My focus has been on investigating the mathematical theory of inverse problems arising in imaging sciences as well as in various areas of physics and engineering. To study these problems, I use microlocal analysis and partial differential equation (PDE) methods. 

• Inverse problems for nonlinear hyperbolic equations. Nonlinear hyperbolic equations apprear in many areas of physics and engineering. Typically, when exploring a medium, we emit waves, such as ultrasound or electromagnetic wave, and take measurements. A challenge lies in reconstructing the metric or nonlinearity based on these measurements. This area of study encompasses questions about unique determination, stability, and the development of effective reconstruction algorithms. 

Our current work focuses on unique determination of the time-dependent metric and nonliearity for nonlinear hyperbolic equations from boundary measurements. The key idea is to use the nonlinearity as a tool that helps us solve the inverse problems. It is the nonlinearity that enables us to prove the recovery, even when the corresponding linear inverse problem remains unresolved (as the boundary control method depends on on a sharp unique continuation result proved by Tataru, which only holds for linear equations with coefficients analytically in time). We use a multi-fold linearization and the nonlinear interaction of waves to produce new waves, analyzing the new singularities using a finer calculi of FIOs and paired Lagrangian distributions, with the help of b-calculus.

Related publications and preprints:

[11] Inverse Boundary Value Problems for Wave Equations with Quadratic Nonlinearities, joint work with Gunther Uhlmann, Journal of Differential Equations.

[10] An inverse boundary value problem arising in nonlinear acoustics, joint work with Gunther Uhlmann, SIAM Journal on Mathematical Analysis,

[9] Nonlinear acoustic imaging with damping, arXiv preprint arXiv:2302.14174.

[8] Inverse problems for a nonlinear Klein-Gordon equation, joint work with Peter Hintz, Katya Krupchyk, Gunther Uhlmann, in preparation. 

• Inverse problems in nonlinear acoustic imaging. Ultrasound waves, commonly used in medical imaging, can be modeled by nonlinear wave equations when propagated at high intensities. A fundamental model to describe the propagation of these waves is the Westervelt equation. Additionally, vairous damping effects exist for wave equations in many fields of physics and engineering. Inverse problems in this context involve reconstructing the metric, nonlinearity, and the damping coefficient.

The models that we consider includes nonlinear wave equations with a weak damping, the Westervelt equation with a space-fractional damping, and the Westervelt equation with a strong damping. 

Related publications and preprints: [10], [9] and 

[7] An inverse problem for the fractionally damped wave equation, joint work with Li Li, arXiv:2307.16065, submitted.

[6] Inverse problems for a quasilinear strongly damped wave equation arising in nonlinear acoustics, joint work with Li Li, arXiv:2309.11775.

[5] On inverse problems for a strongly damped wave equation on compact manifolds, joint work with Li Li, arXiv:2309.16182, submitted.

• Inverse problems in integral geometry and medical imaging. In medical imaging and biomedical research, a crucial inverse problem is to reconstruct an unknown density function (the attenuation coefficient) of a medium from its integral transform over specific family of curves or surfaces. Computed Tomography (CT) scans, for instance, reconstruct this density function from its integral transform over straight lines. Central challenges for these inverse problems encompass determining the injectivity and stability of the integral transform and devising methods for its inversion. 

We use microlocal methods to study integral transforms arising in medical imaging, such as Compton camera imaging and the X-ray luminescence computed tomography. We describe what kind of singularities can be stably recovered from the local data and prove the mapping properties with a microlocal stability estimate. In some cases, the microlocal stability is lost due to the existence of conjugate points, and therefore artifacts are avoidable in the reconstruction. This phenomenon is usually referred to as “cancellation of singularities” by conjugate points, which appears in geodesic X-ray transform and Synthetic Aperture Radar imaging. We perform numerical reconstructions using Landweber’s iteration to illustrate our results for local and global recovery of singularities in the broken ray transform.

Related publications and preprints: 

[4] Recovery of singularities for the weighted cone transform appearing in the Compton camera imaging, Inverse Problems.

[3] The X-ray transform on a generic family of smooth curves, The Journal of Geometric Analysis.

[2] Artifacts in the inversion of the broken ray transform in the plane, Inverse Problems and Imaging.

• Mircolocal analysis in linear elasticity. Rayleigh waves, first studied by Lord Rayleigh, are surface waves in linear elasticity that can be particularly destructive during earthquakes. These waves travel along traction-free boundaries and diminish quickly within the media. They exhibit a retrograde elliptical particle motion in shallow depths, especially with flat and homogeneous media. Stoneley waves, on the other hand, are interface waves between two different solids. Most geophysical studies on these waves address specific scenarios. We use microlocal methods to understand microlocal behaviors of these waves, in an isotropic linear elastic system with variable coefficients and a curved boundary or interface.

Related publications and preprints: 

[1] Rayleigh and Stoneley Waves in Linear Elasticity, Asymptotic Analysis.