arXiv subjects

Jiaqing Yang

Publications and source records attributed to Jiaqing Yang.

At least 19 recordsLinked to original sources

arXiv2026

Inverse Density Problem for Linear Elasticity: Uniqueness from Local Measurements on a Partially Accessible Boundary

We consider the inverse boundary value problem in an elasticity system. It is proved that the density function $ρ$ and its derivatives at the boundary can be uniquely determined from the local Cauchy data. Furthermore, if the density function is analytic, we can uniquely determine the internal buried objects, as well as the unknown boundary and the boundary conditions imposed on it. Our methods mainly based on a precise characterization for the principal part of the difference between a special first-order singular solution and the fundamental solution in the $H^m$ norm, and the blow-up property for the boundary Sobolev norms of the volume potential corresponding to the fundamental solution.

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arXiv2026

Simultaneous Reconstruction of Multiple Unknowns in Stokes-Darcy System from Partial Boundary Data

This paper studies an inverse boundary value problem for a coupled Stokes-Darcy system modeling fluid-porous medium interaction, with an unknown solid object embedded in the free-flow region. We simultaneously recover the viscosity coefficient $μ$, the interface $Γ$, and the internal object $D$ from localized boundary Cauchy data. A novel method based on the construction of an interior transmission problem is introduced, which can amplify the singularity of solutions. We establish a global uniqueness theorem, showing that all three unknowns are uniquely determined by the boundary measurements.

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arXiv2026

Elastic scattering by locally rough interfaces

In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces.

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arXiv2025

Elastic scattering problems by penetrable obstacles with embedded objects

This paper considers 3-D elastic scattering problems by penetrable obstacles with embedded objects. The well-posedness of transmission problem is proved by employing integral equation method. Then the Inverse Problems , which is to recover the obstacle by the far-field pattern measurement, is considered. It is shown that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency.

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arXiv2025

Recovering discontinuous viscosity coefficients for inverse Stokes problems by boundary measurements

In this paper, we investigate the inverse Stokes problem of determining a discontinuous viscosity coefficient $μ$ in a bounded domain $Ω\subset\mathbb{R}^3$. By analyzing the singularity of the Dirichlet Green's functions in $H^1$-norm and constructing a specifically coupled Stokes-Brinkman system in a localized domain, we prove a global uniqueness theorem that the viscosity coefficient $μ$ can be uniquely determined from boundary measurements.

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arXiv2025

Boundary determination of electromagnetic parameters from local data

In this paper, we extend and simplify the methods in [13] to improve the results on uniqueness of the boundary determination for the Maxwell equation. In particular, we show that the electromagnetic parameters are uniquely determined to infinite order at the boundary from the local admittance map, disregarding the presence of an unknown obstacle, where actually only the local Cauchy data of the fundamental solution are used. The proof mainly relies on an elaborate singularity analysis on certain singular solutions to the Maxwell equation.

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arXiv2025

Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data

In this paper, we consider the inverse boundary value problem of the elliptic operator $Δ+q$ in a fixed region $Ω\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely determine $q$ and all of its derivatives at the boundary from the knowledge of the local Dirichlet-to-Neumann map on $\partialΩ$, disregarding the unknown obstacle, where in fact only the local Cauchy data of the fundamental solution is used. Our proof mainly depends on the rigorous singularity analysis on certain singular solutions and the volume potentials of fundamental solution, which is easy to extend to many other cases.

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arXiv2024

On uniqueness of inverse conductive scattering problem with unknown embedded obstacles

This paper is concerning the inverse conductive scattering of acoustic waves by a bounded inhomogeneous object with possibly embedded obstacles inside. A new uniqueness theorem is proved that the conductive object is uniquely determined by the fixed frequency far-field measurements, ignoring its contents. Meanwhile, the boundary informations of several related physical coefficients are also uniquely determined. The proof is mainly based on a detailed singularity analysis of solutions near the interface associated with a family of point sources or hypersingular point sources, which is deduced by the potential theory. Moreover, the other key ingredient in the proof is the well-posedness of the interior transmission problem with the conductivity boundary condition in the L^2 sense, where several sufficient conditions depending on the domain and physical coefficients are provided.

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arXiv2024

The obstacle scattering for the biharmonic equation

In this paper, we consider the obstacle scattering problem for biharmonic equations with a Dirichlet boundary condition in both two and three dimensions. Some basic properties are first derived for the biharmonic scattering solutions, which leads to a simple criterion for the uniqueness of the direct problem. Then a new type far-field pattern is introduced, where the correspondence between the far-field pattern and scattered field is established. Based on these properties, we prove the well-posedness of the direct problem in associated function spaces by utilizing the boundary integral equation method, which relys on a natural decomposition of the biharmonic operator and the theory of the pseudodifferential operator. Furthermore, the inverse problem for determining the obstacle is studied. By establishing some novel reciprocity relations between the far-field pattern and scattered field, we show that the obstacle can be uniquely recovered from the measurements at a fixed frequency.

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arXiv2023

Reverse time migration for inverse acoustic scattering by locally rough surfaces

Consider the inverse scattering of time-harmonic acoustic scattering by an infinite rough surface which is supposed to be a local perturbation of a plane. A novel version of reverse time migration (RTM) is proposed to reconstruct the shape and location of the rough surface. The method is based on a modified Helmholtz-Kirchhoff identity associated with a special rough surface, leading to a modified imaging functional which uses the near-field data generated by point sources as measurements. The modified imaging functional always reaches a peak on the boundary of the rough surface for sound-soft case and penetrable case, and hits a nadir on the boundary of the rough surface for sound-hard case. Furthermore, we also establish the RTM method associated with the far-field data generated by plane waves. As far as we know, this is the first result for the RTM method with the far-filed data. Numerical experiments are presented to show the powerful imaging quality.

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arXiv2022

Simultaneous recovery of a locally rough interface and the embedded obstacle with the reverse time migration

Consider the inverse acoustic scattering of time-harmonic point sources by an unbounded locally rough interface with bounded obstacles embedded in the lower half-space. A novel version of reverse time migration is proposed to reconstruct both the locally rough interface and the embedded obstacle. By a modified Helmholtz-Kirchhoff identity associated with a planar interface, we obtain a modified imaging functional which has been shown that it always peaks on the local perturbation of the interface and on the embedded obstacle. Numerical examples are presented to demonstrate the effectiveness of the method.

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arXiv2021

Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium

Consider the scattering of time-harmonic point sources by an infinite locally rough interface with bounded obstacles embedded in the lower half-space. The model problem is first reduced to an equivalent integral equation formulation defined in a bounded domain, where the well-posedness is obtained in $L^p$ by the classical Fredholm theory. Then a global uniqueness theorem is proved for the inverse problem of recovering the locally rough interface, the embedded obstacles and the wave number in the lower-half space by means of near-field measurements above the interface.

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arXiv2021

Determining conductivity and embedded obstacles from partial boundary measurements

In this paper, we consider an inverse conductivity problem on a bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical Impedance Tomography (EIT), for the case where unknown impenetrable obstacles are embedded into $Ω$. We show that a piecewise-constant conductivity function and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary of the domain $Ω$. The method depends on the well-posedness of a coupled PDE-system constructed for the conductivity equations in the $H^1$-space and some elementary a priori estimates for Harmonic functions.

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arXiv2021

Time domain analysis for electromagnetic scattering by an elastic obstacle in a two-layered medium

In this paper, we consider the scattering of a time-dependent electromagnetic wave by an elastic body immersed in the lower half-space of a two-layered background medium which is separated by an unbounded rough surface. By proposing two exact transparent boundary conditions (TBCs) on the artificial planes, we reformulate the unbounded scattering problem into an equivalent initial-boundary value problem in a strip domain with the well-posedness and stability proved using the Laplace transform, variational method and energy method. A perfectly matched layer (PML) is then introduced to truncate the interaction problem with two finite layers containing the elastic body, leading to a PML problem in a finite strip domain. We further verify the existence, uniqueness and stability estimate of solution for the PML problem. Finally, we establish the exponential convergence in terms of the thickness and parameters of the PML layers via an error estimate on the electric-to-magnetic (EtM) capacity operators between the original problem and the PML problem.

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arXiv2021

Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the $L^2$-norm and $L^{\infty}$-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).

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arXiv2021

Inverse time-harmonic electromagnetic scattering from coated polyhedral scatterers with a single far-field pattern

It is proved that a convex polyhedral scatterer of impedance type can be uniquely determined by the electric far-field pattern of a non-vanishing incident field. The incoming wave is allowed to bean electromagnetic plane wave, a vector Herglotz wave function or a point source wave incited by some magnetic dipole. Our proof relies on the reflection principle for Maxwell's equations with the impedance (or Leontovich) boundary condition enforcing on a hyper-plane. We prove that it is impossible to analytically extend the total field across any vertex of the scatterer. This leads to a data-driven inversion scheme for imaging an arbitrary convex polyhedron.

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arXiv2020

A non-iterative sampling method for inverse elastic wave scattering by rough surfaces

Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-interative sampling technique is proposed for detecting the rough surface by taking elastic wave measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our pervious work for the Helmholtz equation (SIAM J. IMAGING. SCI. 10(3)(2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the inversion algorithm.

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arXiv2020

Near-field imaging of a locally rough interface and buried obstacles with the linear sampling method

Consider the problem of inverse scattering of time-harmonic point sources from an infinite, penetrable rough interface with bounded obstacles buried in the lower half-space, where the interface is assumed to be a local perturbation of a planar surface. A novel version of the sampling method is proposed to simultaneously reconstruct the local perturbation of the rough interface and buried obstacles by constructing a modified near-field equation associated with a special rough surface, yielding a fast imaging algorithm. Numerical examples are presented to illustrate the effectiveness of the inversion algorithm.

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