Research

In general, Dr. Meng is broadly interested in applied and computational mathematics, inverse problems, and data science.  His long-term research goal is to work as an applied mathematician to address interdisciplinary challenges spanning science, engineering, and technology.  

In particular, Dr. Meng is interested in mathematical analysis and numerical methods for inverse scattering problems with complex data. The mathematical theory of wave propagation describes the interaction of waves with scattering object in a given  medium. The goal of inverse scattering (or in short imaging) is to reconstruct the object from measurement of the wave field (i.e. data). Inverse scattering is intrinsically  non-linear and ill-posed which  poses great challenge in its mathematical theory and numerical algorithms. Dr. Meng is mainly interested in shape identification which merits applications in a broad spectrum of scientific and engineering disciplines, including radar,  medical diagnosis, and non-destructive material testing.  He uses tools  from partial differential equations and scientific computing to develop and analyze inversion algorithms for complex media, using measurement data in a variety of application relevant setups. His projects are summarized as follows:  (1) inverse acoustic and electromagnetic scattering,  (2) eigenvalue problems and non-destructive testing in inhomogeneous media, and  (3) wave propagation in periodic and complex media.  

(1) Inverse acoustic and electromagnetic scattering

Classical iterative methods reformulate the inverse scattering problem as a nonlinear optimization problem that required the solution of the direct scattering problem for different domains at each step of the iterative procedure. A prior information such as number of connectivity components and types of boundary conditions was required by such iterative methods. Different from these iterative methods,  the so-called sampling (or qualitative) methods  require less a prior information, and moreover, they provide solid mathematical justifications in addition to efficient numerical algorithms.   One prominent method is the  factorization method which  establishes a sufficient and necessary criterion for shape identification, and suggests efficient numerical algorithms. The factorization method and many other sampling methods apply in the resonance region which do  not assume Born approximation or physical optics approximations. The idea of sampling method is to use the data (or processed data) to introduce  a  data-driven operator with a rich mathematical structure, which leads to a criterion of the support of the scattering object. In specific problems of time-harmonic inverse acoustic and electromagnetic scattering, we encounter usually the following types of data: (1) full multi-static data at a fixed frequency,   (2) limited-aperture data, (3) sparse data (i.e. data measured at a few sparse points/directions), (4) multi-frequency data.  Dr. Meng's research has been focused on development of sampling methods with various types of data in complex media.

Dr. Meng is recently developing data-driven methods for inverse scattering problems.


(2) Eigenvalue problems and non-destructive testing in inhomogeneous media

Eigenvalues associated with certain wave phenomena contain information about the material properties of  the scattering medium and can be determined from measurements of the scattered wave field. Thus, they can be used as target signatures and provide an alternative way for carrying out non-destructive testing and imaging. On the other hand, they play a central role in the analysis of qualitative methods. Dr. Meng's research has been focused on the spectral analysis of such eigenvalues and their corresponding inverse problems.


(3) Wave propagation in periodic and complex media

In recent years, periodic and complex media have been used with remarkable success to manipulate waves toward achieving super-focusing, sub-wavelength imaging, cloaking, and topological insulation. Recent developments on the higher-order effective description of waves in such media have shown the potential to illustrate its dynamic properties and retrieve microscopic information. Dr. Meng's research has been focosed on higher-order homogenization and novel physical phenomena in periodic structures.