Contact :
Address :.
Department of Mathematics,
Purdue University
West Lafayette, IN 479072067
Office :
Math 426
Phone (office) :
(765) 494 1986
Email :
Research Interests :
My research focuses on inverse problems in the broad sense. The topics I have been working on include coupled physics inverse problems in medical and geophysical imaging, , some inverse problems in Lorentzian geometry with connection to astrophysics and the theory of relativity, and inverse boundary value problems from tomography methods.
1. Coupled Physics Inverse Problems :
These problems appear in recent hybrid imaging methods where one combines high resolution modalities and high contrast modalities. Typical examples include Photoacoustic Tomography (PAT) and Thermoacoustic Tomography (TAT) in medical imaging, and Seismoelectrical (SE) method and Electroseismic (ES) method in geophysical imaging. With my collaborators, we

studied the inverse source problem in PAT/TAT with the presence of sound hard reflectors and proposed two Neumann series reconstruction schemes, one based on the averaged time reversal (see the figures below) while the other on the Landweber method.

studied the inverse source problem in PAT/TAT with planar detectors; obtained sharp times for uniqueness and stability, and reconstruction procedures to restore the singularities of the source.
Figure 2: nonaveraged time reversal * using 1 term in the Neumann series
Figure 3: averaged time reversal
using 1 term in the Neumann series
Figure 1: original SheppLogan phantom
Figure 4: averaged time reversal
using 10 term in the Neumann series

considered the inverse diffusion regime in quantitative PAT with partial data and proved uniqueness and stability on the recovery of the diffusion and absorption coefficients.

studied an inverse Maxwell's equations with internal measurement arising in ES imaging and proved uniqueness and stability in the determination of the conductivity and the coupling coefficient.
Related articles :

Quantitative photoacoustic tomography with partial data, (with J. Chen), Inverse Problems, 28 (2012), 115014.

Inverse problem of electroseismic conversion, (with J. Chen), Inverse Problems, 29 (2013), 115006.

Multiwave tomography in a closed domain: averaged sharp time reversal, (with P. Stefanov), Inverse Problems, 31(6) (2015), 065007.

Thermo and Photoacoustic Tomography with variable speed and planar detectors, (with P. Stefanov), SIAM J. Math. Anal. 49(1), (2017), 297310.

Application of CGO solutions on inverse problems of coupled physics imaging methods, (with I. Kocyigit, R.Y. Lai, L. Qiu and T. Zhou), Inverse Problems and Imaging 11(2), (2017), 277304.

Multiwave tomography with reflectors: Landweber's iteration, (with P. Stefanov), Inverse Problems and Imaging 11(2), (2017), 373401.
3. Inverse Boundary Value Problems :
An inverse boundary value problem concern recovery of medium paramters from the DirichlettoNeumann map. We proved

for the perturbed wave operator in Lorentzian geometry, derivatives of the Lorentzian metric, a magnetic field, and a potential can be stably determined on the boundary up to gauge transformations; moreover, the lens relation of the metric and the light ray transforms of the field and the potential can be stably recovered.

for the perturbed polyharmonic operators, a magnetic potential/field and a potential can be identified in the Euclidean geometry and some conformally transversally anisotropic geometry.
Related articles :

Determining the first order perturbation of a biharmonic operator on bounded and unbounded domains from partial data, J. Differ. Equations., 257 (2014), 36073639.

Determining the first order perturbation of a polyharmonic operator on admissible manifolds, (with Y. Assylbekov), J. Differ. Equations, 262(1) (2017), 590614.

The inverse problem for the DirichlettoNeumann map on Lorentzian manifolds, (with P. Stefanov), submitted, arXiv: 1607.08690.

An inverse boundary value problem for the magnetic Schrödinger operator with a bounded magnetic potential in a slab, (with S. Liu), preprint, arXiv: 1311.1576.
2. Inverse Problems in Lorentzian Geometry :
Many inverse problems in astrophysics and the theory of relativity can be naturally formulated in the Lorentzian geometry. We are interested in retrieving information on the Lorentzian metric from timelike or lightlike measurement. We proved

derivatives of the metric in Fermi coordinates can be determined from local travel times of material particles; moreover, if the Lorentzian manifold is realanalytic and satisfies a few topological and geometric conditions, its universal covering space can be determined as well from the measurement.
Related articles :

Determination of the spacetime from local time measurements, (with L. Oksanen and M. Lassas), Math. Ann., (2015), DOI: 10.1007/s0020801512869.

The inverse problem for the DirichlettoNeumann map on Lorentzian manifolds, (with P. Stefanov), submitted, arXiv: 1607.08690.
* This is the time reversal proposed by Stefanov and Uhlmann in the absence of reflectors in their work:
Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011.