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 computational and analytical methods for solving PDEs and inverse problems, and a major theme is on forward and inverse scattering problems. He has been working on reconstruction methods for inverse scattering (sampling methods and dimension reduction techniques based on the generalized prolate spheroidal wave functions), eigenvalues and non-destructive testing (transmission eigenvalue, Stekloff eigenvalue, band structure and other non-selfadjoint eigenvalue problems), and homogenization for periodic media (higher order homogenization at low/high frequencies). His most recent interest is on low rank methods and dimension reduction techniques.

(1) Low rank and dimension reduction for PDEs and inverse problems

Dr. Meng is interested in analyzing and implementing low rank methods and dimension reduction techniques to facilitate efficient and robust algorithms for solving PDEs and inverse problems. Current research topics include: low dimensional methods for inverse scattering (based on generalized prolate spheriodal wave functions), kernel machine, as well as his most recent interest on implicit low rank methods for solving matrix and tensor differential equations and inverse problems.

(2) sampling methods in Inverse scattering

Different from classical iterative methods,  the so-called sampling (or qualitative) methods  require less a prior information (i.e., avoids info such as number of connectivity components and types of boundary conditions), and moreover, they provide solid mathematical justifications in addition to efficient numerical algorithms. The sampling methods utilize only the measurement data and therefore they are data-driven. Dr. Meng's research has been focused on development of sampling methods in complex media with various types of data. In addition to the well known capability of  sampling methods on shape identification, Dr. Meng and his collaborators are recently developing parameter estimation using the sampling methods.

(3) eigenvalue problem in inverse scattering

On the other hand, 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. They play a central role in the analysis of sampling methods. Dr. Meng's research has been focused on the spectral analysis of such (mostly non-selfadjoint) eigenvalues and their corresponding inverse problems.

(4) Wave propagation in periodic media

In recent decades, periodic and complex media have been used with remarkable success to manipulate waves toward achieving super-focusing, sub-wavelength imaging, cloaking, and topological insulation. Dr. Meng's research has been focosed on higher-order homogenization and novel physical phenomena in periodic structures.