Τεχνικές Επίλυσης Αντίστροφων Προβλημάτων Με Βάση Τις Συναρτήσεις Φραγμένης Κύμανσης

Μαρκάκη, Βασιλική; Markaki, Vasiliki

dc.contributor.author	Μαρκάκη, Βασιλική	el
dc.contributor.author	Markaki, Vasiliki	en
dc.date.accessioned	2017-06-12T10:11:42Z
dc.date.issued	2017-06-12
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/45029
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.7181
dc.description	Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Εφαρμοσμένες Μαθηματικές Επιστήμες”	el
dc.rights	Default License
dc.subject	Μερικές διαφορικές εξισώσεις	el
dc.subject	Μέτρο Radon	el
dc.subject	Φραγμένη κύμανση	el
dc.subject	Ελαστογραφία	el
dc.subject	Αγωγιμότητα	el
dc.subject	Αντίστροφο πρόβλημα	el
dc.subject	Partial differential equations	en
dc.subject	Radon measure	en
dc.subject	Bounded variation	en
dc.subject	Conductivity	en
dc.subject	Elastography	en
dc.subject	Inverse problem	en
dc.title	Τεχνικές Επίλυσης Αντίστροφων Προβλημάτων Με Βάση Τις Συναρτήσεις Φραγμένης Κύμανσης	el
dc.title	Investigation Of Inverse Problems On The Basis Of Bounded Variation Functions	en
heal.type	masterThesis
heal.classification	Applied mathematics	en
heal.classificationURI	http://id.loc.gov/authorities/subjects/sh93002523
heal.dateAvailable	2018-06-11T21:00:00Z
heal.language	el
heal.access	embargo
heal.recordProvider	ntua	el
heal.publicationDate	2017-02-21
heal.abstract	The present work has a first part pertaining to the presentation of the fundamental and cornerstone aspects of the inverse conductivity problem: the existence, uniqueness and stability of the solution of this problem. In the sequel, the master thesis involves the development of inversion techniques concerning the solvability of typical inverse dynamic problems in Elastography. The novel concept consists in exploiting and developing analysis from the framework of bounded variation fun\-ctions ($BV$ functions), which represent accurately the discontinuous nature of the inhomogeneities under reconstruction.\ In Chapter $\S2$ \textit{Calderón's Inverse Problem} is presented. The conductivity equation $\nabla \cdot σ\nabla u=0$ is studied for a bounded, two-dimensional domain $Ω$ and an anisotropic $L^{\infty}-$ smooth conductivity $σ.$ The work of \cite{Astala2005a,Astala2006a} is followed, ac\-cor\-ding to which the \textit{Dirichlet-to-Neumann} operator determines uniquely the bounded measurable condu\-cti\-vity $σ$ in a \footnotemark{constructive} manner. This gives a positive answer to a question of \textit{A.P.Calderón} from 1980. In higher dimensions the problem remains open.\ \footnotetext{ through the construction of \textit{complex geometric optics solutions, CGO}} Chapter $\S3$ is devoted to the study of the hyperbolic conductivity equation, as the typical representative of the \textit{Inverse Elastography's Problem}, again for a bounded, two-dimensional domain $Ω.$ The equivalent extremal problem is presented, through a suitable minimizing functional, under the additional assumption that the two-valued conductivity function $γ(x)$ belongs to $BV$ space; $γ\in L^{\infty}\cap BV.$ Inspired by a concept introduced in \cite{Fer2011}, the reconstruction of the function $γ(x)$ is examined from the surface measurements in the form of the \textit{Dirichlet-to-Neumann} operator.	en
heal.advisorName	Χαραλαμπόπουλος, Αντώνιος	el
heal.committeeMemberName	Αρβανιτάκης, Αλέξανδρος	el
heal.committeeMemberName	Γιαννακάκης, Νικόλαος	el
heal.committeeMemberName	Χαραλαμπόπουλος, Αντώνιος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών	el
heal.academicPublisherID	ntua
heal.numberOfPages	83 σ.	el
heal.fullTextAvailability	true