Public defence in the field of Mathematics and Statistics, M.Sc. Antti Autio

Title of the thesis: Approximate solution of a parametric diffusion equation for electrical impedance tomography

Thesis defender: Antti Autio
Opponent: Docent Tomas Vejchodsky, Czech Academy of Sciences, Czech Republic
Custos: Associate Professor Antti Hannukainen, Aalto University School of Science

Electrical impedance tomography is a measurement technique where the inner conductivity distribution of some object is reconstructed from electrode measurements made on the surface of the object. In these measurements, an electric current is fed from the outside and the resulting voltage on the surface is measured. The method has applications both in medicine and in industry. Determining the inner conductivity distribution based on the current input and the measured voltage is a challenging mathematical inverse problem. It requires accurately modelling the electric field inside the object.

In this work, electric potential is modeled by a so-called parametric diffusion equation where the parameter describes the conductivity distribution inside the object. The inverse problem can be solved iteratively. I this case the electric field is modeled with different conductivity parameter values until a value that corresponds to the measurement is found. This thesis primarily focuses on making this step faster and computationally lighter. The diffusion equation is numerically solved using the finite element method. In my work, I study reduced basis methods where the solution is sought from a specific subspace, in other words only certain possible solutions are considered. The method works since the solutions to the equation with different parameters, i.e. the potential fields, have a lot of common structure. This limits the shape of possible solutions and thus can be utilized.

In my thesis, a new method for computing the reduced basis is presented. Additionally, the structure of the solutions in a simple geometry is studied theoretically for understanding the effectiveness of these methods. I also apply the methods to electrical impedance tomography using simulated data. It turns out that the modeling can be considerably sped up without sacrificing quality significantly. Finally, a certain approximate linearized model for impedance tomography is considered. There the solution can be determined from the measurement directly without an iteration.

Keywords: electrical impedance tomography, finite element method, reduced basis method

Thesis available for public display 10 days prior to the defence at Aaltodoc.