Keji Liu

Yat Tin Chow, Kazufumi Ito, Keji Liu et al.

In this work, we are concerned with the diffusive optical tomography (DOT) problem in the case when only one or two pairs of Cauchy data is available. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneities inside a homogeneous background and solve the DOT problem in both full and limited aperture cases. This new method is easy to implement and less expensive computationally. Numerical experiments demonstrate its effectiveness and robustness against noise in the data. This provides a new promising numerical strategy for the DOT problem.

Keji Liu, Yongzhi Xu, Jun Zou

In the reconstruction process of sound waves in a 3D stratified waveguide, a key technique is to effectively reduce the huge computational demand. In this work, we propose an efficient and simple multilevel reconstruction method to help locate the accurate position of a point source in a stratified ocean. The proposed method can be viewed as a direct sampling method since no solutions of optimizations or linear systems are involved. The novel method exhibits several strengths: fast convergence, robustness against noise, advantages in computational complexity and applicability for a very small number of receivers.

Keji Liu, Jun Zou

In the reconstruction process of unknown multiple scattering objects in inverse medium scattering problems, the first important step is to effectively locate some approximate domains that contain all inhomogeneous media. Without such an effective step, one may have to take a much larger computational domain than actually needed in the reconstruction of all scattering objects, thus resulting in a huge additional computational efforts. In this work we propose a simple and efficient multilevel reconstruction algorithm to help locate an accurate position and shape of each inhomogeneous medium. Then other existing effective but computationally more demanding reconstruction algorithms may be applied in these initially located computational domains to achieve more accurate shapes of the scatter and the contrast values over each medium domain. The new algorithm exhibits several strengths: robustness against noise, requiring less incidences, fast convergence, flexibility to deal with scatterers of special shapes, and advantages in computational complexity.

Habib Ammari, Yat Tin Chow, Keji Liu

An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications in the reconstruction process of inverse scattering problems.