A Novel Time-Domain Iterative Method for a Three-Dimensional Inverse Acoustic Obstacle Scattering Problem

Jul 16, 2026

A new time-domain approach addresses the three-dimensional forward and inverse acoustic obstacle scattering problem. The forward model uses a retarded potential formulation discretized by convolution quadrature and Galerkin methods. For inversion, a novel time-domain convolution-quadrature iterative method reconstructs both obstacle shape and location via retarded boundary integrals on homothetic surfaces, then converts them into fast-to-evaluate non-singular s-domain integrals. The method provides a direct Fréchet derivative and proves time-domain convergence, with stability improved through incremental truncation validated by numerical experiments.

LanguageEnglish

CategoryResearch & Report

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Pages28

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Introduction

Problem formulation and measurements

Related work and background

arXiv :2607 .02286v1 [math .NA] 2 Jul 2026 A NOVEL TIME-DOMAIN ITERATIVE METHOD FOR A THREE-DIMENSIONAL INVERSE ACOUSTIC OBSTACLE SCATTERING PROBLEM LU ZHAO, HEPING DONG, AND ZHIYONG CHENG Abstract. This paper concerns the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem, a retarded potential formulation discretized by convolution quadrature and Galerkin methods is introduced. By introducing the retarded boundary integral defined on a homothetic surface, we propose a novel time-domain convolution quadrature based iterative method to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integralsin the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fr´echet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability of the inversion algorithm, an incremental truncation technique is proposed, and numerical experiments confirm the effectiveness and robustness of our method. 1. Introduction In this paper, we consider an inverse acoustic scattering problem of reconstructing a bounded obstacle from time-domain scattered field. Such inverse problems have attracted considerable attention due to their wide range of practical applications, including geophysical exploration [9], biomedical imaging [2], and nondestructive evaluation [3] . We begin by formulating the mathematical model for the time-dependent acoustic obstacle scattering problem. Let D ⊂ R3 be a bounded open domain with a smooth boundary ΓD . The exterior region R3 \ D is filled with homogeneous medium of unit mass density. We assume that the incident wave uinc(x, t) = ϱ (t − | x − x0 |)/(4π|x − x0 |) emitted from a point x0 ∈ R3 \ D with a causal and smooth signal ϱ . The forward scattering problem consists in determining the scattered field usc(x, t) that satisfies the following initial-boundary problem (1) 􀀸 􀀾 􀀼 􀀾 􀀺 ∂2tusc − ∆usc = 0 usc = −uinc usc(·, 0) = ∂tusc(·, 0) = 0 in 􀀀R3 \ D 􀀁 × (0 , ∞), on ΓD × (0 , ∞), in R3 \ D. Let R > 0 be a fixed constant and define the ball B :=􀀈x ∈ R3 : | x| < R 􀀉 such that D ⊂ B. The boundary ΓB of B is chosen as the observation surface, and let T > 0 be the truncated final time. The near-field measurements are given by Λ := {usc(x, t) : (x, t) ∈ ΓB × [0, T]} . Figure 1 provides a schematic illustration of the time-domain scattering problem in R3 . With these settings, we formulate the inverse scattering problem as follows. 2020 Mathematics Subject Classification. 65M32, 78A46, 65M70 . Key words and phrases. inverse acoustic scattering, time domain, homothetic surface, convolution quadrature, iterative method. Corresponding author: Heping Dong ([dhp@jlu.edu.cn](dhp@jlu.edu.cn)). 2 LU ZHAO, HEPING DONG, AND ZHIYONG CHENG Problem 1 (Inverse acoustic obstacle scattering) . Given the near-field measurements Λ on Γ B ×[0, T], determine the shape and location of the obstacle D ⊂ R3 . observation surface ΓB x0 incident wave uinc scattered field usc Figure 1 . An illustration of acoustic obstacle scattering. Time-domain broadband signals typically contain richer information and are often easier to capture in practice than frequency-domain data. In recent years, significant mathematical and computational progress has been achieved on time-domain acoustic obstacle scattering and the associated inverse scattering problems. Concerning the well-posedness analysis of the direct scattering problem, the theoretical foundations have been established in [12, 23] . Numerically, a common starting point is to formulate a boundary integral equation via retarded potential theory, whic

FAQs

What inverse problem does the paper address?

The paper reconstructs a bounded rigid obstacle’s shape and location in R3 from time-domain near-field scattered data measured on an observation surface over a finite time interval.

How is the forward time-domain acoustic problem formulated?

The forward problem is posed as an initial-boundary value problem for the scattered field in the exterior domain, driven by a causal incident wave and enforced with boundary conditions on the obstacle surface.

What is the core idea of the proposed iterative inversion method?

It builds on a retarded potential boundary integral on a homothetic surface, reformulates the time-domain convolution-quadrature terms into s-domain integrals for efficient computation, and derives the Fréchet derivative to drive the iterative reconstruction. Stability is enhanced using incremental truncation.

Jul 16, 2026

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