Abstract

Electrical resistance tomography and electrical capacitance tomography are two imaging techniques used in medical, process and geophysical exploration to image the electrical conductivity and permittivity distribution within a medium, from electrical measurements made on the surface of the medium. Both the techniques involve the solution of a specific inverse problem, which is ill-posed, and also highly non-linear.

In this presentation I focus on the mathematical aspects of the numerical algorithms used for the solution of these inverse problems. Given the non-linearity of the problems, the most popular numerical algorithms rely on the Output Least Squares (OLS) framework. The OLS framework formulates the inverse problems in order to minimize the discrepancy between the measured and simulated data, by adjusting the unknown distribution through a sequence of local linear or quadratic approximations. Finally, the solution of the inverse problem arises from the solution of a sequence of systems of linear equations with large and dense matrices.

Differently from OLS, the recently appeared Residual Least Squares (RLS) framework, because of a different formulation of the inverse problem, involves the solution of a sequence of systems of linear equations with large and sparse matrices. The memory saving due to the sparsity of these matrices represents for RLS a big advantage in comparison with OLS, because RLS can tackle problems that would be more memory demanding with OLS. I also show that the sparsity in these matrices follows specific patterns, which I analyze in order to develop a fast numerical solution method. With regard to the ill-posedness of the inverse problems, I also present the results from the analysis of several methods that stabilize the solution obtained from RLS.

 

 

Biography

Luca Dimiccoli was born in Barletta (Italy) in 1976. He was educated in Informatics, with a computational modelling orientation, at the Università degli Studi di Bari (Italy), where he received the MSc degree. Soon after, he begun a postgraduate study at the Department of Electronics and Informatics of the Vrije Universiteit Brussel (Belgium), which he completed successfully, obtaining the PhD degree in Engineering. Currently he is a Postdoctoral researcher at the Department of Electronics and Informatics of the Vrije Universiteit Brussel. His research interests include numerical linear and non-linear algebra, computations with sparse matrices, linear and non-linear programming, boundary values problems, inverse problems and computational electromagnetics.