Research

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 numerical analysis, applied analysis, scientific computing and machine learning, with applications to solving PDEs and inverse problems.  A major theme of his research is on wave propagation and inverse scattering problems. His most recent interest is exploring low rank structures in PDEs and inverse problems.


(1) Low rank for inverse problems and PDEs

While low-rank approximations using singular value decomposition have been well understood for over a century, a comprehensive framework for low-rank structures in the context of inverse scattering has yet to be developed. To lay the mathematical and algorithmic foundation of data science, Dr. Meng's research focuses on low rank structures for inverse scattering and machine learning. 

Dr. Meng is also interested in low rank methods for solving matrix and tensor differential equations.



(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 can be seen as 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 collaborations recently discovered the capability of sampling methods in parameter estimation  and are developing such parameter estimation for a variety of problems.



(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 reduced-order modelling, higher-order homogenization and novel physical phenomena in periodic structures.