Applied Inverse Problems: Select Contributions from the by Anatoly B. Bakushinsky, Alexandra B. Smirnova, Hui Liu

By Anatoly B. Bakushinsky, Alexandra B. Smirnova, Hui Liu (auth.), Larisa Beilina (eds.)

This complaints quantity is predicated on papers provided on the First Annual Workshop on Inverse difficulties which was once held in June 2011 on the division of arithmetic, Chalmers college of expertise. the aim of the workshop was once to provide new analytical advancements and numerical equipment for strategies of inverse difficulties. state of the art and destiny demanding situations in fixing inverse difficulties for a large diversity of functions was once additionally mentioned.

The contributions during this quantity are reflective of those issues and should be worthy to researchers during this area.

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2 of [9] and Sect. 2 in [6]. Assuming that (16) holds, we obtain ∞ q (x, τ ) d τ . ” Using (12), we obtain an equivalent formula for the tail, V (x, s) = ln w (x, s) . s2 (19) Approximate Global Convergence in Imaging of Land Mines from Backscattered Data 21 Using (12), (13), and (17) we obtain the following nonlinear integral-differential equation: ⎤2 ⎡ s s ∇q (x, τ ) d τ + 2s ⎣ ∇q (x, τ ) d τ ⎦ + 2s2 ∇q∇V Δ q − 2s ∇q · 2 s s (20) s ∇q (x, τ ) d τ + 2s (∇V ) = 0, x ∈ Ω , s ∈ [s, s] , 2 −2s∇V · s q |∂ Ω = ψ (x, s) := ∂s ϕ (x, s) .

Assume that the asymptotic behavior (21) takes place. 2. Assume that the functions V ∗ and q∗ have the following asymptotic behavior: p∗ (x) , s → ∞, s 1 p∗ (x) p∗ (x) q∗ (x, s) = ∂sV ∗ (x, s) = − 2 + O 3 ≈ − 2 , s → ∞. s s s V ∗ (x, s) = 1 p∗ (x) +O 2 s s ≈ (25) / Ω. We assume that Ω ⊂ R3 is a convex bounded domain with the boundary x0 ∈ Setting in (23) s = s we get Δ q∗ + 2s2 ∇q∗ ∇V ∗ + 2s(∇V ∗ )2 = 0, x ∈ Ω , q∗ |∂ Ω = ψ ∗ (x, s) ¯ ∀x ∈ ∂ Ω . (26) Then, using the first terms in the asymptotic behavior (25) for the exact tail p∗ (x) p∗ (x) V ∗ (x, s) = s and for the exact function q∗ (x, s) = − s2 , we have − ∗ ∗ (∇p∗ )2 Δ p∗ 2 ∇p ∇p + 2 s ¯ − 2 s ¯ = 0, x ∈ Ω , s2 s2 s s2 q∗ |∂ Ω = ψ ∗ (x, s) ¯ ∀x ∈ ∂ Ω .

However, the case of MCIPs is a more challenging one. Based on our recent numerical experience we can conclude that approximate globally convergent method is numerically efficient and can be applied in real-life reconstruction resulted from a single measurement data. In our numerical experiments of this paper we concentrate on imaging of plastic land mines inside slowly changing background medium from backscattered data. We are not interested in imaging of slowly changing backgrounds and we do not use a priori knowledge of the background medium.

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