The Light Ray Transform in Stationary and Static Lorentzian Geometries
Feizmohammadi, A., Ilmavirta, J., & Oksanen, L. (2021). The Light Ray Transform in Stationary and Static Lorentzian Geometries. Journal of Geometric Analysis, 31(4), 3656-3682. https://doi.org/10.1007/s12220-020-00409-y
Published in
Journal of Geometric AnalysisDate
2021Discipline
MatematiikkaInversio-ongelmien huippuyksikköMathematicsCentre of Excellence in Inverse ProblemsCopyright
© The Authors 2020
Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.
...


Publisher
SpringerISSN Search the Publication Forum
1050-6926Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/35315024
Metadata
Show full item recordCollections
Related funder(s)
Academy of FinlandFunding program(s)
Postdoctoral Researcher, AoF
Additional information about funding
A.F. was supported by EPSRC Grant EP/P01593X/1, J.I. was supported by the Academy of Finland (decision 295853) and L.O. was supported by the EPSRC Grants EP/P01593X/1 and EP/R002207/1.License
Related items
Showing items with similar title or keywords.
-
Applications of Microlocal Analysis in Inverse Problems
Salo, Mikko (MDPI AG, 2020)This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, ... -
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
Covi, Giovanni; Mönkkönen, Keijo; Railo, Jesse (American Institute of Mathematical Sciences (AIMS), 2021)We prove a unique continuation property for the fractional Laplacian (−Δ)s when s∈(−n/2,∞)∖Z where n≥1. In addition, we study Poincaré-type inequalities for the operator (−Δ)s when s≥0. We apply the results to show that ... -
Optimal recovery of a radiating source with multiple frequencies along one line
Brander, Tommi; Ilmavirta, Joonas; Piiroinen, Petteri; Tyni, Teemu (American Institute of Mathematical Sciences (AIMS), 2020)We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam ... -
Quantitative Runge Approximation and Inverse Problems
Rüland, Angkana; Salo, Mikko (Oxford University Press, 2019)In this short note, we provide a quantitative version of the classical Runge approximation property for second-order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We ... -
Inverse problems for elliptic equations with power type nonlinearities
Lassas, Matti; Liimatainen, Tony; Lin, Yi-Hsuan; Salo, Mikko (Elsevier, 2021)We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for ...