- Aalto University
- School of

Art and Design - School of Chemical Technology
- School of

Economics - School of Electrical Engineering
- School of

Engineering - School of

Science

Department of Mathematics and Systems Analysis

Matias Dahl | Jérémi Dardé | Jenni Heino | Tapio Helin | Matti Lassas | Maia Lesosky |

Hanna Pikkarainen |
Samuli Siltanen | Erkki Somersalo | Hari M. Varma |

Inverse problems constitute an active and expanding research field of mathematics and its applications. Inverse problems are encountered in several areas of applied sciences such as biomedical engineering and imaging, geosciences, vulcanology, remote sensing, and non-destructive material evaluation. Roughly speaking, a forward problem is to deduce consequences of a cause, while the inverse problem is to find the causes of a known consequence. Typically, inverse problems emerge when one has indirect observations of the quantity of interest.

A typical feature of inverse problems is that they are ill-posed: Small errors in the measured data can cause arbitrarily large errors in the estimates of the parameters of interest, or can even render the problem unsolvable. It may also occur that an inverse problem is non-unique, i.e., there are several different parameter values that could produce the same observed data. Therefore, to solve inverse problems successfully, one needs a good understanding of the uniqueness and stability of the solution and methods for incorporating prior information into the inverse solver algorithms.

Below is a brief description of the types of inverse problems that are studied in the TKK research group.

In diffuse tomographic methods, the objective is to make inference of the internal structure of a body by observations of diffuse fields at the boundary. There are two main applications that are studied:

- Electrical impedance tomography (EIT)
- Near-infrared optical absorption and scattering tomography (OAST)

A commonly used method of making inference of an inaccessible region is to use wave fields for sounding. The wave fields may be acoustic, electromagnetic or elastic waves. The research is mostly focused on the following topics concerning wave fields.

- Electromagnetic inverse boundary value problems
- Inverse scattering problems
- Remote sensing, imaging

Consider a Riemannian manifold and a differential operator. The inverse problem is to determine the manifold and the coefficients when the boundary of the manifold and the Cauchy-data on the boundary of all solutions of the corresponding differential equation are known. The formulation of inverse problems for manifolds is not done solely in order to render the real problem in an invariant form. Often all the information of interest is indeed hidden in the structure of the manifold and the metric of the manifold corresponds to natural physical parameters. As an example, one may consider an object with variable wave propagation speed. The travel time of the wave between two points defines a natural distance between the points. In this particular case, the problem of determining the metric of the manifold means simply the determination of the propagation speed. In the same fashion, electric and heat conductivities can be interpreted as metrics on certain manifolds. The related problems studied in the group are:

- Inverse travel time problem, called also boundary rigidity problem.
- Boundary control methods.
- Stability of inverse problems
- Applications for anisotropic inverse problems in Euclidean space.

In inverse source problems, one seeks to estimate the source of a field that is observed outside the source area. Clearly, the inverse source problems are intimately related to diffuse tomographic and wave field inverse problems as the observations are measurements of diffuse or wave fields. Inverse source problems are encountered e.g. in biomedical applications such as Electroencephalography/cardiography (EEG/ECG) and Magnetoencephalography/cardiography (MEG/MCG). Another area where these problems appear is seismology. The research in the area of inverse source problems is focused on biomedical problems, in particular the MEG/MCG modalities.

A very generic and versatile method of investigating inverse problems is to recast the problem in a Bayesian problem of statistical inference. Powerful methods for solving very complicated inverse problems numerically can be developed by using various Monte Carlo sampling algorithms. These methods are also related to dynamical problems that are traditionally referred to as filtering problems.

The research group participates in various national research programmes and graduate schools. Below is a list of links to these programmes

The group has numerous domestic and international research contacts. Domestic collaboration is organized through the Finnish Inverse Problems Society.

- Finnish Centre of Excellence in Inverse Problems Research
- Finnish Inverse Problems Society
- Inverse Problems International Association (IPIA)
- Inverse Problems Network (IPNET)
- Journal Inverse Problems
- Journal Inverse Problems and Imaging (IPI)
- SIAM Journal on Imaging Sciences (SIIMS)