# 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.

Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides, arXiv:2207.04645 (2022), accepted SIAM J. Appl. Math.

Data completion algorithms and their applications in inverse acoustic scattering with limited-aperture backscattering data, Journal of Computational Mathematics 469, 111550 (2022) (with F.Dou, X. Liu, and B. Zhang)

Modified sampling method with near field measurements, SIAM J. Appl. Math 82 (1), 244-266 (2022) (with X. Liu and B. Zhang)

A multi-frequency sampling method for the inverse source problems with sparse measurements, arXiv:2109.01434 (2021) (With X. Liu)

A Sampling Type Method in an Electromagnetic Waveguide, Inverse Problems & Imaging doi: 10.3934/ipi.2021012 (2021)

The Interior Inverse Electromagnetic Scattering for an Inhomogeneous Cavity, Inverse Problems 37 025007 (2021) (With F. Zeng)

Factorization method versus migration imaging in a waveguide, Inverse Problems 35 (2019), no. 12, 124006, 33 pp (With L. Borcea)

A direct approach to imaging in a waveguide with perturbed geometry, Journal of Computational Physics, 392, 556–577 (2019) (with L. Borcea and F. Cakoni)

The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30, 045008 (2014) (with F. Cakoni and H. Haddar)

The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemporary Mathematics, 615, 71-88 (2014) (with F. Cakoni and D. Colton)

## (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.

A Note on Transmission Eigenvalues in Electromagnetic Scattering Theory, Inverse Problems & Imaging doi: 10.3934/ipi.2021025 (2021) (with F. Cakoni and J. Xiao)

Spectral analysis of the transmission eigenvalue problem for Maxwell’s equations, arXiv:1707.04815, Journal de Mathématiques Pures et Appliquées 120, 1-32 (2018) (with H. Haddar)

Modified transmission eigenvalues in inverse scattering theory, Inverse Problems, 33, 125002 (2017) (with S. Cogar, D. Colton and P. Monk)

Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math. 76 (4), 1737–1763 (2016) (with D. Colton, F. Cakoni and P. Monk)

Boundary integral equations for the transmission eigenvalue problem for Maxwell’s equations, J. Int. Eqns. Appl. 27 (3), 375-406 (2015) (with F. Cakoni and H. Haddar)

Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31, 035006 (2015) (with D. Colton and Y.J. Leung)

Spectral properties of the exterior transmission eigenvalue problem, Inverse Problems, 30, 105010 (2014) (with D. Colton)

The inverse spectral problem for exterior transmission eigenvalues, Inverse Problems, 30, 055010 (2014) (with D. Colton and Y.J. Leung)

## (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.

Asymptotic anatomy of the Berry phase for scalar waves in 2D periodic continua, Proceedings of the Royal Society A 478, 2262, (2022) (with B. Guzina and O. Oudghiri-Idrissi)

On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions, The Quarterly Journal of Mechanics and Applied Mathematics, https://doi.org/10.1093/qjmam/hbab003 (2021) (with B. Guzina and O. Oudghiri-Idrissi)

A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term, Applicable Analysis, https://doi.org/10.1080/00036811.2021.1929932 (2021) (with B. Guzina and O. Oudghiri-Idrissi)

A rational framework for dynamic homogenization at finite wavelengths and frequencies, Proceedings of Royal Society A 475: 20180547 (2019) (with B. Guzina and O. Oudghiri-Idrissi)

Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium, Journal of Elasticity (2019) (with Y. Lin)

Determination of electromagnetic Bloch modes in a medium with frequency-dependent coefficients, Journal of Computational and Applied Mathematics 358, 359-373 (2019) (with C. Lackner and P. Monk)

On the dynamic homogenization of periodic media: Willis’ approach versus two-scale paradigm, Proceedings of Royal Society A 474 20170638 (2018) (with B. Guzina)