Resumen de The inverse problem on finite networks
Cristina Araúz Lombardía
The aim of this thesis is to contribute to the field of discrete boundary value problems on finite networks. Boundary value problems have been considered both on the continuum and on the discrete fields. Despite working in the discrete field, we use the notations of the continuous field for elliptic operators and boundary value problems. The reason is the importance of the symbiosis between both fields, since sometimes solving a problem in the discrete setting can lead to the solution of its continuum version by a limit process. However, the relation between the discrete and the continuous settings does not work out so easily in general. Although the discrete field has softness and regular conditions on all its manifolds, functions and operators in a natural way, some difficulties that are avoided by the continuous field appear. Specifically, this thesis endeavors two objectives. First, we wish to deduce functional, structural or resistive data of a network taking advantage of its conductivity information. The actual goal is to gather functional, structural and resistive information of a large network when the same specifics of the subnetworks that form it are known. The reason is that large networks are difficult to work with because of their size. The smaller the size of a network, the easier to work with it, and hence we try to break the networks into smaller parts that may allow us to solve easier problems on them. We seek the expressions of certain operators that characterize the solutions of boundary value problems on the original networks. These problems are denominated direct boundary value problems, on account of the direct employment of the conductivity information. The second purpose is to recover the internal conductivity of a network using only boundary measurements and global equilibrium conditions. For this problem is poorly arranged because it is highly sensitive to changes in the boundary data, at times we only target a partial reconstruction of the conductivity data or we introduce additional conditions to the network in order to be able to perform a full internal reconstruction. This variety of problems is labelled as inverse boundary value problems, in light of the profit of boundary information to gain knowledge about the inside of the network. Our work tries to find situations where the recovery is feasible, partially or totally. One of our ambitions regarding inverse boundary value problems is to recuperate the structure of the networks that allow the well-known Serrin's problem to have a solution in the discrete setting. Surprisingly, the answer is similar to the continuous case. We also aim to achieve a network characterization from a boundary operator on the network. With this end we define a new class of boundary value problems, that we call overdetermined partial boundary value problems. We describe how the solutions of this family of problems that hold an alternating property on a part of the boundary spread through the network preserving this alternance. If we focus in a family of networks, we see that the above mentioned operator on the boundary can be the response matrix of an infinite family of networks associated to different conductivity functions. By choosing an specific extension, we get a unique network whose response matrix is equal to a previously given matrix. Once we have characterized those matrices that are the response matrices of certain networks, we try to recover the conductances of these networks. With this end, we characterize any solution of an overdetermined partial boundary value problem and describe its resolvent kernels. Then, we analyze two big groups of networks owning remarkable boundary properties which yield to the recovery of the conductances of certain edges near the boundary. We aim to give explicit formulae for the acquirement of these conductances. Using these formulae we are allowed to execute a full conductivity recovery under certain circumstances.
