Recent progress on the explicit inversion of geodesic X-ray transforms

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• Francois Monard (University of Washington)
• Monday 05 May 2014, 16:15-17:15
• CMS, MR13.

If you have a question about this talk, please contact Prof. Neshan Wickramasekera.

We review recent progress made by the author on some inverse problems involving geodesic X-ray transforms on Riemannian surfaces with boundary. We are concerned with the reconstruction of functions, or more generally, of symmetric solenoidal tensor fields from knowledge of their X-Ray transform.

Recalling some results known in the simple case (Fredholm equations for functions and solenoidal vector fields, s-injectivity of the ray transform for tensors of any order), we then explain how to reconstruct other sections of certain bundles (k-differentials for k an integer), which in some cases coincide with solenoidal tensor fields, from knowledge of their ray transform. Such reconstruction formulas take the form of Fredholm equations when the metric is simple. Furthermore, the error is proved to be a contraction when the gaussian curvature is small in C^1 norm, in which case the unknowns can be exactly reconstructed via Neumann series.

Second, we present numerical implementation of these formulas. We observe that, while the borderline cases where the error operators cease to be contractions are not well known quantitatively, numerics indicate that, on the examples treated, the Neumann series converges for a family of metrics that is arbitrarily close to non-simple.

The numerical code is finally used to briefly illustrate some recent instability results of this transform in cases where the metric has conjugate points, established in a joint work with Plamen Stefanov and Gunther Uhlmann.

This talk is part of the Partial Differential Equations seminar series.

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