BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Recent progress on the explicit inversion of geodesic X-ray transf
 orms - Francois Monard (University of Washington)
DTSTART:20140505T151500Z
DTEND:20140505T161500Z
UID:TALK52456@talks.cam.ac.uk
CONTACT:Prof. Neshan Wickramasekera
DESCRIPTION:We review recent progress made by the author on some inverse p
 roblems\ninvolving \ngeodesic X-ray transforms on Riemannian surfaces with
  boundary. \nWe are concerned with the reconstruction of functions\, or mo
 re generally\, of\nsymmetric solenoidal tensor fields from knowledge of th
 eir X-Ray transform. \n\nRecalling some results known in the simple case (
 Fredholm equations for\nfunctions\nand solenoidal vector fields\, s-inject
 ivity of the ray transform for tensors\nof\n any order)\, we then explain 
 how to reconstruct other sections of certain\nbundles\n(k-differentials fo
 r k an integer)\, which in some cases coincide with\nsolenoidal tensor fie
 lds\, from knowledge of their ray transform. Such reconstruction formulas 
 take the\nform\nof Fredholm equations when the metric is simple. Furthermo
 re\, the error is \nproved to be a contraction when the gaussian curvature
  is small in C^1 norm\,\nin which \ncase the unknowns can be exactly recon
 structed via Neumann series. \n\nSecond\, we present numerical implementat
 ion of these formulas. We observe\nthat\, \nwhile the borderline cases whe
 re the error operators cease to be\ncontractions are not\nwell known quant
 itatively\, numerics indicate that\, on the examples treated\,\nthe Neuman
 n \nseries converges for a family of metrics that is arbitrarily close to\
 nnon-simple. \n\nThe numerical code is finally used to briefly illustrate 
 some recent\ninstability results\nof this transform in cases where the met
 ric has conjugate points\,\nestablished \nin a joint work with Plamen Stef
 anov and Gunther Uhlmann.\n\n\n
LOCATION:CMS\, MR13
END:VEVENT
END:VCALENDAR