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Localization of Defects and Applications to Parameter Identification

Yann Grisel1,2, P.A. Mazet1,2, V. Mouysset1, J.P Raymond2

1ONERA Toulouse, DTIM, M2SN, 2Universit´

e Toulouse III, Paul-Sabatier.

June 26, 2012

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Physical background

Object Plane wave sources Far-field measurements Figure: Acoustic scattering: plane wave incidence directions and far-field measurements.

Problem : recover information about a scatterer from far field data Goals

1 Reconstruct the scatterer’s refraction index through an iterative numerical

method

2 Build a fast numerical method to locate defects in some reference

refraction index.

3 Investigate the coupling of these methods.

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Mathematical setting

Γe m Γ x ^ Ui U 8 n θ Plane-wave sources Far-field measurements Object

Figure: Inhomogeneous medium (O) studied at a fixed frequency

Plane wave sources : ui(x) := eikx·

θ, x ∈ Rd,

θ ∈ Γe Helmholtz equation for inhomogeneous media in an unbounded domain:    ∆us + k2n(x)us = −k2(n(x) − 1)ui, x ∈ Rd, lim

|x|→∞|x|

d−1 2 (∂|x|us − ikus) = 0.

Far-field pattern : us(x) =

eik|x| |x|

d−1 2

u∞(ˆ x) + o

• 1

|x|

d−1 2

• , x ∈ Rd, ˆ

x ∈ Γm Problem: extract some information about the actual medium’s index n⋆ ∈ L∞(O) from far-field measurements u∞ ∈ C∞(Γe, Γm) and a reference medium’s index n ∈ L∞(O). Difficulties: non-linear and ill-posed inverse problem

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Localization of defects

Theorem n(x), n⋆(x) ∈ R (n − n⋆)(x) > 0 or < 0 Incoming and measurement directions covering the whole unit sphere n(x) = n⋆(x) ⇐ ⇒ 0 < M{n,n⋆}(x) := W − 1

2 u(·, x)−2

L2(Γe)

where W is an operator built from the measurements, and u is the total field for the reference index n.

Figure: Plot of M{n,n⋆}(x) for a 2D object with two defects

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Application 1: reconstruction of a perturbed index

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.95 2 2.05 2.1 2.15

(a) Reference index n(x)

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(b) Perturbed index n⋆(x)

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.1 0.2 0.3 0.4

(c) Level lines of M{n,n⋆}

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(d) Selective reconstruction Figure: Reconstruction of a perturbed index

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Application 2: adaptive refinement

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(a) Reconstruction n1(x) with 4

parameters

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.05 0.1 0.15 0.2 0.25

(b) Plot of M{n1,n⋆}(x)

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6

(c) Selection of a zone to divide

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(d) Reconstruction n4(x) with

13 selected parameters

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(e) Reconstruction with 13

parameters uniformly distributed

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

(f) Actual index n⋆(x) Figure: Adaptive refinement

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Uniqueness of the solution

Usual reconstruction of n⋆(x) : min J(n) := Simulation(n) − Observations(n⋆)2

L2(Γm)

Theorem n(x), n⋆(x) ∈ R (n − n⋆)(x) > 0 or < 0 Incoming and measurement directions covering the whole unit sphere M{n,n⋆}(x) = 0 ⇐ ⇒ n(x) = n⋆(x).

−0.5 0.5 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 1.8 1.9 2 2.1 2.2

Figure: Reconstruction of n⋆(x) by minimization of JM(n) := M{n,n⋆}2

L2(O)

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Achievements Localization of defects Reconstruction of a perturbed index Adaptive refinement New reconstruction approach Perspectives Extension of the localization to limited aperture data and absorbing media Motion detection in inhomogeneous media Free domain decomposition through the new reconstruction approach L1-norm minimisation

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Thank you for your attention

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