The effective resistance as a tool for the study of the inverse problem of the conductances and the analysis of the perturbations on networks
Total activity: 3
Type of activity
Competitive project
Acronym
REPIPER
Funding entity
MIN DE ECONOMIA Y COMPETITIVIDAD
Funding entity code
MTM2014-60450-R
Amount
63.404,00 €
Start date
2015-01-01
End date
2018-06-30
Abstract
The project raises progress in two kind of problems in finite networks: the inverse problem of recovering of the conductances and the problem of determining the effect of perturbations on the basic parameters of a network. The results in the first one are of potential application in electrical impedance tomography which is currently one of the non-invasive methods of clinical diagnosis with more development opportunities. The second provides advances in the analysis of structural properties of chemical compounds and will contribute to the study of the inverse problem, as well. Most of the techniques we use, come from the discrete potential theory which together with the discrete vector calculus developed by our research group allowed us to use strategies and tools that represent the discrete counterpart of those used in the continuous case. The appropriate theoretical framework to address discrete inverse problem is the partial and ovedetermined boundary value problems, while the fundamental tool is the Dirichlet-to-Robin application. This is a map that measures the difference of voltages between boundary vertices when electrical currents are applied to them. In this project, we analyze the resolvent kernels associated with these problems, which will be the main tool for building complex recovery algorthms, such as n-dimensional lattices and cylindrical networks. The goal is to stablish the relationship between the Dirichlet-to-Robin map and the effective resistance between boundary nodes of the network. The Dirichlet-to-Robin map is highly sensitive to small changes in the structure of the network, therefore it is of great interest to analyze the behavior of the map, the kernel of partial and overdetermined problems, and the effective resistances of a perturbed network and to describe transformations that do not modify these parameters, as well. We will tackle perturbed networks both from a general theoretical framework, understanding the perturbations in the theory of discrete elliptic operators, and from the point of their application to the analysis of families of networks that may be of interest in the context of Organic Chemistry. Due to the nature of the problems, it is of special importance to obtain closed formulas for the inverse, or generalized inverse, of a block tridiagonal matrix. The context for this situation is the resolution of boundary value problems for second-order difference equations with coefficients in a non-commutative unitary ring. This project represents the contribution to a line of work internationally appreciated and can bring a breakthrough in the research oriented challenge " health, demographic change and wellness".
Scope
Adm. Estat
Plan
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Call year
2015
Funcding program
Programa Estatal de I+D+i Orientada a los Retos de la Sociedad
Funding call
Retos de Investigación: Proyectos de I+D+i
Grant institution
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco
Carmona, A.; Mitjana, M.; Monso, E. Linear and multilinear algebra p. 1-15 DOI: 10.1080/03081087.2016.1256945 Date of publication: 2016-11-12 Journal article
Carmona, A.; Mitjana, M.; Monso, E. Electronic notes in discrete mathematics Vol. 54, p. 295-300 DOI: 10.1016/j.endm.2016.09.051 Date of publication: 2016-10 Journal article