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Titel: Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations
VerfasserIn: Vierus, Lukas
Schuster, Thomas
Sprache: Englisch
Titel: Electronic Transactions on Numerical Analysis
Bandnummer: 57
Seiten: 80-100
Verlag/Plattform: Kent State University Press
Erscheinungsjahr: 2022
Freie Schlagwörter: attenuated refractive dynamic ray transform of tensor field
geodesics
transport equation
viscosity solutions
DDC-Sachgruppe: 510 Mathematik
Dokumenttyp: Journalartikel / Zeitschriftenartikel
Abstract: We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as inverse source problem for the associated transport equation. Following Fermat’s principle the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static euclidean setting, whereas there are only few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well-defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.
DOI der Erstveröffentlichung: 10.1553/etna_vol57s80
Link zu diesem Datensatz: urn:nbn:de:bsz:291--ds-394174
hdl:20.500.11880/35538
http://dx.doi.org/10.22028/D291-39417
ISSN: 1068-9613
Datum des Eintrags: 30-Mär-2023
Fakultät: MI - Fakultät für Mathematik und Informatik
Fachrichtung: MI - Mathematik
Professur: MI - Prof. Dr. Thomas Schuster
Sammlung:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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