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Matti Lassas - Examples of research projects

Inverse problems

Inverse problems are a branch of exact mathematics, pure and applied, where the goal is to find unknown parameters or structures by indirect measurements. A typical inverse problem is the inverse conductivity problem, called in medical imaging the Electrical Impedance Tomography. 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, this means the problem of finding of the unknown parameter functions of a partial differential equation from the knowledge of the boundary values of the solutions.

The inverse problems are studied all around Finland and we have founded the Finnish Inverse Problems Society which promotes the research on the area.

Absorbing boundary conditions in 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.


Matti Lassas <Matti.Lassasathut.fi>
19.8.2004