# Research

My current research interest is in mathematical physics and in particular inverse (scattering) problems.
Recently I have been interested in Riemannian geometry and I study nonlinear hyperbolic equations in this abstract setting.
I did my PhD work on elliptic biharmonic problems i.e. differential and integral, linear and non-linear, equations involving the bi-Laplacian $\Delta^2$.
I have studied operators of 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

- Brander T, Ilmavirta J, Piiroinen P and Tyni T,
*Optimal recovery of a radiating source with multiple frequencies along one line*, To appear in Inverse Problems & Imaging (arXiv preprint) - Tyni T,
*Numerical results for Saito's uniqueness theorem in inverse scattering theory*, Inverse Problems, 2020,**35**(6), 065002 - Serov V and Tyni T,
*Inverse medium problem for a singular contrast*, Journal of Mathematical Physics**60**(2019), 111508. - Tyni T and Serov V,
*Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line*, Inverse Problems & Imaging**13**:1 (2019), 159-175. - Tyni T,
*Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D*, Inverse Problems**34**(2018), 045007. - Tyni T and Serov V,
*Scattering problems for perturbations of the multidimensional biharmonic operator*, Inverse Problems & Imaging**12**:1 (2018), 205-227. - Tyni T and Harju M,
*Inverse backscattering problem for perturbations of biharmonic operator*, Inverse Problems**33**(2017), 105002. - Tyni T, Harju M and Serov V,
*Recovery of singularities in a fourth-order operator on the line from limited data*, Inverse Problems**32**(2016), 075001.

## Conference proceedings and preprints

- Lassas M, Liimatainen T, Potenciano-Machado L and Tyni T,
*Uniqueness and stability of an inverse problem for a semi-linear wave equation*ArXiv - Serov V and Tyni T,
*Inverse scattering problems for perturbed bi-harmonic operator*, AIP Conference Proceedings 1776 (2016)

## Texts in preparation

- Harju M and Tyni T,
*Numerical solution of the direct and inverse scattering problems in 3D*, Manuscript.

## Conference talks

- 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

- The Finnish Inverse Prize, 2018
- Part of local organizing committee of Inverse Days 2017 (Oulu).