The inverse backscattering problem
Speaker 
Prof Rakesh
University of Delware, USA


When 
Jan 04, 2017
from 04:00 PM to 05:00 PM 
Where  LH111 
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Abstract
An acoustic medium is probed by plane waves from all directions and the medium response is measured back in the same directions. The goal is the recovery of the acoustic properties of the medium from this backscattered data. Specifically, suppose q(x) is a compactly supported smooth function on R 3 , representing the acoustic property of a medium. For each unit direction ω in R 3 , let u(x, t; ω) be the solution of the initial value problem
utt − ∆xu + q(x)u = 0, (x, t) ∈ R 3 × R
u(x, t; ω) = δ(t − x · ω), x ∈ R 3 , t << 0.
The backscattering data, in the direction ω, with delay s, is
β(s, ω) = lim r→∞ ru(rω, r − s, ω), s ∈ R, ω ∈ R 3 , ω = 1.
The inverse backscattering problem is the study of the nonlinear map
F : q(·) → β(·, ·),
particularly the injectivity and the inversion of F. We quickly survey the results for this longstanding (still) unsolved problem and then give details behind our proof of our partial result.