Helmholtz system (1) 􀀸 􀀾 􀀾 􀀼 􀀾 􀀾 􀀺 ∆usc + κ2 usc = 0 Busc = −Buinc lim r (∂rusc − iκusc) = 0, r→∞ in R3 \ D , on Γ , r = | x| , where κ > 0 is the real wavenumber. For any sufficiently smooth function u defined on Γ, the boundary operator B is given by (2) Bu = 􀀚 ν, u + iηu, Dirichlet boundary condition, impedance boundary condition. Here, ν is the outward unit normal vector to Γ, and η denotes the impedance function with ℜ(η) ≥ 0. The impedance condition reduces to the Neumann case when η = 0 . The scattered field admits the 2020 Mathematics Subject Classification. 78A46 . Key words and phrases. Helmholtz equation, boundary integral equation, iterative approach, inverse acoustic obstacle scattering, phaseless data. Corresponding author: Heping Dong ([dhp@jlu.edu.cn](dhp@jlu.edu.cn)). 2 ZHIYONG CHENG, HEPING DONG, AND LU ZHAO following asymptotic behavior [11] usc(x) = eκxxx|~~ ~~x|~~ ~~| 􀀚 u∞ (xˆxx) + O 􀀒 ~~ ~~|~~ ~~1xxx|~~ ~~ 􀀓􀀛 , xˆxx = |~~ ~~xxxxxx| ∈ S2 , | x| → ∞ , where u∞ denotes the far-field pattern of usc , and S2 := {x ∈ R3 : | x| = 1} is the unit sphere centered at the origin. It is worth mentioning that the modulus of the far-field data has translation invariance for a shift domain, when an incident plane wave is employed [20] . That is, the location of the obstacle cannot be determined from phaseless far-field data. To address this issue, we choose a point source as incident wave, and collect the phaseless far-field data |utot,∞ | = |G∞ + u∞ |, where utot,∞ denotes the far-field pattern of the total field utot, and G∞(xˆxx, z) = 4~~1~~πe−iκxˆxx ·z is the far-field pattern of the point source incident field G (x, z) with xˆxx ∈ S2 . In this work, we concern with three kinds of inverse problems: Problem 1 . Given the scattered field usc measured on (3) ΓBR :=􀀈x ∈ R3 : | x| = R 􀀉 for the incident plane waves uinc , determine the shape and locatio

FAQs

What inverse acoustic scattering task does the paper focus on?

It reconstructs a bounded obstacle in three dimensions from scattered field data or phased/phaseless far-field data under Dirichlet or impedance boundary conditions.

How does the proposed method avoid singularity handling?

It builds the boundary integral formulation on a homothetic surface and develops an iterative reconstruction scheme based on that formulation, thereby completely avoiding dealing with singularity.

What theoretical guarantees are provided for the iterative reconstruction?

The paper proves injectivity and dense-range properties of the Fréchet derivative, ensuring solvability of the linearized equivalent data equation, and shows that the homothetic-surface generated scattered field can approximate the exact one arbitrarily.

Jul 16, 2026

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