# Inverse problems

Inverse problems research concentrates on the mathematical theory and practical implementation of indirect measurements. Applications are found in numerous research fields involving scientific, medical or industrial imaging; familiar examples include X-ray computed tomography and ultrasound imaging. Inverse problems have a rich mathematical theory employing modern methods in partial differential equations, harmonic analysis, and differential geometry.

The inverse problems research group focuses on fundamental theoretical aspects of inverse problems such as the Calderón problem in electrical imaging and travel time tomography in seismic imaging. The group is part of the Finnish Centre of Excellence in Inverse Problems Research, is involved in activities of the Finnish Inverse Problems Society, and is supported by the European Research Council.

## Current events

- Inverse problems reading group
- Inverse days (Oulu, 12-14 December)
- Finnish mathematical days (Joensuu, 4-5 January)
- Past events

## People

Mikko Salo | professor |

Joonas Ilmavirta | postdoctoral researcher |

Aleksis Koski | postdoctoral researcher |

Jere Lehtonen | postdoctoral researcher |

Valter Pohjola | postdoctoral researcher |

Giovanni Covi | doctoral student |

Keijo Mönkkönen | doctoral student |

Jesse Railo | doctoral student |

## Research

Examples of research topics in the field are given on the research highlights webpage of the Centre of Excellence.

## Recent publications

- J. Ilmavirta,
*On Radon transforms on finite groups*. Preprint (2014), arXiv:1411.3829. - J. Ilmavirta,
*On Radon transforms on compact Lie groups*. Proceedings of the American Mathematical Society (to appear). - G. Paternain, M. Salo and G. Uhlmann,
*Invariant distributions, Beurling transforms and tensor tomography in higher dimensions*. Preprint (2014), arXiv:1404.7009. - P. Caro and M. Salo,
*Stability of the Calderón problem in admissible geometries*. Preprint (2014), arXiv:1404.6652. - L. Päivärinta, M. Salo and E. Vesalainen,
*Strictly convex corners scatter*. Preprint (2014), arXiv:1404.2513. - J. Ilmavirta and M. Salo,
*Broken ray transform on a Riemann surface with a convex obstacle*. Communications in Analysis and Geometry (to appear). - T. Brander,
*Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary*. Proceedings of American Mathematical Society (to appear), preprint arXiv:1403.0428. - J. Ilmavirta,
*On Radon transforms on tori*. Journal of Fourier Analysis and Applications**21**(2015), no. 2, 370–382. - G. Paternain, M. Salo and G. Uhlmann,
*Spectral rigidity and invariant distributions on Anosov surfaces*. J. Diff. Geom. (to appear). - T. Liimatainen and M. Salo,
*n-harmonic coordinates and the regularity of conformal mappings*. Math. Res. Lett. (to appear). - M. Hubenthal,
*The Broken Ray Transform in n Dimensions*. Preprint (2013), arXiv:1310.7156. - F. Chung, M. Salo and L. Tzou,
*Partial data inverse problems for the Hodge Laplacian*. Preprint (2013), arXiv:1310.4616. - M. Salo and T. Liimatainen,
*Local gauge conditions for ellipticity in conformal geometry*. Preprint (2013), arXiv:1310.3666. - A. Garcia,
*L^p-L^q estimates for Electromagnetic Helmholtz equation. Singular potentials*. Preprint (2013), arXiv:1310.2457. - M. Lassas, M. Salo and L. Tzou,
*Inverse problems and invisibility cloaking for FEM models and resistor networks*. Preprint (2013), arXiv:1307.1539. - A. Waters,
*Stable determination of X-ray transforms of time dependent potentials from the dynamical Dirichlet-to-Neumann map*. Preprint (2013), arXiv:1306:0052. - D. Dos Santos Ferreira, Y. Kurylev, M. Lassas and M. Salo,
*The Calderón problem in transversally anisotropic geometries*. Preprint (2013), arXiv:1305.1273. - M. Kar and M. Sini,
*Reconstruction of interfaces using CGO solutions for the Maxwell equations*. Journal of Inverse and Ill-posed problems (to appear). - G. Paternain, M. Salo and G. Uhlmann,
*On the range of the attenuated ray transform for unitary connections*. Int. Math. Res. Not. (to appear). - C. Kenig and M. Salo,
*The Calderón problem with partial data on manifolds and applications*. Analysis & PDE (to appear). - G. Paternain, M. Salo and G. Uhlmann,
*Tensor tomography: progress and challenges*. Chinese Ann. Math. Ser. B (to appear). - C. Kenig and M. Salo,
*Recent progress in the Calderón problem with partial data*. Contemp. Math. (to appear). - D. Dos Santos Ferreira, C. Kenig and M. Salo,
*On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds*. Forum Math. (to appear). - J. Ilmavirta,
*Boundary reconstruction for the broken ray transform*. Annales Academiae Scientiarum Fennicae Mathematica**39**(2014), no. 2, 485–502. - J. Ilmavirta,
*A reflection approach to the broken ray transform*. Mathematica Scandinavica (to appear). - G.P. Paternain, M. Salo, and G. Uhlmann,
*Tensor tomography on surfaces*. Invent. Math.**193**(2013), no. 1, 229-247. - D. Dos Santos Ferreira, C. Kenig and M. Salo,
*Determining an unbounded potential from Cauchy data in admissible geometries*. Comm. PDE 38 (2013), no. 1, 50-68. - M. Hubenthal,
*Numerical recovery of source singularities via the radiative transfer equation with partial data*. SIAM J. on Imaging Sciences, vol.**6**(2013), no. 3, pp. 1175-1198. - J. Ilmavirta,
*Broken ray tomography in the disk*. Inverse Problems, vol.**29**(2013), no. 3, pp. 035008.

## Links

- Centre of Excellence in Inverse Problems Research
- Finnish Inverse Problems Society
- Inverse Problems Doctoral Programme
- Inverse Problems and Imaging (journal published by AIMS)
- Conferences in inverse problems

This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 307023.