This project is devoted to achieve advances in two types of problems in finite networks: the analysis of the robustness and the centrality of electrical networks and the design and implementation of algorithms for the inverse problem of !he recovery of the conductances. The first one will lead to a better analysis of connectivity in electrical circuits, which is of great interest in many fields such as Organic Chemistry, pattern recognition or computer vision. The results in the second one can be applied in the electrical impedance tomography analysis, which is one of the non-invasive clinical diagnosis methods with more possibilities of development. Together with the academic advances, we will mainly focus on the prattical implementation of the theor9tical algorithms already obtained for our group in former projects. The main techniques come mostly from the discrete potentiat theory, which jointly with !he discrete vectorial calculus developed by the group, allows us to use strategies and toots that represen! the discrete counterpart of !hose used in !he continuous case. The appropriate theoretical framework to address !he discrete inverse problem is the overdetermined partial boundary value problems posed on boundary networks, whereas the fundamental tool is the Dirichlet-to-Robin map. This application measures the curren! generated at the boundary due to a potential applied there, but it is highly sensitive to small modifications of the network structure. For this reason, once the recovery problem to salve is known to have a solution, the construction of stable algorithms for its determination will be a significan! progress, with high impact in real applications. The project also faces the spectral characterization of the response matrix associated to !he Dirichlet-to- Robín map and its relation with he spectrum of !he corresponding Scréidinger operator. Actually, as both of them are M- matrices, Newton inequalities will play a crucial role in their relationship and its characterization. Since, this kind of inequalities are closely related with the so-called transitional measures, we will study the role of the different distances to which the inequalities give rise and we will extend this analysis to the metrics determined by superharmonic functions. AII !he results will be useful to determine measures of connectivity and robustness of the networks. In short, this project represents the culmination of an internationally recognized line of work whose achievements will provide progress into the "reto salud, cambio demográfico y bienestar" project.
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de I+D+i Orientada a los Retos de la Sociedad