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 back-scattered data. Specifically, suppose q(x) is a compactly supported smooth function on ℝ3, representing the acoustic property of a medium. For each unit direction ω in ℝ3, let u(x, t; ω) be the solution of the initial value problem
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utt - Δxu + q(x)u = 0, |
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(x,t) ∈ ℝ3 x ℝ |
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u(x, t; ω) = δ(t- x · ω), |
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x ∈ ℝ3, t << 0. |
The back-scattering data, in the direction ω, with delay s, is
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β(s, ω) = |
lim
r→∞ |
u(r ω, r – s ω), s∈ ℝ, ω ∈ ℝ3 , |ω| = 1. |
The inverse back-scattering problem is the study of the non-linear map
F: q( · ) → β( ·, ·),
particularly the injectivity and the inversion of F. The speaker and his group survey the results for this long-standing unsolved problem, based on work done with Gunther Uhlmann.
About the speaker
Prof Rakesh Rakesh received his PhD from Cornell University in 1986. He then joined the University of Delaware as an Assistant Professor. He went up through the ranks and is currently the Professor of Mathematical Science there.
Prof Rakesh is mainly interested in inverse problems for hyperbolic PDEs, in the one dimensional and the multidimensional cases. He studies the uniqueness, continuous dependence, inversion, and related theoretical questions, for these inverse problems.
About the program
For more information, please refer to the program website http://iasprogram.ust.hk/inverseproblems for details.
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