# Research

My current research interest is in mathematical physics and in particular inverse (scattering) 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 $(\partial_t^2-\Delta_g) u + qu^p=0$ and various Schrödinger type equations of the form $\Delta_g u + qu^p=0$, where $\Delta_g$ is the Laplace-Beltrami operator.
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 also work on numerical implementations of the theoretical results.

More information about the Inverse Problems research group is available on the group's website.

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

# Publications and other things

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## Journal articles

- 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 - 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 - Tyni T,
*Numerical results for Saito's uniqueness theorem in inverse scattering theory*, Inverse Problems,**35**:6, 065002, 2020. pdf - Serov V and Tyni T,
*Inverse medium problem for a singular contrast*, Journal of Mathematical Physics**60**, 111508, 2019. pdf - 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 - Tyni T,
*Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D*, Inverse Problems**34**, 045007, 2018. pdf - Tyni T and Serov V,
*Scattering problems for perturbations of the multidimensional biharmonic operator*, Inverse Problems & Imaging**12**:1, 205-227, 2018. pdf - Tyni T and Harju M,
*Inverse backscattering problem for perturbations of biharmonic operator*, Inverse Problems**33**, 105002, 2017. pdf - 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

## Conference proceedings and preprints

- Liimatainen T, Lin Y-H, Salo M, Tyni T,
*Inverse problems for elliptic equations with fractional power type nonlinearities*, Submitted December 2020, ArXiv - Lassas M, Liimatainen T, Potenciano-Machado L and Tyni T,
*Uniqueness and stability of an inverse problem for a semi-linear wave equation*, Submitted August 2020, ArXiv - Serov V and Tyni T,
*Inverse scattering problems for perturbed bi-harmonic operator*, AIP Conference Proceedings 1776 (2016) pdf

## Conference talks

- Inverse Days 2020 (Helsinki)
- Mathematical Days 2020 (Oulu)
- Inverse Days 2019 (Jyväskylä)
- AIP 2019 (Grenoble, France)
- Inverse Days 2018 (Helsinki)
- Mathematical Days 2018 (Joensuu)
- Inverse Days 2016 (Kuopio)
- Inverse Days 2015 (Lappeenranta)

## Other

- I took part in the local organizing committee of Inverse Problems and Nonlinearity 2020 (Helsinki).
- In 2018 I received The Finnish Inverse Prize.
- I took part in the local organizing committee of Inverse Days 2017 (Oulu).