Research

I am interested in wave propagation and its inverse problems. My long-term research goal is to address interdisciplinary challenges spanning science, engineering, and technology.

During my Ph.D. from the University of Delaware, I have been working extensively on the development of qualitative methods in inverse scattering and the associated eigenvalue problems. During my employment at University of Minnesota and University of Michigan, I have developed my research in three main topics: (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.

In the following, I describe the highlights of my previous and current work on these three research topics. 

(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. I use tools  from partial differential equations, functional analysis and scientific computing to develop and analyze inversion algorithms for complex media, in a variety of application relevant setups.

During my Ph.D. work, I have investigated the inverse scattering problem for a cavity that is bounded by a penetrable anisotropic inhomogeneous medium of compact support, where the goal is to determine the shape of the cavity from internal measurements on a curve or surface inside the cavity. During my employment at the University of Michigan, I started working on imaging in waveguides with  variable and unknown geometry. Currently, I am working on imaging in waveguide using time domain measurements.

  1. Factorization method versus migration imaging in a waveguide, Inverse Problems, accepted (2019) (With L. Borcea)
  2. 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)
  3. The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30, 045008 (2014) (with F. Cakoni and H. Haddar)
  4. 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. Those eigenvalues are also related to non-scattering incident fields.

During my Ph.D. work, I  have investigated the transmission eigenvalues and Stekloff eigenvalues. I have extended my eigenvalue research to include modified transmission eigenvalue and  Bloch varieties during my employment at the University of Minnesota and the University of Michigan.  

  1. Spectral analysis of the transmission eigenvalue problem for Maxwell’s equations, Journal de Mathématiques Pures et Appliquées 120, 1-32 (2018) (with H. Haddar)
  2. Modified transmission eigenvalues in inverse scattering theory, Inverse Problems, 33, 125002 (2017) (with S. Cogar, D. Colton and P. Monk)
  3. Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math. 76 (4), 1737–1763 (2016) (with D. Colton, F. Cakoni and P. Monk)
  4. 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)
  5. Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31, 035006 (2015) (with D. Colton and Y.J. Leung)
  6. Spectral properties of the exterior transmission eigenvalue problem, Inverse Problems, 30, 105010 (2014) (with D. Colton)
  7. 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. The Willis’ approach has attracted much attention in the engineering community.

During my employment at the University of Minnesota and the University of Michigan, I have worked on Willis’ approach, higher-order asymptotic methods, and homogenization for bounded periodic media.

  1. 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)
  2. Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium, Journal of Elasticity (2019) (with Y. Lin)
  3. 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)
  4. On the dynamic homogenization of periodic media: Willis’ approach versus two-scale paradigm, Proceedings of Royal Society A 474 20170638 (2018) (with B. Guzina)