Haibing Wang

Linear sampling method for identifying cavities in a heat conductor

We consider an inverse problem of identifying the unknown cavities in a heat conductor. Using the Neumann-to-Dirichlet map as measured data, we develop a linear sampling-type method for the heat equation. A new feature is that there is a freedom to choose the time variable, which suggests that we have more data than the linear sampling methods for the inverse boundary value problem associated with EIT and inverse scattering problem with near field data.

Linear sampling method for the heat equation with inclusions

We are concerned with the reconstruction of unknown inclusions inside a heat conductor from boundary measurements, which is modeled as an inverse boundary value problem for a parabolic equation. Taking the Neumann-to-Dirichlet map as measured data, we establish a linear sampling-type method to identify the inclusions. As in inverse scattering problems, the so-called forward interior transmission problem naturally arises in the linear sampling method for identifying inclusions. The unique solvability of the forward interior transmission problem for parabolic equations is also investigated.

On uniqueness of an inverse problem in electromagnetic obstacle scattering for an impedance cylinder

We consider an inverse problem for the scattering of an obliquely incident electromagnetic wave by an impedance cylinder. In previous work, we have shown that the direct scattering problem is governed by a pair of Helmholtz equations subject to coupled oblique boundary conditions, where the wave number depends on the frequency and the incident angle with respect to the axis of the cylinder. In this paper, we are concerned with the inverse problem of uniquely identifying the cross-section of an unknown cylinder and the impedance function from the far-field patterns at fixed frequency and a range of incident angles. A uniqueness result for such an inverse scattering problem is established. Our method is based on the analyticity of solution to the direct scattering problem, which is justified by using the Lax–Phillips method together with the perturbation theory of Fredholm operators.

Reconstruction of an unknown cavity with Robin boundary condition inside a heat conductor

Active thermography is a non-destructive testing technique to detect the internal structure of a heat conductor, which is widely applied in industrial engineering. In this paper, we consider the problem of identifying an unknown cavity with Robin boundary condition inside a heat conductor from boundary measurements. To set up the inverse problem mathematically, we first state the corresponding forward problem and show its well-posedness in an anisotropic Sobolev space by the integral equation method. Then, taking the Neumann-to-Dirichlet map as mathematically idealized measured data for the active thermography, we present a linear sampling method for reconstructing the unknown Robin-type cavity and give its mathematical justification by using the layer potential argument. In addition, we analyze the indicator function used in this method and show its pointwise asymptotic behavior by investigating the reflected solution of the fundamental solution. From our asymptotic analysis, we can establish a pointwise reconstruction scheme for the boundary of the cavity, and can also know the distance to the unknown cavity as we probe it from its inside.

Inverse scattering for obliquely incident polarized electromagnetic waves

We consider an inverse problem for the scattering of an obliquely incident electromagnetic wave by an impedance cylinder. We show that the cross-section of the cylinder can be uniquely determined from the far-field patterns of only the electric or magnetic field, through the use of special polarized incident plane waves. The proof is based on the generalized mixed reciprocity principle and the singularity analysis of the reflected solutions for point sources. Our result would be helpful in practical situations where only one type of measurement is available. Moreover, our argument is reconstructive, and can be used directly to set up a numerical scheme. In fact, the singular behavior of the reflected solutions for point sources enables us to reconstruct the shape of the cross-section numerically. To recover the impedance, we use a more singular source to derive a pointwise calculation formula.

Reconstruction of an Impedance Cylinder at Oblique Incidence from the Far-Field Data

Consider scattering of obliquely incident plane electromagnetic waves by an impedance cylinder embedded in a homogeneous background medium. Under some assumptions, the direct scattering problem is governed by a coupled system of the two-dimensional Helmholtz equations for

The integral equation method for electromagnetic scattering problem at oblique incidence

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The direct electromagnetic scattering problem from an imperfectly conducting cylinder at oblique incidence

components of the electric field and the magnetic field. In this paper, we study a corresponding inverse scattering problem, which is to reconstruct an unknown scattering object from measurement data of the

Blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain

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An inverse scattering problem with generalized oblique derivative boundary condition

component of only the electric field. A reconstruction scheme called the linear sampling method is established with rigorous mathematical analysis. Due to the coupled oblique derivative boundary condition, some new characteristics appear in our analysis. Numerical examples are also presented to illustrate the feasibility of the reconstruction algorithm.