Localization of Defects and Applications to Parameter Identification

Jun 26, 2012 - (∂|x|us − ikus )=0. Far-field pattern : us (x) = eik|x|. |x| d−1. 2 u∞(x) + o. (. 1. |x| d−1. 2. ) , x ∈ R d , x ∈ Γm. Problem: extract some information ...
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Localization of Defects and Applications to Parameter Identification Yann Grisel1,2 , P.A. Mazet1,2 , V. Mouysset1 , J.P Raymond2

2

1 ONERA Toulouse, DTIM, M2SN, Universit´ e Toulouse III, Paul-Sabatier.

June 26, 2012

Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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Physical background

Plane wave sources

Object

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

Yann Grisel

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. Localization of Defects, and, Applications to Parameter Identification

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Mathematical setting

Plane-wave sources

8

^ Γ x m Un

Ui

Far-field measurements

Object

θ

Γe

Figure: Inhomogeneous medium (O) studied at a fixed frequency ~ Plane wave sources : u i (x) := e ikx·θ , x ∈ Rd , θ~ ∈ Γe Helmholtz equation for inhomogeneous media in an unbounded domain:  s 2 s 2 i d  ∆u + k n(x)u = −k (n(x) − 1)u , x ∈ R ,

 lim |x|

d−1 2

|x|→∞

Far-field pattern : u s (x) =

(∂|x| u s − iku s ) = 0. e ik|x| d−1

|x| 2

u ∞ (ˆ x) + o



1 d−1

|x| 2



, x ∈ Rd , xˆ ∈ 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 Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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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 1

n(x) 6= n? (x) ⇐⇒ 0 < M{n,n? } (x) := kW − 2 u(·, x)k−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 Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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Application 1: reconstruction of a perturbed index

2.15

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(b) Perturbed index n? (x)

(a) Reference index n(x) 0.6

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(c) Level lines of M{n ,n? }

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(d) Selective reconstruction

Figure: Reconstruction of a perturbed index Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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Application 2: adaptive refinement

0.6 0.6

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(b) Plot of M{n1 ,n? } (x)

parameters

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(a) Reconstruction n1 (x) with 4

0

−0.2

−0.2 1.9 −0.4 1.8−0.6

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(c) Selection of a zone to divide

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(d) Reconstruction n4 (x) with (e) Reconstruction with 13 13 selected parameters parameters uniformly distributed

2

0

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0

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(f) Actual index n? (x)

Figure: Adaptive refinement

Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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Uniqueness of the solution Usual reconstruction of n? (x) : min J(n) := kSimulation(n) − Observations(n? )k2L2 (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). 0.6

2.2

0.4

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0.2 2

0

1.9

−0.2 −0.4

1.8

−0.6 −0.5

0

0.5

1

Figure: Reconstruction of n? (x) by minimization of JM (n) := kM{n,n? } k2L2 (O) Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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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

Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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Thank you for your attention

Yann Grisel

Localization of Defects, and, Applications to Parameter Identification

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