RT info:eu-repo/semantics/doctoralThesis
T1 Some inverse problems on finite networks
A1 Samperio Valdivieso, Álvaro
A2 Universidad de Valladolid. Escuela de Doctorado
K1 Cálculo vectorial
K1 Inverse problems
K1 Problemas inversos
K1 Stable algorithms
K1 Algoritmos estables
K1 Electrical networks
K1 Redes eléctricas
K1 Discrete vector calculus
K1 Cálculo vectorial discreto
K1 12 Matemáticas
AB We propose a version of the discrete vector calculus for Alternating Current (AC) and Direct Current (DC) networks. From the definition of the tangent space at each vertex of the network, we can define discrete concepts that mimic the continuous differential operators, vector fields or boundary value problems. This discrete vector calculus provides a framework to explain the relation between potential, current injected and power injected at the vertices of the network. In particular, it allows us to formulate two inverse problems in electrical networks and also to introduce concepts and results that we use to solve them.The first inverse problem that we study is the inverse conductance problem, which is the discrete version of Calderon’s problem. The inverse conductance problem consists in determining the conductance of a DC network from its Dirichlet-to-Neumann map. This map gives us the linear relationship between the potential and the respective injected current at a subset of vertices of the network under the condition that there is zero injected current at the rest of the vertices.Despite for several families of networks it is known that this problem has a unique solution and there are algorithms based on explicit formulas to solve it, the problem is severely ill-posed. In order to get a numerically stable recovery of the conductance, we propose to reformulate the inverse conductance problem as a polynomial optimization one, in which we incorporate a regularization term that penalizes the deviation respect to an a priori known hypothesis. That hypothesis is that the conductance is piecewise constant on a partition of the edge set with few subsets.We show several examples in which we recover with stability the conductance of well-connected spider networks with up to 47 boundary vertices following our approach, improving the known methods that show instabilities when the number of boundary vertices is greater than 16. Even in some cases in which the hypothesis is false, i.e., the real conductance is not exactly piecewise constant on the partition, we are able to recover a good approximation to the conductance. We study the variation of the error in the problem with the penalty parameter, and the guarantees that a minimum obtained of the problem is global with techniques of Sum of Squares (SOS) decompositions of polynomials.The second inverse problem that we study is the simultaneous recovery of the admittance and topology of an AC or DC network from a set of measurements of voltage and its corresponding power injected at all vertices. We show that this problem is also ill-posed. As a consequence, even if the network is sparse, i.e., has few edges, we usually recover a network which has many edges, which is not efficient for applications. Therefore, we reformulate that inverse problem. We seek to recover a sparse network such that the fitting error to the data is below a fixed tolerance. We propose an algorithm to solve it, which is based on techniques of spectral graph sparsification, and it is based on original results bounding the fitting error of a sparse approximation of a network. We show several experimental results in which we are able to recover a sparse network.
YR 2024
FD 2024
LK https://uvadoc.uva.es/handle/10324/75087
UL https://uvadoc.uva.es/handle/10324/75087
LA eng
NO Escuela de Doctorado
DS UVaDOC
RD 22-feb-2025