Research
Inverse Problems
-- Inferring causal factors from observations
My research team focuses on integration of inverse problem analysis and computation, with the objective to develop fundamental mathematical and data exploration tools for the next generation of medical and geophysical imaging technologies.
Research support from the National Science Foundation (NSF) and National Institute of Health (NIH) are highly appreciated.
Inverse Problem Analysis
On the theoretical side, we develop analytic tools based on theory of partial differential equations, microlocal analysis, and differential geometry to establish uniqueness, stability, and reconstruction results for determination of medium parameters. These results provide unique mathematical insight for understanding computational challenges such as parameter identifiability, impact of noises, and algorithm development. Problems of particular interest include:
Inverse Boundary Value Problems
Inverse Boundary Value Problems concerns determination of medium parameters from infinite boundary Cauchy data. We have analyzed inverse boundary value problems in acoustics, optics, thermodynamics, and elasticity.
Selected Publications:
Reconstruction of the collision kernel in the nonlinear Boltzmann equation. (with R.-Y. Lai and G.
Uhlmann), SIAM Journal on Mathematical Analysis, 53(1), (2021), 1049-1069.
Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity (with J. Zhai). Inverse Problems and Imaging, 13(6), (2019), 1309–1325.
The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds (with P. Stefanov). Analysis & PDE, 11(6), (2018), 1381-1414.
Determining the first order perturbation of a polyharmonic operator on admissible manifolds (with Y. Assylbekov). Journal of Differential Equations., 262(1) (2017), 590-614.
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data. Journal of Differential Equations., 257 (2014), 3607–3639.
Geometric Inverse Problems
Geometric Inverse Problems aim to identify geometric quantities such as Riemannian metrics and vector fields. We have analyzed geometric inverse problems in Riemannian and Lorentzian geometries.
Selected Publications:
Travel time tomography in stationary spacetimes (with G. Uhlmann and H. Zhou). Journal of Geometric Analysis, 31(10) (2021): 9573-9596.
The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds (with P. Stefanov). Analysis & PDE, 11(6), (2018), 1381-1414.
Determination of the spacetime from local time measurements (with L. Oksanen and M. Lassas). Mathematische Annalen, 365(1), (2016), 271–307.
An inverse radiative transfer in refractive media equipped with a magnetic field (with Y. Assylbekov). Journal of Geometric Analysis, 25(4) (2015), 2148–2184.
Inverse Problem Computation & Tomographic Image Reconstruction
On the application side, we employ the insight gained from theoretical analysis to provide fundamental innovation for scientific computing and data science, resulting in novel tomographic image reconstruction algorithms with enhanced efficiency, improved accuracy, and mathematically-provable convergence. Problems of particular interest include:
Ultrasound Computed Tomography
Ultra-Sound Computed Tomography (USCT) utilizes ultrasound waves generated by active sources to create images of sound speed and acoustic attenuation. In contrast to 2D medical ultrasound, USCT provides 3D quantitative images with richer anatomical information and higher diagnostic value. Our team aims to develop non-iterative boundary control algorithms that can mitigate the "cycle-skipping" challenge, based on our theoretical advancement of acoustic inverse boundary value problems.
Selected Publications:
A linearized boundary control method for the acoustic inverse boundary value problem. (with L. Oksanen and T. Yang*), Inverse Problems, (2022) (accepted).
A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem. (with L. Oksanen and T. Yang), Inverse Problems & Imaging 16, no. 1 (2022): 1-18.
Photoacoustic Tomography
Photoacoustic Tomography, or Photo-Acoustic Computed Tomography (PACT), measures ultrasound responses triggered by laser illumination to image optical properties of biological tissue. PACT has the potential to break through the optical diffusion limit by coupling the ultrasonic resolution and optical contrast. Our team aims to develop mathematically-justified PACT imaging algorithms for (1) media with unknown acoustic properties, and (2) reduction of reflection artifacts, based on our theoretical advancement of acoustic inverse source problems.
Selected Publications:
Piecewise acoustic source imaging with unknown speed of sound. (with G. Huang* and J. Qian),, (2022), submitted.
Source independent velocity recovery using imaginary FWI (S. Qin, Y. Yang, and R. Wang). 82nd EAGE Annual Conference & Exhibition, Vol. (2021), no. 1, pp. 1-5. European Association of Geoscientists & Engineers.
Accelerated correction of reflection artifacts by deep neural networks in photo-acoustic tomography (H. Shan, G. Wang. and Y. Yang,). Applied Sciences, 9(13), (2019), 2615–2632.
Thermo and Photoacoustic Tomography with variable speed and planar detectors (with P. Stefanov). SIAM Journal of Mathematical Analysis, 49(1), (2017), 297-310.
Multiwave tomography with reflectors: Landweber’s iteration (with P. Stefanov). Inverse Problems and Imaging, 11(2), (2017), 373-401.
Multiwave tomography in a closed domain: averaged sharp time reversal (with P. Stefanov). InverseProblems, 31(6) (2015), 065007.
Optical Tomography
Optical Tomography images spatial distribution of optical parameters from light transmission and scattering. It is non-invasive and economical for superficial tissue imaging. Our research studies multiple optics-based imaging methods in the presence of ultrasound modulation.
Selected Publications:
Inverse source problem for acoustically-modulated electromagnetic waves. (with W. Li, J. Schotland, and Y. Zhong), SIAM Applied Mathematics, (2022), accepted.
An acousto-electric Inverse Source Problem. (with W. Li, J. Schotland, and Y. Zhong), SIAM Journal of Imaging Sciences, 14(4), (2021), 1601–1616.
Inverse transport problem in fluorescence ultrasound modulated optical tomography with angularly-averaged measurements. (with W. Li and Y. Zhong), Inverse Problems, 36(2), (2020), 025011.
Ultrasound modulated bioluminescence tomography with a single optical measurement. (with F. Chung and T. Yang), Inverse Problems, 37(1), (2020), 015004.
A hybrid inverse problem in the fluorescence ultrasound modulated optical tomography in the diffusive regime (with W. Li and Y. Zhong). SIAM Applied Mathematics, 79(1), (2019), 356-376.
