Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis.

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed.

A multiscale optimization framework for reconstructing binary images using multilevel PCA-based control space reduction.

An efficient computational approach for optimal reconstructing parameters of binary-type physical properties for models in biomedical applications is developed and validated. The methodology includes gradient-based multiscale optimization with multilevel control space reduction by using principal component analysis (PCA) coupled with dynamical control space upscaling. The reduced dimensional controls are used interchangeably at fine and coarse scales to accumulate the optimization progress and mitigate side effects at both scales.