# Research

In general, Dr. Meng is broadly interested in computational mathematics, inverse problems, machine learning, 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 wave propagation and inverse problems: such as inverse scattering problem, inverse source problem, eigenvalue problem, homogenization, periodic structure, reduced order modeling, and machine learning.

## (1) Learning theory and algorithm for inverse problems

In recent decades, machine learning has demonstrated its enormous success in recognition, computer vision, and many problems in science, engineering, economics and technology. Dr. Meng is interested in exploring powerful learning theory and algorithm to develop more efficient and robust imaging algorithms with better accuracy evaluated within a reduced order model setting. Current research topics include: learning data-driven basis, feature learning, kernel machine, and deep learning.

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

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

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)

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 method and eigenvalue problem 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 such as (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), and (4) multi-frequency data.

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

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)

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 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 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.

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)