Jian Zhai's Homepage

I am an Associate Professor in the School of Mathematical Sciences at Fudan University, My research interests lie in inverse problems, partial differential equations, microlocal analysis and scientific computing.

Employment

Asssociate Professor, School of Mathematical Sciences, Fudan University, Sep. 2021 - Now
Postdoctoral Fellow, IAS, The Hong Kong University of Science and Technology, Mar. 2019 - Sep. 2021
Visiting Lecturer, Department of Mathematics, University of Washington, Sep. 2018 - Mar. 2019
Postdoctoral Fellow, Department of Computational and Applied Mathematics, Rice University, July 2018 - Aug. 2018

Education

Ph. D. in Computational and Applied Mathematics, May 2018, Rice University
Ph. D. Student in Mathematics, Aug. 2013 - Aug. 2015, Purdue University
M. S. in Mathematics, June 2013, Fudan University
B. S. in Mathematics, June 2010, Sichuan University

Publications

Determination of the density in the linear elastic wave equation, preprint, arXiv:2503.12825.
(with G. Uhlmann) Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors, preprint, arXiv:2402.12640
(with G. Uhlmann) Invertibility of local geodesic transverse and mixed ray transforms I: basic cases, preprint, arXiv:2401.09017
(with G. Uhlmann) Determination of the density in a nonlinear elastic wave equation, Math. Ann., 390 (2024) 2825–2858.
(with S. Lu)Increasing stability of a linearized inverse boundary value problem for a nonlinear Schrödinger equation on transversally anisotropic manifolds, Inverse Problems, 40 (2024) 045003.
(with Y. Zhao)Determination of piecewise homogeneous sources for elastic and electromagnetic waves, Inverse Probl. Imaging, 3 (2023) 614-628.
(with R. Lai, G. Uhlmann and H. Zhou)Single pixel X-ray transform and related inverse problems, SIAM J. Imaging Sci., 15 (2022) 1894-1909.
(with S. Acosta and G. Uhlmann) Nonlinear ultrasound imaging modeled by a Westervelt equation, SIAM J. Appl. Math., 82 (2022) 408–426.
(with P. Hintz and G. Uhlmann) The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds, Comm. Partial Differential Equations, 47 (2022) 1894-1909.
(with G. Bao and X. Xu) Inverse spectral problem for a damped wave operator, SIAM J. Appl. Math., 81 (2021) 1799-1820.
(with P. Li and Y. Zhao) Lipschitz stability for an inverse source scattering problem at a fixed frequency, Inverse Problems, 37 (2021) 025003.
(with P. Hintz and G. Uhlmann) An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds, Int. Math. Res. Not., 2022 (2022) 13181-13211.
(with G. Uhlmann) Inverse problems for nonlinear hyperbolic equations, Discrete Contin. Dyn. Syst., 41 (2021) 455-469.
(with P. Li and Y. Zhao) Stability for the acoustic inverse source problem in inhomogeneous media, SIAM J. Appl. Math., 80 (2020) 2547–2559.
(with G. Uhlmann) On an inverse boundary value problem for a nonlinear elastic wave equation, J. Math. Pures Appl., 153 (2021) 114-136.
(with M. V. de Hoop, T. Saksala and G. Uhlmann) Generic uniqueness and stability for the mixed ray transform, Trans. Amer. Math. Soc., 374 (2021), 6085-6144.
(with M. V. de Hoop, A, Iantchenko and R. D. van der Hilst) Semiclassical inverse spectral problem for seismic surface waves in isotropic media: part II. Rayleigh waves, Inverse Problems, 36 (2020) 075016.
(with M. V. de Hoop, A, Iantchenko and R. D. van der Hilst) Semiclassical inverse spectral problem for seismic surface waves in isotropic media: part I. Love waves, Inverse Problems, 36 (2020) 075015.
(with X. Xu) Inversion of trace formulas for a Sturm-Liouville operator, J. Comput. Math., 140 (2022) 398-416.
(with M. V. de Hoop and G. Uhlmann) Inverting local geodesic ray transform of higher rank tensors, Inverse Problems, 35 (2019) 115009.
(with M. V. de Hoop and T. Saksala) Mixed ray transform on simple 2-dimensional Riemannian manifolds, Proc. Amer. Math. Soc., 147 (2019) 4901-4913.
(with Y. Yang) Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity, Inverse Probl. Imaging, 13 (2019) 1309-1325.
(with M. V. de Hoop and G. Nakamura) Unique recovery of piecewise analytic density and stiffness tensor from the elastic-wave Dirichlet-to-Neumann map, SIAM J. Appl. Math., 79 (2019) 2359-2384
(with M. V. de Hoop, A. Iantchenko and G. Nakamura) Semiclassical analysis of elastic surface waves, preprint, arXiv:1709.06521.
(with M. V. de Hoop and G. Nakamura) Reconstruction of Lamé moduli and density at the boundary enabling directional elastic wavefield decomposition, SIAM J. Appl. Math., 77 (2017) 520-536.
(with E. Beretta, M. V. de Hoop, E. Francini and S. Vessella) Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017) 035013.

Expository work

Notes on tensor tomography on Riemannian manifolds with boundaries, arXiv:2009.01459

Selected Presentations

SciCADE 2024, Minisymposium on Theory and Numerics of Inverse Problems, Singapore, July 2024
The 11th International Conference "Inverse Problems: Modeling and Simulation", Minisymposium on Microlocal Analysis and Waves: Progress and Challenges, Malta, May 2024
Differential Geometry and PDE Seminar, University of Washington, Seattle, WA, USA, May 2024
IAS workshop on Inverse Problems, Imaging and Partial Differential Equations, HKUST, Hong Kong, China, Dec. 2023.
ICIAM 2023 Tokyo, Minisymposium on Advances in Inverse Problems and Imaging, Tokyo, Japan, Aug. 2023
The 12th Conference on Inverse Problems, Imaging and Applications, Southern University of Science and Technology, Shenzhen, China, May 2023.
CSIAM Annual Meeting, Minisymposium on Inverse Problems and Imaging, Nov. 2022.
Geometrical Inverse Problems, Linz, Austria, Nov. 2022.
Joint Workshop on Applied and Computational Mathematics, online, Nov. 2021
Joint Fudan-RICAM Seminar on Inverse Problems, online, Dec. 2020.
Seminar, Southern University of Science and Technology, online, Jun. 2020.
Minisymposium on Recent Advances in Geometric Inverse Problems, Applied Inverse Problems Conference, Grenoble, France, July 2019.
Minisymposium on Inverse Problems in Elastic Medium, Applied Inverse Problems Conference, Grenoble, France, July 2019.
Seminar, Zhejiang University, Hangzhou, China, Jun. 2019.
The 11th Conference on Inverse Problems, Imaging and Applications, Lanzhou, China, Jun. 2019.
The 5th East Asia Section of IPIA Young Scholars Symposium, Beijing, China, Jun. 2019.
Seminar, Hong Kong University of Science and Technology, Hong Kong, China, May. 2019.
Canadian Mathematical Society Winter Meeting, Vacouver, Canada, Dec. 2018.
Differential Geometry and PDE Seminar, University of Washington, Seattle, WA, USA, Oct. 2018.
International Workshop on Inverse Problems for PDEs, Nanjing, China, Sep. 2018.
SIAM Annual Meeting, Portland, OR, USA, July 2018.
IAS Workshop on Inverse Problems, Imaging and PDEs, HKUST, Hong Kong, China, Dec. 2016.
Seminar, Fudan University, Shanghai, China, May 2016.