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Consistent Inversion of Noisy Non‐Abelian X‐Ray Transforms
Author(s) -
Monard François,
Nickl Richard,
Paternain Gabriel P.
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21942
Subject(s) - mathematics , invertible matrix , matrix (chemical analysis) , algorithm , inverse , ode , inverse problem , algebraic number , mathematical analysis , pure mathematics , geometry , materials science , composite material
For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field Φ : M → so n from discrete, noisy measurements of the SO ( n ) ‐valued scattering data C Φ of a solution of a matrix ODE is considered ( n ≥ 2 ). Injectivity of the map Φ ↦ C Φ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of C Φ increases, the statistical error in the recovery of Φ converges to 0 in L 2 ( M ) ‐distance at a rate that is algebraic in 1/ N and approaches 1 / N for smooth matrix fields Φ . The proof relies, among other things, on a new stability estimate for the inverse map C Φ → Φ . Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography . © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC