Research

Dr. Meng is broadly interested in applied and computational mathematics, inverse problems, and data science. 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

1. Preconditioning low rank Generalized Minimal Residual Method (GMRES) for implicit discretizations of matrix differential equations, SIAM J. Matrix Anal. Appl 46 (4), 2475-2500 (with D Appelo, and Y Cheng)

2. Exploring low-rank structure for an inverse scattering problem with far-field data, in print, SIAM J. Appl. Math. (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. 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)

5. 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) Direct imaging method for Inverse scattering

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

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

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

4. A multi-frequency sampling method for the inverse source problems with sparse measurements, CSIAM Trans. Appl. Math. (2021) (With X. Liu)

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)

(3) eigenvalue problem in inverse scattering

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 and Metamaterials

1. On the surface Bloch waves in truncated periodic media, arxiv preprint (with B Guzina, P Salasiya, and Long Nguyen)

2. 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)

3. 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)

4. 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)

5. 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)

6. 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)

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

8. 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)

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