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
1/ 9
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
2/ 9
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
3/ 9
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
4/ 9
Application 1: reconstruction of a perturbed index
2.15
0.6 0.4
2.1
0.6
2.2
0.4
2.1
0.2
0.2 2.05
0 −0.2
2
2
0
1.9
−0.2 −0.4
−0.4
1.95
−0.6 −0.5
0
0.5
−0.5
1
0.4
0.4 0.3
0.2 0
0.2
−0.2 −0.4
0.1
−0.6
0
0.5
1
(b) Perturbed index n? (x)
(a) Reference index n(x) 0.6
1.8
−0.6
0.6
2.2
0.4
2.1
0.2 2
0
1.9
−0.2 −0.4
1.8
−0.6 −0.5
0
0.5
1
(c) Level lines of M{n ,n? }
−0.5
0
0.5
1
(d) Selective reconstruction
Figure: Reconstruction of a perturbed index Yann Grisel
Localization of Defects, and, Applications to Parameter Identification
5/ 9
Application 2: adaptive refinement
0.6 0.6
0.25 0.4
2.2 0.6
0.4 2.1
0.2 0
2
−0.2
0.4
0.15
0
−0.6 −0.5
0
0.5
1
0.1
−0.6
0
0.5
−0.5
1
(b) Plot of M{n1 ,n? } (x)
parameters
−0.4
0.05 −0.5
(a) Reconstruction n1 (x) with 4
0
−0.2
−0.2 1.9 −0.4 1.8−0.6
−0.4
0.2
0.2
0.2
0
0.5
1
(c) Selection of a zone to divide
0.6
2.2 0.6
2.2 0.6
2.2
0.4
2.1 0.4
2.1 0.4
2.1
0.2
0.2 2
0
0.2 2
0
−0.2
1.9−0.2
1.9−0.2
−0.4
−0.4 1.8 −0.6
−0.4 1.8 −0.6
−0.6 −0.5
0
0.5
1
−0.5
0
0.5
1
(d) Reconstruction n4 (x) with (e) Reconstruction with 13 13 selected parameters parameters uniformly distributed
2
0
1.9 1.8 −0.5
0
0.5
1
(f) Actual index n? (x)
Figure: Adaptive refinement
Yann Grisel
Localization of Defects, and, Applications to Parameter Identification
6/ 9
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
2.1
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
7/ 9
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
8/ 9
Thank you for your attention
Yann Grisel
Localization of Defects, and, Applications to Parameter Identification
9/ 9