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Inversion of abelian and non-abelian ray transforms in the presence of statistical noise

Banff International Research Station for Mathematical Innovation and Discovery

We will discuss two problems associated with ray transforms on simple surfaces: (1) how to reconstruct a function from its noisy geodesic X-ray transform (with applications to X-ray tomography) (2) how to reconstruct a skew-hermitian Higgs field from its noisy scattering data (with applications to Neutron Spin Tomography) For (1), the derivation of new mapping properties for the normal operator I*I, based on a generalization of the transmission condition, allows to prove a Bernsteinâ von Mises theorem, about the statistical reliability of the Maximum A Posteriori as a reconstruction candidate in a Bayesian statistical inversion framework, including a reliable assessment of the credible intervals. For (2), a non-linear problem whose injectivity for the noiseless case was established by Paternainâ Saloâ Uhlmann, the derivation of a new stability estimate allows one to prove a consistency result for the mean of the posterior distribution in the large data sample limit. Numerical illustrations will be presented. Joint works with Gabriel Paternain and Richard Nickl (Cambridge).

Mathematics; Global analysis, analysis on manifolds; Dynamical systems and ergodic theory; Global analysis

video/mp4

Author affiliation: University of California Santa Cruz

2019-10-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71904

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/