Research

My current research interest is broadly in mathematical physics and in particular inverse problems. Recently I have been interested in Riemannian and Lorentzian geometry and I study nonlinear elliptic and hyperbolic equations in this abstract setting. Some model equations (and their variants) that I study are the nonlinear wave equation $(\partial_t^2-\Delta_g) u + q(u)=0$ and various Schrödinger type equations of the form $\Delta_g u + q(u)=0$, where $\Delta_g$ is the Laplace-Beltrami operator. I also work on numerical implementations of the theoretical results. I did my PhD work on elliptic biharmonic problems i.e. differential and integral, linear and non-linear, equations involving the bi-Laplacian $\Delta^2$. During my PhD I studied operators of the form $H_4:= \Delta^2 + \vec{q}\cdot\nabla + V$ where $\vec{q}$ and $V$ are functions from some suitable function spaces.

I teach and supervise theses at the Applied and Computational Mathematics (ACM) group in Oulu.

The following subjects (in no particular order) are close to my heart

  • Mathematical physics: scattering theory
  • Inverse problems
  • Topics of partial differential equations, operator theory, integral equations
  • Numerical implementations
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Publications and other things

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Please find preprint versions of the articles by clicking the link titled "pdf" after each paper. Please note that there can be changes and corrections to the preprints in the published versions.

Peer-reviewed journal articles

  1. Lassas M, Liimatainen T, Potenciano-Machado L, and Tyni T, Stability and Lorentzian geometry for an inverse problem of a semi-linear wave equation, To appear in Analysis and PDE, 2024, pdf
  2. Tyni T, Stinchcombe A R, and Alexakis S, A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on real-world data, SIAM J. Imag. Sci., 17:1, 2024., pdf
  3. Covi G, Railo J, Tyni T, and Zimmerman P, Stability estimates for the inverse fractional conductivity problem, SIAM J. Math. Anal. 56:2, DOI: 10.1137/22M1533542, 2024. pdf
  4. Lassas M, Liimatainen T, Potenciano-Machado L, and Tyni T, An inverse problem for a semi-linear wave equation: a numerical study, Inverse Problems & Imaging, 18:1, 62-85, DOI:10.3934/ipi.2023022, 2024. pdf
  5. Lassas M, Liimatainen T, Potenciano-Machado L, and Tyni T, Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation, Journal of Differential Equations, 337, 395-435, DOI:10.1016/j.jde.2022.08.010, 2022. pdf
  6. Liimatainen T, Lin Y-H, Salo M, and Tyni T, Inverse problems for elliptic equations with fractional power type nonlinearities, Journal of Differential Equations, DOI:10.1016/j.jde.2021.10.015, 2022. pdf
  7. Harju M, Kultima J, Serov V, and Tyni T, Two-dimensional inverse scattering for quasi-linear biharmonic operator, Inverse Problems & Imaging, DOI:10.3934/ipi.2021026, 2021. pdf
  8. Brander T, Ilmavirta J, Piiroinen P, and Tyni T, Optimal recovery of a radiating source with multiple frequencies along one line, Inverse Problems & Imaging, 14:6, 967-983, 2020. pdf
  9. Tyni T, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 35:6, 065002, 2020. pdf
  10. Serov V and Tyni T, Inverse medium problem for a singular contrast, Journal of Mathematical Physics 60, 111508, 2019. pdf
  11. Tyni T and Serov V, Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems & Imaging 13:1, 159-175, 2019. pdf
  12. Tyni T, Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D, Inverse Problems 34, 045007, 2018. pdf
  13. Tyni T and Serov V, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems & Imaging 12:1, 205-227, 2018. pdf
  14. Tyni T and Harju M, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems 33, 105002, 2017. pdf
  15. Serov V and Tyni T, Inverse scattering problems for perturbed bi-harmonic operator, AIP Conference Proceedings 1776 (2016) pdf
  16. Tyni T, Harju M, and Serov V, Recovery of singularities in a fourth-order operator on the line from limited data, Inverse Problems 32, 075001, 2016. pdf

Preprints

  1. Lin Y-H, Tyni T, and Zimmerman P, Well-posedness and inverse problems for semilinear nonlocal wave equations, Preprint Feb 2024 ArXiv

Grants and awards

  1. Since 2024 I am a PI in the FAME Flagship of Advanced Mathematics for Sensing, Imaging and Modelling.
  2. In 2023 I was awarded the Emil Aaltonen project grant.
  3. In 2021 I was awarded the Emil Aaltonen Young researcher grant.
  4. In 2018 I received The Finnish Inverse Prize.

Selected conference talks

  1. Applied Inverse Problems AIP2023 (Göttingen, Germany)
  2. International Workshop on Operator Theory and its Applications IWOTA2023 (Finland)
  3. Inverse Problems: Modeling and Simulation 2022 (Malta)
  4. International Zoom Inverse Problems seminar 2022 (UCI) (Link to the talk)
  5. Inverse Problems and Nonlinearity 2021 (Helsinki, Finland) (Link to the talk)

Organizing conferences

  1. Together with prof. A. Hauptmann I am a main organizer of Inverse Days 2024 in Oulu.
  2. I am one of the organizers of the minisymposim M46 at the AIP2023 conference in Göttingen, Germany
  3. I organized minisymposium M27 at the IPMS2022 conference in Malta
  4. I took part in the local organizing committee of Inverse Problems and Nonlinearity 2020 (Helsinki).
  5. I took part in the local organizing committee of Inverse Days 2017 (Oulu).