Yukun Guo

Toward a time domain approach to the linear sampling method

We consider an inverse scattering problem for time-dependent acoustic waves in an inhomogeneous medium. We wish to determine the support of the inhomogeneity from measurements of causal scattered waves by using the linear sampling method in the time domain. The performance of the algorithm is illustrated with several numerical examples, including the first examples of the use of the time domain linear sampling method in three dimensions.

Fourier method for solving the multi- frequency inverse source problem for the Helmholtz equation

We consider an inverse source problem for the Helmholtz equation. This is concerned with the reconstruction of an unknown source from multi-frequency data obtained from the radiated fields. Based on a Fourier expansion of the source, a numerical method is proposed to solve the inverse problem. Stability is analyzed and numerical experiments are presented to show the effectiveness of our method.

The reciprocity gap functional method for the inverse scattering problem for cavities

In this paper, we concern with determining the shape of a perfectly conducting cavity from the Cauchy data on a curve inside the cavity. The near-field linear sampling method (LSM), i.e. the reciprocity gap (RG) functional method, is employed to reconstruct the shape of the cavity. The equivalence of the RG method and the linear sampling method with mere the scattered field is established. But from the examples, we can see that the reconstructions are as satisfactory as the exterior scattering problems. I think this behavior is due to our reconstruction method since this method is due to the Cauchy data, but the LSM with mere the scattered field is used. Numerical tests show that the methods can provide qualitative information on the cavity. The numerical influence of the proposed method with respect to the wave numbers, the curve for the Cauchy data on which are measured, and the curve which is used to construct the single-layer potential function, respectively, are also analyzed with some examples. In particular, we give the examples of determining the cavity from the Cauchy data measured on a portion of the curve inside the cavity.

The linear sampling method for sparse small aperture data

In an effort to improve the performance of the linear sampling method in situations involving sparse data-sets, this method in inverse scattering has recently been extended from the frequency domain to the time domain. In this paper, we consider the relative merits of the time and multifrequency linear sampling methods for sparse, limited aperture, and data-sets. Among our conclusions are that, for limited aperture measurements single-frequency data can fail to reconstruct the scatterer, whereas both time and multifrequency domain data perform satisfactorily. On the other hand, if the aperture is too small all the sampling methods fail and increasing the number of measurements in a fixed size aperture is of no help.

Numerical analysis for the scattering by obstacles in a homogeneous chiral environment

The scattering of time-harmonic electromagnetic waves propagating in a homogeneous chiral environment by obstacles is studied. The problem is simplified to a two-dimensional scattering problem, and the existence and the uniqueness of solutions are discussed by a variational approach. The diffraction problem is solved by a finite element method with perfectly matched absorbing layers. Our computational experiments indicate that the method is efficient.

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.

An optimization method for acoustic inverse obstacle scattering problems with multiple incident waves

The inverse problem considered in this article is to determine the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of some far field patterns. Based on the optimization method due to Kirsch and Kress for the inverse scattering problem, we propose a new scheme by reformulating the cost functional via a technique of piecewise integration with respect to incident directions. Convergence analysis of this method is given. Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions for the full-aperture problems. The method is extended to the limited-aperture case by weighting the total fields with special factors. Numerical examples for limited-aperture problems are also presented which show that our method produces satisfactory results efficiently in the illuminated regions.

Uniqueness results on phaseless inverse acoustic scattering with a reference ball

This paper is devoted to the uniqueness in inverse scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a plane wave and a fixed point source as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Third, only a single frequency is needed. Finally, it provides a very simple proof of the uniqueness.

Retrieval of acoustic sources from multi-frequency phaseless data

This paper is concerned with the inverse source problem of reconstructing an unknown acoustic excitation from phaseless measurements of the radiated fields away at multiple frequencies. It is well known that the non-uniqueness issue is a major challenge associated with such an inverse problem. We develop a novel strategy to overcome this challenging problem by recovering the radiated fields via adding some reference point sources as extra artificial sources to the inverse source system. This novel reference source technique requires only a few extra data, and brings in a simple phase retrieval formula. The stability of this phase retrieval approach is rigorously analyzed. After the reacquisition of the phase information, the multi-frequency inverse source problem with recovered phase information is solved by the Fourier method, which is non-iterative, fast and easy to implement. Several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

Abstract This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.