Direct Sampling Methods for Inverse Medium Scattering Problems of Elastic Waves

Jul 16, 2026

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Inverse elastic scattering aims to determine an unknown penetrable obstacle from far-field measurements. The work reformulates the coupled boundary value problem using the Helmholtz decomposition into a coupled scalar Helmholtz system, establishing uniqueness for both the coupled Helmholtz formulation and its boundary integral equation counterpart. A Nyström discretization enables efficient computation. Using the relationship between compressional/shear far-field patterns for the Navier and coupled Helmholtz systems, three indicators reconstruct obstacle location and shape. Decay behavior and stability estimates of the indicators are proved, and numerical experiments confirm robustness under limited-aperture data.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages29

Reading Time1 h 13 min

Introduction

Direct and inverse elastic scattering background

Numerical methods for direct elastic-wave scattering

Quantitative and qualitative methods for inverse scattering

arXiv :2607 .03745v1 [math .NA] 4 Jul 2026 DIRECT SAMPLING METHODS FOR INVERSE MEDIUM SCATTERING PROBLEMS OF ELASTIC WAVES LU ZHAO, YILING LI, AND ZHIYONG CHENG Abstract. This paper concerns the inverse elastic scattering problem of determining an unknown penetrable obstacle from far-field data. Using the Helmholtz decomposition, the coupled boundary value problem is reformulated as a coupled scalar Helmholtz system. We prove the uniqueness of the associated coupled Helmholtz boundary value problem and of the corresponding boundary integral equation system, which is then discretized by a Nystr¨om method for efficient numerical computation. Leveraging the relation between the compressional and shear far-field patterns of the Navier system and those of the coupled Helmholtz system, we employ three indicators to reconstruct the location and shape of the obstacle. Furthermore, we analyze the decay properties of these indicators and establish corresponding stability estimates. Numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method, even for limited-aperture data. 1. Introduction In this paper, we consider an inverse elastic scattering for penetrable obstacle from far-field data. In various fields of science and technology, such as noninvasive testing, geophysical exploration, and medical imaging, this type of inverse problem has attracted substantial interest [1, 2, 3] . For the direct elastic wave scattering problem, two principal numerical methods have been extensively developed. The first is a boundary integral equation (BIE) method [4], which restricts computations to the boundary of obstacle. When combined with appropriate discretization schemes, this method effectively reduces computational costs. In [5, 6], the Helmholtz decomposition was used to transform the elastic wave equation to a coupled system of Helmholtz equations, and a Nystr¨om discretization of the associated boundary integral equations was developed to solve the resulting system. For the multiple obstacle scattering problem, a GMRES iterative solver accelerated by the fast multipole method was proposed in [7] to efficiently solve the boundary integral equations. Moreover, Galerkin BIE methods [8, 9] have been successfully applied to elastic scattering in both two and three dimensions. For the three-dimensional elastic-wave scattering problem, a high-order combined-field spectral method was introduced in [10] . Building on the Galerkin framework and spectral techniques, [11] developed a fully discrete spectral scheme for the three-dimensional elastic obstacle scattering problem that reduces the evaluation of the integral operators to scalar operations, thereby greatly simplifying the numerical implementation. Boundary integral formulations have also been used for elastic transmission problems. In two dimensions, coupled singular integral equations on the interface were derived and shown to be uniquely solvable [12], while regularized BIE formulations with high-order Nystr¨om discretizations were later developed for two-dimensional elastodynamic transmission problems [13] . The second major class comprises artificial boundary methods [14], which introduce either a Dirichlet-to-Neumann (DtN) map or a perfectly matched layer (PML) truncation to reformulate the scattering problem as a boundary value problem on a bounded computational domain; the resulting problem is then solved by finite-element or finite-difference methods. In [15], a priori and a posteriori error estimates were established for finite element discretizations coupled with the DtN map. For the elastic obstacle scattering problem, an adaptive 2020 Mathematics Subject Classification. 74J25, 78A46 . Key words and phrases. inverse medium scattering, elastic wave, Helmholtz decomposition, direct sampling method. Corresponding author: Zhiyong Cheng. 2 LU ZHAO, YILING LI, AND ZHIYONG CHENG finite element DtN method guided by a poster

FAQs

What inverse problem does the paper address?

It addresses inverse elastic scattering for determining an unknown penetrable obstacle from far-field data.

How is the elastic system reformulated for analysis and computation?

The coupled boundary value problem is reformulated via Helmholtz decomposition into a coupled scalar Helmholtz system.

What technique reconstructs the obstacle from measured far-field patterns?

Three indicators are derived using the relationship between compressional and shear far-field patterns of the Navier system and those of the coupled Helmholtz system.

Jul 16, 2026

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