Research

Overview

My research interests include inverse problems for linear, nonlinear, fully nonlinear, and nonlocal partial differential equations, as well as their connections with integral geometry and geometric analysis.

Inverse problems

Inverse problems seek to recover hidden structures or physical properties from indirect observations. Examples include satellite remote sensing, LiDAR, seismic exploration, wave scattering, and medical imaging. The central questions concern uniqueness, stability, and reconstruction: what can the data determine, how sensitive is the answer to measurement errors, and how can the unknown information be recovered?

Satellite sensing, airborne LiDAR, wave scattering, and subsurface exploration in a cutaway landscape.
Indirect measurements across different scales.

Nonlinear and fully nonlinear inverse problems

Nonlinear equations describe responses that need not be proportional to the applied input. Their inverse problems ask how measurements can identify unknown nonlinearities, coefficients, or sources.

My work ranges from semilinear equations to fully nonlinear models, including Monge–Ampère, prescribed Gaussian curvature, k-Hessian, and infinity Laplacian equations. A recurring question is how different boundary inputs can reveal information encoded in the nonlinear response.

A schematic solution surface with highlighted local curvatures, boundary inputs, and nonlinear responses.
Boundary inputs, nonlinear responses, and interior information.

Nonlocal inverse problems

Nonlocal equations describe interactions across distances. In fractional models, inputs and measurements in an exterior region can reveal information about an inaccessible interior.

My work includes recovery of leading coefficients and sources, identification of angular kernels of stable operators, and monotonicity-based reconstruction. Stability estimates, including Lipschitz stability under suitable finite-dimensional assumptions, address how recovery is affected by measurement errors.

Exterior input and response regions coupled to an unknown interior through nonlocal interactions.
Probing an unknown interior from the exterior.

Geometric inverse problems

Geometric inverse problems ask whether measurements determine an unknown interior geometry. A central example is the anisotropic Calderón problem, motivated by electrical impedance tomography in media with direction-dependent conductivity.

Boundary fixing changes of coordinates leave the measurements unchanged, creating a natural ambiguity known as gauge freedom. Determining geometry up to this freedom remains a central open problem for general smooth metrics in dimensions three and higher.

Two coordinate representations with matching boundary nodes and different interior grids.
Different interior coordinates can give identical boundary measurements.

Scattering, elasticity and cloaking

Scattering problems study how waves interact with obstacles and inhomogeneous materials. Their inverse counterparts use the observed fields to recover hidden objects, sources, or material parameters.

My interests include acoustic and electromagnetic scattering, elastic systems, and near-cloaking. Cloaking explores the complementary question: can a surrounding medium make an object undetectable, or nearly undetectable, from exterior measurements? Together these problems examine the possibilities and limits of wave-based probing.

Comparison of a detectable scattered field around an obstacle and an idealized cloak guiding waves around it.
Detectable scattering and an idealized cloaking response.

Preprints and accepted manuscripts are listed below as PDFs. Please consult the published journal versions for the final articles. My preprints are also available on arXiv; further publication information can be found on MathSciNet and Google Scholar.

Submitted articles
  1. The anisotropic Calderón problem: rigidity near the Euclidean metric
    preprint. [arXiv]
  2. The Calderón problem for the infinity Laplacian with large affine boundary data
    preprint. [arXiv]
  3. Affine section tomography for inverse source problems in k-Hessian equations with restricted large boundary data
    preprint. [arXiv]
  4. Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation
    preprint. [arXiv]
  5. (with C.-L. Lin and J.-N. Wang) An inverse source problem for a fully nonlinear elliptic equation
    preprint. [arXiv]
  6. Recovering stable kernels from exterior measurements
    preprint. [arXiv]
  7. Entanglement principle for the fractional Laplacian on hyperbolic spaces and applications to inverse problems
    preprint. [arXiv]
  8. (with R.-Y. Lai and L. Yan) Entanglement principle and fractional Calderón problem for nonlocal parabolic operators
    preprint. [arXiv]
  9. (with T. Liimatainen) An inverse problem for the Monge–Ampère equation
    preprint. [arXiv]
  10. Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation
    preprint. [arXiv]
  11. (with A. Feizmohammadi) Entanglement principle for the fractional Laplacian with applications to inverse problems
    preprint. [arXiv]
  12. (with G. Nakamura and P. Zimmermann) Partial data Calderón problem for a transversally anisotropic Schrödinger equation
    preprint. [arXiv]
  13. (with P. Zimmermann) Approximation and uniqueness results for the nonlocal diffuse optical tomography problem
    preprint. [arXiv]
  14. (with C.-L. Lin and G. Uhlmann) A nonlocal-to-local transfer principle for parabolic Calderón problems
    preprint. [arXiv]
  15. (with P. Zimmermann) Unique determination of coefficients and kernel in nonlocal porous medium equations with absorption term
    preprint. [arXiv]
  16. (with C.-L. Lin and G. Uhlmann) The Calderón problem for nonlocal parabolic operators
    preprint. [arXiv]
Monographs
  1. (with H. Liu) Inverse Problems for Integro-differential Operators
    Applied Mathematical Sciences Series, Volume 222, Springer, Switzerland (2025).
    Springer
Articles in refereed scientific journals
  1. (with D.-H. Chen and I. Yousept) Finite element error analysis for elliptic parameter identification with power-type nonlinearity
    SIAM J. Numer. Anal., to appear.
    2. [arXiv]
  2. The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds
    Commun. Anal. Comput., Vol. 7 (2026), 60-75.
    3. [Journal] [arXiv]
  3. (with T. Tyni and P. Zimmermann) Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities
    Cal. Var. Partial Differential Equations, Vol. 64, article number 280 (2025), 1-43.
    4. [Journal] [arXiv]
  4. (with B. Harrach and T. Weth) The Calderón problem for the logarithmic Schrödinger equation
    J. Differential Equations, Vol. 444, 113665 (2025), 1-25.
    5. [Journal] [arXiv]
  5. (with J. Railo and P. Zimmermann) The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients
    Rev. Mat. Iberoam., Vol. 41, No. 3 (2025), 1129–1172.
    6. [Journal] [arXiv]
  6. Determining both leading coefficient and source in a nonlocal elliptic equation
    J. Inverse Ill-Posed Probl., Vol. 33, No. 2 (2025), 207-216.
    7. [Journal] [arXiv]
  7. (with M. Kar and P. Zimmermann) Determining coefficients for a fractional p-Laplace equation from exterior measurements
    J. Differential Equations, Vol. 406 (2024), 338-365.
    8. [Journal] [arXiv]
  8. (with T. Tyni and P. Zimmermann) Well-posedness and inverse problems for semilinear nonlocal wave equations
    Nonlinear Anal., Vol. 24 (2024), 113601, 1-19.
    9. [Journal] [arXiv]
  9. (with T. Liimatainen) Uniqueness results for inverse source problems of semilinear elliptic equations
    Inverse Problems, Vol. 40, No. 4 (2024), 045030, 1-32.
    10. [Journal] [arXiv]
  10. (with Y. Kian and T. Liimatainen) On determining and breaking the gauge class in inverse problems for reaction-diffusion equations
    Forum Math. Sigma, Vol. 12, e25 (2024), 1-42.
    11. [Journal] [arXiv]
  11. (with H. Liu and X. Liu) Determining a nonlinear hyperbolic system with unknown sources and nonlinearity
    J. Lond. Math. Soc., Vol. 109, Issue 2, e12865 (2024), 1-39.
    12. [Journal] [arXiv]
  12. (with A. Feizmohammadi and T. Liimatainen) An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds
    Ann. PDE, Vol. 9, Article number: 12 (2023), 1-54.
    13. [Journal] [arXiv]
  13. (with H. Liu) Inverse problems for fractional equations with a minimal number of measurements
    Commun. Anal. Comput., Vol. 1 (2023), 72-93.
    14. [Journal] [arXiv]
  14. (with B. Harrach) Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation
    Nonlinear Anal., Vol. 228 (2023), 113188, 1-14.
    15. [Journal] [arXiv]
  15. (with H. Liu, X. Liu and S. Zhang) Simultaneous recoveries for semilinear parabolic systems
    Inverse Problems, Vol. 38, No. 11 (2022), 115006, 1-39.
    16. [Journal] [arXiv]
  16. Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities
    Cal. Var. Partial Differential Equations, Vol. 61, Article number: 188 (2022), 1-30.
    17. [Journal] [arXiv]
  17. (with P.-Z. Kow and J.-N. Wang) The Calderón problem for the fractional wave equation: Uniqueness and optimal stability
    SIAM J. Math Anal., Vol. 54, Issue 3 (2022), 3379-3419.
    18. [Journal] [arXiv]
  18. (with R.-Y. Lai) Inverse problems for fractional semilinear elliptic equations
    Nonlinear Anal., Vol. 216 (2022), 112699, 1-21.
    19. [Journal] [arXiv]
  19. (with T. Liimatainen, M. Salo, and T. Tyni) Inverse problems for elliptic equations with fractional power type nonlinearities
    J. Differential Equations, Vol. 306, Issue 5 (2022), 189-219.
    20. [Journal] [arXiv]
  20. (with P. Caro, R.-Y. Lai and T. Zhou) Boundary determination of electromagnetic and Lamé parameters with corrupted data
    Inverse Probl. Imaging, Vol. 15, Issue 5 (2021), 1171-1198.
    21. [Journal] [arXiv]
  21. (with M. Lassas, T. Liimatainen and M. Salo) Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations
    Rev. Mat. Iberoam., Vol. 37, Issue 4 (2021), 1553–1580.
    22. [Journal] [arXiv]
  22. (with G. Nakamura, R. Potthast and H. Wang) Duality between range and no-response tests and its application for inverse problems
    Inverse Probl. Imaging, Vol. 15, No. 2 (2021), 367-386.
    23. [Journal] [arXiv] [Corrigendum]
  23. (with M. Lassas, T. Liimatainen and M. Salo) Inverse problems for elliptic equations with power type nonlinearities
    J. Math. Pures Appl., Vol. 145 (2021), 44-82.
    (Dedicated to the memory of Yaroslav Kurylev)
    24. [Journal] [arXiv]
  24. (with R. Murthy, K. Shin and J. Mueller) A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón method in the plane
    Inverse Problems, Vol. 36 (2020), 125008, 1-21.
    25. [Journal] [arXiv]
  25. (with R.-Y. Lai and A. Rüland) The Calderón problem for a space-time fractional parabolic equation
    SIAM J. Math Anal., Vol. 52, Issue 3 (2020), 2655-2688.
    26. [Journal] [arXiv]
  26. (with M. Cekić and A. Rüland) The Calderón problem for the fractional Schrödinger equation with drift
    Cal. Var. Partial Differential Equations, Vol. 59, No. 3 (2020), 1-46.
    27. [Journal] [arXiv]
  27. (with B. Harrach) Monotonicity-based inversion of the fractional Schrödinger equation II. General potentials and stability
    SIAM J. Math Anal., Vol. 52, Issue 1 (2020), 402-436.
    28. [Journal] [arXiv]
  28. (with B. Harrach) Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials
    SIAM J. Math Anal., Vol. 51, Issue 4 (2019), 3092-3111.
    29. [Journal] [arXiv]
  29. (with S. Meng) Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium
    J. Elasticity, Vol. 137, Issue 2 (2019), 177-217.
    30. [Journal] [arXiv]
  30. (with X. Cao and H. Liu) Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
    Inverse Probl. Imaging, Vol. 13, Issue 1 (2019), 197-210.
    31. [Journal] [arXiv]
  31. (with R.-Y. Lai) Global uniqueness for the fractional semilinear Schrödinger equation
    Proc. Amer. Math. Soc., Vol. 147, No. 3 (2019), 1189-1199.
    32. [Journal] [arXiv]
  32. (with E. Blåsten) Radiating and non-radiating sources in elasticity
    Inverse Problems, Vol. 35, No. 1 (2019), 1-16.
    33. [Journal] [arXiv] [Corrigendum]
  33. (with B. Harrach and H. Liu) On localizing and concentrating electromagnetic fields
    SIAM J. Appl. Math, Vol. 78, Issue 5 (2018), 2558-2574.
    34. [Journal] [arXiv]
  34. Nearly cloaking for the elasticity system with residual stress
    Asymptot. Anal., Vol. 106, Issue 1 (2018), 1-23.
    35. [Journal] [arXiv]
  35. (with T. Ghosh and J. Xiao) The Calderón problem for variable coefficient nonlocal elliptic operators
    Comm. Partial Differential Equations, Vol. 42, No. 12 (2017), 1923-1961.
    36. [Journal] [arXiv]
  36. (with G. Nakamura) Boundary determination of the Lamé moduli for the isotropic elasticity system
    Inverse Problems, Vol. 33, No. 12 (2017), 1-23.
    37. [Journal] [arXiv]
  37. Reconstruction of penetrable obstacles in the anisotropic acoustic scatterings
    Inverse Probl. Imaging, Vol. 10, Issue 3 (2016), 765-780.
    38. [Journal] [arXiv]
  38. Strong unique continuation for a residual stress system with Gevrey coefficients
    Proc. Amer. Math. Soc. 144 (2016), 2171-2183.
    39. [Journal] [arXiv]
  39. (with R. Kuan and M. Sini) The enclosure method for the anisotropic Maxwell system
    SIAM J. Math Anal., Vol 47 (2015), 3488-3527.
    40. [Journal] [arXiv]
Articles in refereed conference proceedings
  1. A local uniqueness theorem for the fractional Schrödinger equation on closed Riemannian manifolds
    A chapter of the book Extended Abstracts IPMS 2024 Conference - "Inverse Problems: Modeling and Simulation” (2025), 175-181.
    2. [Conference proceedings] [arXiv]
PhD Thesis
  1. The development of the enclosure method in an anisotropic background and the strong unique continuation for the elasticity with residual stress
    doi: 10.6342 (2016), NTU/201600158 (National Taiwan University)
Collaborators
(40 collaborators in alphabetical order).
  • Blåsten, Emilia (Associate Professor, LUT School of Engineering, LUT University, Finland)
  • Cao, Xinlin (Assistant Professor, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong)
  • Caro, Pedro (Ikerbasque Research Associate Professor, Basque Center for Applied Mathematics (BCAM), Spain)
  • Cekić, Mihajlo (Chaire de professeur junior, CNRS, Université Paris-Est Créteil, France)
  • Chen, De-Han (Associate Professor, Department of Mathematics, Central China Normal University, China)
  • Feizmohammadi, Ali (Assistant Professor, University of Toronto, Canada)
  • Ghosh, Tuhin (Reader, Harish-Chandra Research Institute, India)
  • Harrach, Bastian (Professor & Dean of the Department of Computer Science and Mathematics, Institute of Mathematics, Goethe University Frankfurt, Germany)
  • Kar, Manas (Assistant Professor, Department of Mathematics, IISER Bhopal, India)
  • Kian, Yavar (Professor, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie, France)
  • Kow, Pu-Zhao (Assistant Professor, Department of Mathematical Sciences, National Chengchi University, Taiwan)
  • Kuan, Rulin (Associate Professor, Department of Mathematics, National Cheng Kung University, Taiwan)
  • Lai, Ru-Yu (Associate Professor, University of Minnesota, USA)
  • Lassas, Matti (Professor, Department of Mathematics and Statistics, University of Helsinki, Finland)
  • Liimatainen, Tony (Researcher, Department of Mathematics and Statistics, University of Jyväskylä, Finland)
  • Lin, Ching-Lung (Professor, Department of Mathematics, National Cheng Kung University, Taiwan)
  • Liu, Hongyu (Chair Professor, Department of Mathematics, City University of Hong Kong, Hong Kong)
  • Liu, Xu (Professor, School of Mathematics and Statistics, Northeast Normal University, China)
  • Meng, Shixu (Senior Research Associate, Department of Mathematics, Virginia Tech, USA)
  • Muller, Jennifer (Albert C. Yates Endowed Chair in Mathematics & Professor, Department of Mathematics, Colorado State University, USA)
  • Murthy, Rashmi (AI Researcher, SandLogic Technologies Ltd, India)
  • Nakamura, Gen (Professor Emeritus, Hokkaido University, Japan)
  • Ola, Petri (University Lecturer, Department of Mathematics and Statistics, University of Helsinki, Finland)
  • Potthast, Roland (Professor & Director Meteorological Analysis and Modeling (Department FE1) at DWD/BMDV, Germany)
  • Railo, Jesse (Associate Professor, LUT School of Engineering, LUT University, Finland)
  • Rüland, Angkana (Professor, Institute for Applied Mathematics, Hausdorff Center for Mathematics, University of Bonn, Germany)
  • Salo, Mikko (Professor, Department of Mathematics and Statistics, University of Jyväskylä, Finland)
  • Shin, Kwancheol (Research Assistant Professor, Institute of Mathematical Science at Ehwa Womans University, Korea)
  • Sini, Mourad (Professor, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria)
  • Tyni, Teemu (Assistant Professor, Unit of Applied and Computational Mathematics, University of Oulu, Finland)
  • Uhlmann, Gunther (Robert R. Phelps and Elaine F. Phelps Endowed Professor of Mathematics, University of Washington, USA)
  • Wang, Haibing (Professor, Department of Mathematics, Southeast University, China)
  • Wang, Jenn-Nan (Distinguished Professor, Institute of Applied Mathematics, National Taiwan University, Taiwan)
  • Weth, Tobias (Professor, Institute of Mathematics, Goethe University Frankfurt, Germany)
  • Xiao, Jingni (Assistant Professor, Department of Mathematics, Drexel University, USA)
  • Yan, Lili (Postdoctoral Research, North Carolina State University, USA)
  • Yousept, Irwin (Professor, Faculty of Mathematics, Universität Duisburg-Essen, Germany)
  • Zhang, Shen (Visiting Assistant Professor, Department of Mathematics, Michigan State University, USA)
  • Zhou, Ting (Professor, Department of Mathematics, Zhejiang University, China)
  • Zimmermann, Philipp (Postdoctoral Researcher, Institute of Mathematics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland)
International Visitors
  • Salo, Mikko (Professor, Department of Mathematics and Statistics, University of Jyväskylä, Finland). April 23-30, 2026.
  • Okasanen, Lauri (Professor, Department of Mathematics and Statistics, University of Helsinki, Finland). June 19-23, 2024.
  • Takase, Hiroshi (Lecture, Okayama University, Japan). June 24-28, 2024.
  • Lee, Hyundae (Professor, Department of mathematics, Inha university, Korea). March 20-24, 2024.
  • Yu, Sanghyeon (Associate Professor, Department of Mathematics, Korea University, Korea). March 20-24, 2024.
  • Nakamura, Gen (Professor Emeritus, Hokkaido University, Japan). January 14-21, 2024.
  • Liimatainen, Tony (Researcher, Department of Mathematics and Statistics, University of Jyväskylä, Finland). November 30-December 8, 2023.
  • Nakanishi Kenji (Professor, Research Institute for Mathematical Sciences, Kyoto University, Japan). March 6-13, 2023.
  • Harrach, Bastian (Professor & Dean of the Department of Computer Science and Mathematics, Institute of Mathematics, Goethe University Frankfurt, Germany). May 10-19, 2023.
  • Nakamura, Gen (Professor Emeritus, Hokkaido University, Japan). November 15-December 1, 2022.
Some Lecture Notes
These notes (only for personal records) might be incorrect or contain errors. Please be careful in using them.
  • Harmonic Analysis course @ NTU, 2016.02-2016.06 PDF
  • NCTS Distinguished Scholar and NTU visiting chair
    • Spectral Graph Theory and its Applications PDF
  • IAS workshop on Inverse Problems, Imaging and PDE
    • Calderon's problem for elliptic system PDF
  • NCTS Summer Courses on Analysis and PDEs
    • Introduction to EIT PDF
    • Classical DeGiorgi method to NS equation PDF
  • NCTS Minicourse on Periodic Homogenization of Elliptic Problems
    • Introduction to elliptic homogenization problem PDF
  • NCTS seminar - The DeGiorgi method and its application to parabolic regularity
    • Vasseur, DeGiorgi method PDF
  • On the inviscid limit problem for incompressible flows in the half-plane
    • Incompressible Navier-Stokes equation PDF
  • NCTS Distinguished Lectures Series
    • Theory and Applications to homogenization - A revisit with recent progress PDF
  • Other Notes:
    • Method of moving plane PDF
    • Dirichlet-Neumann relations PDF
    • Radon transform PDF
    • Liapunov-Schmidt reduction PDF