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 computational and analytical methods for solving PDEs and inverse problems, and a major theme is on forward and inverse scattering problems. He has been working on reconstruction methods for inverse scattering (sampling methods and dimension reduction techniques based on the generalized prolate spheroidal wave functions), eigenvalues and non-destructive testing (transmission eigenvalue, Stekloff eigenvalue, band structure and other non-selfadjoint eigenvalue problems), and homogenization for periodic media (higher order homogenization at low/high frequencies). His most recent interest is on low rank methods and dimension reduction techniques.

(1) Low rank and dimension reduction for PDEs and inverse problems

Dr. Meng is interested in analyzing and implementing low rank methods and dimension reduction techniques to facilitate efficient and robust algorithms for solving PDEs and inverse problems. Current research topics include: low dimensional methods for inverse scattering (based on generalized prolate spheriodal wave functions), kernel machine, as well as his most recent interest on implicit low rank methods for solving matrix and tensor differential equations and inverse problems.

1. Kernel machine learning for inverse source and scattering problems, preprint (with B. Zhang)

2. Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator, ArXiv preprint arXiv:2306.16199 (with L. Audibert)

3. 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)

4. 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) 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 are 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 collaborators are recently developing parameter estimation using the sampling methods.

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

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

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

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

7. 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)

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

9. 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. 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.

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 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, https://doi.org/10.1093/qjmam/hbab003 (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, https://doi.org/10.1080/00036811.2021.1929932 (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)