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. 

(1) Low rank STRUCTURE 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.

1. Preconditioning low rank Generalized Minimal Residual Method (GMRES) for implicit discretizations of matrix differential equations, arXiv preprint (with D Appelo, and Y Cheng)

2. Exploring low-rank structure for an inverse scattering problem with far-field data, arXiv preprint (with Y Zhou, L Audibert, and B Zhang)

3. A Kernel machine learning for inverse source and scattering problems, SIAM J. Numer. Anal. 62 (2024), no. 3, pp. 1443–1464  (with B. Zhang)

4. Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator, Inverse Problems 40 (2024), 095007, 30pp (with L. Audibert)

5. Data-driven Basis for Reconstructing the Contrast in Inverse Scattering: Picard Criterion, Regularity, Regularization, and Stability, SIAM J. Appl. Math. 83. (5), 2003-2026 (2023)

6. 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)

(2) numerical method and mathematical analysis for Inverse scattering

Inverse scattering merits important applications in medical imaging, non-destructive testing, marine seismic imaging and many other areas. Dr. Meng's research has been focused on the numerical method and mathematical analysis for inverse scattering in complex media with various types of data. 

1. An inverse obstacle scattering problem with random sources in the time domain, preprint (with X Liu, J Tian, and B Zhang)

2. Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator, Inverse Problems 40 (2024), 095007, 30pp (with L. Audibert)

3. Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides, SIAM J. Appl. Math. 83. (2), 394-417 (2023)

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

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

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

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

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

9. 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)

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

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

(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. Dr. Meng's research has been focused on the spectral analysis of such (mostly non-selfadjoint) 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)

(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.

1. Plug-and-play analytical paradigm for the scattering of plane waves by “layer-cake” periodic systems, Proc. R. Soc. A. 480 (2024), no. 2294, 2024118 (with P Salasiya and B Guzina)

2. 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)

3. 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)

4. 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)

5. 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)

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

7. 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)

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