Jiaqing Yang

The Factorization Method for Reconstructing a Penetrable Obstacle with Unknown Buried Objects

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle with buried objects inside. We prove under certain conditions that the factorization method can be applied to reconstruct the penetrable obstacle from far-field data without knowing the buried objects inside. Numerical examples are also provided illustrating the inversion algorithm.

A sampling method for the inverse transmission problem for periodic media

This paper is concerned with the inverse scattering problem of reconstructing the support of a periodic inhomogeneous medium from knowledge of the scattered field measured on a straight line above and below the periodic structure. A linear sampling method is proposed to reconstruct the support of the periodic inhomogeneous medium based on a linear operator equation. The mathematical analysis of the sampling method is developed and numerical examples are given showing the practicality of the reconstruction algorithm.

The inverse electromagnetic scattering problem in a piecewise homogeneous medium

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method. In Liu and Zhang (2009 Appl. Anal. 88 1339–55) it was proved, under the condition that the wave numbers in the innermost and outermost homogeneous layers coincide and S0 is known in advance, that the obstacle with its physical property can be uniquely determined from knowledge of the electric far-field pattern for incident plane waves. In this paper, we will remove this restriction by establishing a new mixed reciprocity relation. Furthermore, inspired by Hahners idea in Hahner (1993 Inverse Problems 9 667–78), we prove that the penetrable interface between layers can also be uniquely determined.

An inverse transmission scattering problem for periodic media

Consider the problem of scattering of time-harmonic waves by a penetrable periodic structure. The structure separates the whole space into three regions: the medium above and below the structure is assumed to be homogeneous, and the medium inside the structure is assumed to be inhomogeneous with the refractive index q(x). Having established the well posedness of the scattering problem, we show that the scattered field measured only above the structure, corresponding to a countably infinite number of quasi-periodic waves, uniquely determine the two grating profiles. Furthermore, we prove that the refractive index q(x) can be uniquely determined from the scattered field measured above and below the structure, corresponding to a countably infinite number of quasi-periodic waves, if q depends only on one direction and if the two grating profiles are known constants. The proofs are based on a priori estimates of the solutions of the direct problem and new mixed reciprocity relations.

Reconstruction of Complex Obstacles with Generalized Impedance Boundary Conditions from Far-Field Data

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic plane waves from a complex obstacle in the sense that a so-called generalized impedance boundary condition (GIBC) is satisfied on the boundary of the obstacle. The factorization method is established to reconstruct the complex obstacle from the far-field data at a fixed frequency. Numerical examples are also provided to illustrate the practicability of the inversion algorithm.

Near-field imaging of scattering obstacles with the factorization method

In this paper we establish a factorization method for recovering the location and shape of an acoustic bounded obstacle with using the near-field data, corresponding to infinitely many incident point sources. The obstacle is allowed to be an impenetrable scatterer of sound-soft, sound-hard or impedance type or a penetrable scatterer. An outgoing-to-incoming operator is constructed for facilitating the factorization of the near-field operator, which can be easily implemented numerically. Numerical examples are presented to demonstrate the feasibility and effectiveness of our inversion algorithm, including the case where limited aperture near-field data are available only.

An inverse scattering problem for a layered periodic structure

This article is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable periodic structure in a periodic layered medium. Having established the well-posedness of the direct problem by the integral equation method, we show that the measurements, corresponding to a countably infinite number of incident quasi-periodic waves above the structure, uniquely determine both the penetrable periodic interface and the impenetrable periodic structure together with its physical property. To this end, an important role is played by a priori estimates of the solution and a mixed reciprocity relation for the direct problem.

Near-field imaging of periodic interfaces in multilayered media

This paper is concerned with the inverse problem of scattering of time-harmonic electromagnetic waves by a penetrable multilayered periodic structure. The structure separates the whole space into three regions: the medium above and below the structure is assumed to be homogeneous but with different wave numbers, and the medium inside the structure is assumed to be inhomogeneous characterized by the refractive index. We prove, under certain conditions, that the factorization method can be used to reconstruct the upper interface from the scattered near-field data measured only above the structure and generated by a countably infinite number of downward propagating incident waves merely from the top region, leading to a fast imaging algorithm. A similar result can be obtained for reconstructing the lower interface from the scattered near-field data measured only below the structure, corresponding to a countably infinite number of upward propagating incident waves also merely from the bottom region. Thus, our approach is applicable, even if one merely has access to the object under investigation from one side. Finally, numerical examples are presented to illustrate the effectiveness of the inversion algorithm.

Detecting buried wave-penetrable scatterers in a two-layered medium

Abstract In a two-layered medium, we prove that a buried inhomogeneous scatterer is uniquely determined from the wave field data measured in the upper half-space with respect to many incident point sources. Moreover, we extend the multilevel sampling method in Liu and Zou (2013) to numerically reconstruct the buried scatterer applying only few incident fields and partial scattered data. The extended recovery scheme only involves matrix–vector operations and does not need to solve any large-scale ill-posed linear systems or any optimization process. It is feasible to deal with the scatterers of different features and easy to implement, highly tolerant to noise and computationally quite cheap. We can regard it as an effective yet simple computational method to provide a reliable initial guess for the implementation in existing more accurate and refined optimization-type reconstruction algorithms.