Dr. Meng is interested in wave propagation and its inverse problems. His long-term research goal is to address interdisciplinary challenges spanning science, engineering, and technology. His research has been focused on (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

The mathematical theory of wave scattering describes the interaction of waves (e.g., acoustic, electromagnetic, or elastic) with natural or manufactured perturbations of the medium through which they propagate. The goal of inverse scattering (or in short imaging) is to estimate the medium from observations of the wave field. It has applications in a broad spectrum of scientific and engineering disciplines, including seismic imaging, radar, astronomy, medical diagnosis, and non-destructive material testing. Dr. Meng's research has been focused on robust and efficient sampling methods, such as linear sampling method, factorization method, and other related imaging methods.

  1. Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides, arXiv:2207.04645 (2022)

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

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

  4. Data completion algorithms and their applications in inverse acoustic scattering with limited-aperture backscattering data, arXiv:2106.11101 (2021) (with F.Dou, X. Liu, and B. Zhang)

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

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

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

  8. 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)

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

  10. 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.

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

  2. 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)

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

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

  5. 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)

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

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

  8. 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.

  1. 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)

  2. On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions, The Quarterly Journal of Mechanics and Applied Mathematics, (2021) (with B. Guzina and O. Oudghiri-Idrissi)

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

  4. 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)

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

  6. 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)

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