A sharp stability estimate for tensor tomography in non-positive curvature
Paternain, G. P., & Salo, M. (2021). A sharp stability estimate for tensor tomography in non-positive curvature. Mathematische Zeitschrift, 298(3-4), 1323-1344. https://doi.org/10.1007/s00209-020-02638-x
Published inMathematische Zeitschrift
DisciplineInversio-ongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematics
© The Author(s) 2020
We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form L2↦H1/2TL2↦HT1/2, where the H1/2THT1/2-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.
Publication in research information system
MetadataShow full item record
Related funder(s)Academy of Finland; European Commission
Funding program(s)Centre of Excellence, AoF; Academy Project, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingWe are very grateful to the referee for several comments that improved the presentation and in particular for suggesting a simplified proof of Lemma 4.4. GPP was supported by EPSRC Grant EP/R001898/1 and the Leverhulme trust. MS was supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, Grant Numbers 312121 and 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924). This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the authors were in residence at MSRI in Berkeley, California, during the semester on Microlocal Analysis in 2019. ...
Showing items with similar title or keywords.
Ilmavirta, Joonas; Mönkkönen, Keijo (Birkhäuser, 2022)We prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set ...
Ilmavirta, Joonas; Uhlmann, Gunther (Academic Press, 2018)The X-ray transform on the periodic slab [0, 1]×Tn, n ≥ 0, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor ﬁelds gauge freedom increases the kernel further, and ...
Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther (Springer, 2014)We survey recent progress in the problem of recovering a tensor field from its integrals along geodesics. We also propose several open problems.
Brander, Tommi (University of Jyväskylä, 2016)We investigate a generalization of Calderón’s problem of recovering the conductivity coeﬃcient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation div σ ...
Kow, Pu-Zhao; Wang, Jenn-Nan (American Institute of Mathematical Sciences (AIMS), 2022)Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache . In this work, based on Mandache's idea, we refine the ...