Inverse problems

Inverse problems are the problems of finding unknown parameters or structures by indirect measurements. A typical inverse problem is the inverse conductivity problem. Its practical setting is the following: Assume that you want to find the inner structure of your torso by doing resistivity measurements at your skin.
In mathematical terms, the inverse problems usually mean the finding of the unknown parameter functions of a partial differential equation from the knowledge of the boundary values of the solutions. Recently I have worked with the inverse problems in Riemannian geometry, e.g. how to find an unknown Riemannian manifold and an unknown elliptic operator defined on it from 'boundary data'. This has some connections to my thesis concerning electromagnetic inverse spectral problem (which was in playfully terms the problem 'Can you hear the shape of a radio?') Recently, my research has turned more towards inverse problems in differential geometry and stochastical inverse problems.

The inverse problems are studied all around Finland, we have founded the Finnish Inverse Problems Society which promotes the research on the area. Below is a list of some of my collaborators and friends studying inverse problems in Finland.

• Rolf Nevanlinna Institute
  Lassi Päivärinta
  Petri Ola.
  Kenrick Bingham,
  Hannu Mäkiö,
  Simopekka Vänskä,
  Pekka Tietäväinen.

• University of Helsinki, Department of mathematics
  Kari Astala.

• Helsinki University of Technology
  Erkki Somersalo and his group, in particular, Mustapha Mataich.
  

• University of Jyväskylä
  Eero Saksman and his group.

• General Electrics Medical systems/Instrumentarium Imaging
  Samuli Siltanen.

• Geological Survey of Finland
  Jukka Liukkonen.

• University of Kuopio
  Jari Kaipio and his group.

• Sodankylä Observatory
  Markku Lehtinen, his group and Inverse Ltd.

• Lappeenranta University of Technology
  Heikki Haario and his group.

Here are some of my collaborators around the world

• Michael Anderson, State University of New York, USA.
• Margaret Cheney, Rensselare Polytechnic Institute, USA.
• Allan Greenleaf, Rochester, USA.
• David Isaacson, Rensselaer, USA.
• Atsushi Katsuda, Okayama, Japan.
• Alexander Katchalov, Steklov Institute, Russia.
• Slava Kurylev, Loughborough, UK.
• Niculae Mandache, Loughborough, UK.
• Jennifer Mueller, Colorado State, USA.
• Vladimir Sharafutdinov, Novosibirsk, Russia.
• Michael E. Taylor, University of North Carolina, USA.
• Gunther Uhlmann, Washington, USA.

In Finland there are also various institutes solving inverse problems in practice.

• Check out the 3D-movies of reconstructed viruses and human chromosomes made by using X-ray tomography in the Department of Virology in University of Helsinki and CSC-center.

• Instrumentarium Imaging Inc. is one of worlds biggest manufacturer of dental X-ray and mammography devices. The collaboration with Finnish inverse problems researcher and this company has been very active.

• Invers Ltd is producer of world fastest stochastical inverse problem solvers as well as sonar and radar systems.

• The finnish company Neuromag is specialized in noninvasively study the human brain by using electromagnetic measurements.

You can look also the list of the universities with Inverse Problems research groups .

Mathematical methods for electromagnetism

We have studied absorbing boundary conditions, particularly so-called Perfectly Matched Layer (PML)-condition. Absorbing boundary conditions are used in computer simulations for scattering problems, for simulating radar or cellular phones etc. When simulating the waves in infinite space one faces the problem that any computer has only finite memory. Thus the domain of simulation has to be cut finite. The boundary of this new domain should cause as little echo as possible. The echo-less boundary structures implemented at the boundary are called absorbing boundary conditions. Mathematically, the PML-structure is equivalent for interpreting the real space Rⁿ as a submanifold of the complex space Cⁿ and stretching the real space into the complex direction. In following videos the PML absorbing boundary condition is demonstrated: In Video1 the is the scattered wave when a plane wave scatters to a ball. In Video2 the a solution is shown in the presence of absorbing boundary layer. Note how the waves propagate into the absorbing media and fade there without giving any echo. Thus the solution coincides with the true solution near the ball. ps

We have have also done some applied research of radar systems for a Finnish company Vaisala.

General mathematical interests

Inverse problems are an area of mathematics applying various results from several different fields. My own fields of interests contain the areas of real, complex and functional analysis, particularly theory of PDE:s and microlocal analysis, Riemannian geometry and stochastics.

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