Localization of Defects and Applications to Parameter - PowerPoint PPT Presentation

Localization of Defects and Applications to Parameter Identification Yann Grisel 1 , 2 , P.A. Mazet 1 , 2 , V. Mouysset 1 , J.P Raymond 2 1 ONERA Toulouse, DTIM, M2SN, 2 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 Far-field sources measurements Object 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. Yann Grisel Localization of Defects, and, Applications to Parameter Identification 2/ 9

Mathematical setting ^ x Γ m 8 Plane-wave U Far-field n Ui sources measurements Object θ Γ e Figure: Inhomogeneous medium (O) studied at a fixed frequency Plane wave sources : u i ( x ) := e ikx · � θ , x ∈ R d , � θ ∈ Γ e Helmholtz equation for inhomogeneous media in an unbounded domain: ∆ u s + k 2 n ( x ) u s = − k 2 ( n ( x ) − 1) u i , x ∈ R d ,   2 ( ∂ | x | u s − iku s ) = 0 . d − 1 | x |→∞ | x | lim  � � e ik | x | Far-field pattern : u s ( x ) = u ∞ (ˆ 1 , x ∈ R d , ˆ x ) + o x ∈ Γ m d − 1 d − 1 | x | 2 | x | 2 Problem: extract some information about the actual medium’s index n ⋆ ∈ L ∞ ( O ) from far-field measurements u ∞ ∈ C ∞ (Γ e , Γ m ) 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 ⇒ 0 < M { n , n ⋆ } ( x ) := � W − 1 n ( x ) � = n ⋆ ( x ) ⇐ 2 u ( · , x ) � − 2 L 2 (Γ e ) 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 2.2 2.15 0.6 0.6 0.4 2.1 0.4 2.1 0.2 0.2 2 2.05 0 0 −0.2 1.9 −0.2 2 −0.4 −0.4 1.8 1.95 −0.6 −0.6 −0.5 0 0.5 1 −0.5 0 0.5 1 (a) Reference index n ( x ) (b) Perturbed index n ⋆ ( x ) 2.2 0.6 0.6 0.4 0.4 0.4 2.1 0.3 0.2 0.2 2 0 0 0.2 1.9 −0.2 −0.2 −0.4 −0.4 0.1 1.8 −0.6 −0.6 −0.5 0 0.5 1 −0.5 0 0.5 1 (c) Level lines of M { n , n ⋆ } (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 0.4 2.2 0.6 0.25 0.4 0.4 0.2 2.1 0.2 0.2 0.2 0 0.15 0 2 0 −0.2 −0.2 −0.2 0.1 1.9 −0.4 −0.4 −0.4 0.05 1.8 −0.6 −0.6 −0.6 −0.5 0 0.5 1 −0.5 0 0.5 1 −0.5 0 0.5 1 (a) Reconstruction n 1 ( x ) with 4 (b) Plot of M { n 1 , n ⋆ } ( x ) (c) Selection of a zone to divide parameters 2.2 2.2 2.2 0.6 0.6 0.6 0.4 0.4 0.4 2.1 2.1 2.1 0.2 0.2 0.2 2 2 2 0 0 0 −0.2 1.9 −0.2 1.9 −0.2 1.9 −0.4 −0.4 −0.4 1.8 1.8 1.8 −0.6 −0.6 −0.6 −0.5 0 0.5 1 −0.5 0 0.5 1 −0.5 0 0.5 1 (d) Reconstruction n 4 ( x ) with (e) Reconstruction with 13 (f) Actual index n ⋆ ( x ) 13 selected parameters parameters uniformly distributed 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 ) := � Simulation ( n ) − Observations ( n ⋆ ) � 2 L 2 (Γ m ) 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 ) . M { n , n ⋆ } ( x ) = 0 ⇐ 2.2 0.6 0.4 2.1 0.2 2 0 −0.2 1.9 −0.4 1.8 −0.6 −0.5 0 0.5 1 Figure: Reconstruction of n ⋆ ( x ) by minimization of J M ( n ) := �M { n , n ⋆ } � 2 L 2 ( 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 L 1 -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