Defects localization applied to the inverse medium problem Waves 2013
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 3, 2013
Yann Grisel
Defects localization applied to the inverse medium problem
1/ 25
Physical background
Plane wave sources
Object
Far-field measurements
Figure: Acoustic scattering: plane wave incidence directions and far-field measurements.
Goals
Yann Grisel
1
Completely characterize the non-homogeneous medium through an iterative numerical method
2
Focus on the (most) useful parameters
3
Build a fast numerical method to locate defects in some reference refraction index.
Defects localization applied to the inverse medium problem
2/ 25
Mathematical setting
Plane-wave sources
8
^ Γ x m Un
Ui θ
Far-field measurements
Object
Γe
Figure: Inhomogeneous medium (O) studied at a fixed frequency
Inhomogeneous medium: n(x), with n(x) = 1 outside O (bounded) ~ x) := e ikx·θ~ , x ∈ Rd , θ~ ∈ Γe Plane wave sources: u i (θ, Helmholtz equation for inhomogeneous media in an unbounded domain: s 2 s 2 i d ∆un + k n(x)un = −k (n(x) − 1)u , x ∈ R , lim |x| |x|→∞
d−1 2
(∂|x| uns − ikuns ) = 0.
Far-field pattern: ∞ ~ ~ x) = e ik|x| uns (θ, x) + o d−1 un (θ, ~ |x| 2
1 d−1 |x| 2
~ ~x ) ∈ (Γe × Γm ). , x ∈ Rd , (θ,
Problem: extract some information about the actual medium’s index n? ∈ L∞ (O) from (noisy) far-field measurements uε ∈ C ∞ (Γe , Γm ) Difficulties: non-linear and ill-posed inverse problem Yann Grisel
Defects localization applied to the inverse medium problem
3/ 25
Full reconstruction of the unknown index n? Index to far-field mapping: F : n 7→ un∞ Usual Tykhonov regularized cost-functional: J(n) := kF(n) − uε k2L2 (Γe ×Γm ) + λ kn − n0 k2L2 (O) . P Piece-wise constant indices: O = ∪Zi , i = 1 . . . N, n(x) = ηi 1Zi (x) Derivative of the far-field mapping: Z ~ ~x ) 7→ ~ x) un (−~x , x) dn(x) dx, θ~ ∈ Γe , ~x ∈ Γm . DF(n) dn : (θ, k 2 un (θ, x∈O
Gauss-Newton (G.-N.) algorithm for the regularized cost-functional: 1 2 3
Input: n0 ∈ L2 (O) p ← 0; repeat Compute np+1 by solving the linear system DF(np )? DF(np ) +
4
Yann Grisel
λ id (np+1 − n0 ) = 2 −DF(np )? F(np ) − uε − DF(np )(np − n0 ) ,
p ← p + 1; until knp − np−1 k2 /(1 + knp−1 k2 ) < ; Output: npEnd Defects localization applied to the inverse medium problem
4/ 25
Final relative error (e p En d) in %
Full reconstruction of the unknown index n?
11 10 9 8 7 6 1 10
Figure: N 7→ epEnd :=
Yann Grisel
2
3
10 10 Numb er of parameters (N )
kn? −npEnd k kn? k
(percentage)
Defects localization applied to the inverse medium problem
5/ 25
1 - Localisation of defects
Yann Grisel
Defects localization applied to the inverse medium problem
6/ 25
A sampling approach U
n
Ui
O
Ω
U
8
O
8
Ui
n*
Ω
Figure: Initial and perturbed sates of measured object
Green’s function for an inhomogeneous medium: ∆ + k 2 n(x) Gn (z, x) = −δ(z − x). Superposition of Green’s function far-fields: C : L2 (Ω) → L2 (Γm ) Z C (h)(~x ) := Gn∞ (z, ~x )h(z). z∈Ω
Proposition For each x ∈ R3 , we have x ∈ Ω ⇐⇒ Gn∞ (x, ·) ∈ R (C ) . Yann Grisel
Defects localization applied to the inverse medium problem
7/ 25
The factorization method Z Far-field operator: Fn (g )(~x ) := ~ e θ∈Γ
g ∈ L2 (Γe ) Sum of plane waves
u i ∈ L2 (O)
~ ~x )g (θ), ~ un∞ (θ, Fn
~x ∈ Γm .
/ un∞ ∈ L2 (Γm ) O Far field
Hn Helmholtz
/ un ∈ L2 (O)
Figure: Factorization of the far field operator
Factorization of the far-field subtraction: Z ~ ~x )g (θ) ~ Fn? − Fn (g )(~x ) = un∞? − un∞ (θ, ~ e θ∈Γ Z Z ~ z)g (θ) ~ = Gn∞ (z, ~x )k 2 n? − n (z)un? (θ, ~ θ∈Γe z∈Ω Z Z ~ z)g (θ) ~ = Gn∞ (z, ~x )k 2 n? − n (z)Hn? Hn−1 un (θ, ~ e z∈Ω θ∈Γ Z Z ~ (θ) ~ = Gn∞ (z, ~x )k 2 n? − n (z)Hn? Hn−1 Gn∞ (z, −θ)g Yann Grisel
z∈Ω
Defects localization applied to the inverse medium problem
~ e θ∈Γ
8/ 25
The factorization method Scattering operator: S := id + 2ik |γ|2 Fn . Proposition If Γe = Γm = S d−1 , n(x) ∈ R, n? (x) ∈ R, then ~ = S ? Gn∞ (z, ·) (θ), ~ Gn∞ (z, −θ)
θ~ ∈ S d−1
Factorization of the far-field subtraction (continued): Z Z Fn? − Fn (g )(~x ) = Gn∞ (z, ~x )k 2 n? − n (z)Hn? Hn−1
~ d−1 θ∈S
z∈Ω
~ (θ) ~ Gn∞ (z, −θ)g
= CAC ? S(g )(~x ). =⇒ (Fn? − Fn )S ? = CAC ? . F# transform: F# := Re (Fn? − Fn )S ? + Im (Fn? − Fn )S ?
|A| := (A? A)1/2 ,
Re A := (A + A? )/2,
Im A := (A − A? )/2i
Proposition (Kirsch, . . . , Lechleiter) F# = CA# C ? , with A# positive and self adjoint and 1/2 R F# = R (C ) . Yann Grisel
Defects localization applied to the inverse medium problem
9/ 25
The factorization method Theorem If Γe = Γm = S d−1 , n(x) ∈ R, n? (x) ∈ R, then n(x) 6= n? (x) ⇐⇒ ∃g ∈ L2 (S d−1 )
/
⇐⇒ 0 < S{n,n? } (x) :=
1/2
F# g = un (−·, x) !−1 X hun (−·, x), ψj iL2 (S d−1 ) 2 σj j
1/2
where (ψj , σj ) is an eigen-system for F# , and un is the total field induced by n.
Figure: Plot of S{n,n? } (x) for a 2D object with two defects Yann Grisel
Defects localization applied to the inverse medium problem
10/ 25
2 - Enhancements of the Gauss-Newton method via defects localization
Yann Grisel
Defects localization applied to the inverse medium problem
11/ 25
Application 1: selective reconstruction of a perturbed index
1 2 3
Input: n0 ∈ L2 (O) = known initial state Si ← maxZi S{n0 ,n? } (x); OT ← the set of zones for which Si > T max(Si ); npEnd ← G.-N.(n0 OT ) (all indices are extended by n0 outside OT ); Output: npEnd Algorithm 1: Selective reconstruction
Yann Grisel
Defects localization applied to the inverse medium problem
12/ 25
Application 1: selective reconstruction of a perturbed index (real part) 0.6
0.6
0.6 1.6
0.4 0.2
1.6
0.4 0.2
1.4
1.4
0
0
−0.2
1.2−0.2
1.2−0.2
−0.4
−0.4
−0.4
−0.6
1 −0.6
1 −0.6
0
0.5
1
−0.5
0
0.5
−0.5
1.6
Selected 0.4
0.4
0.2
0.2
1.4
0
0
−0.2
−0.2
1.2
−0.4
−0.4
Rejected
1
−0.6
−0.6 −0.5
0
0.5
1
(d) Thresholded selection Yann Grisel
−0.5
0
0.5
0.1
0
0.5
1
(c) Level lines of S{n0 ,n? }
0.6 0.6
0.2
1
(b) Initial guess n0 (x) Final relative error (e p En d) in %
−0.5
0.3
0.2
0
(a) Perturbed index n? (x)
0.4
0.4
10
Selective reconstruction error Selection ratio (/10)
9 8 7 6 5 4 3 2 1 0
1
(e) Selective reconstruction Defects localization applied to the inverse medium problem
(f)
10 20 Selection Threshold (T )
?
n −np
End epEnd := kn? k
30
13/ 25
Application 2: adaptive refinement
1 2 3 4 5 6 7 8 9 10 11
Input: n0 ∈ L2 (O) = a single constant (no a priori information) p ← 0; repeat Si ← maxZi S{np ,n? } (x); I ← {i such that Zi contains more than 16 mesh elements}; iSplit ← i such that SiSplit = maxi∈I Si ; Update the set of zones by splitting ZiSplit into four sub-zones; Update the set of parameters accordingly by duplicating ηiSplit three times; N ← N + 3; np+pEnd ← G.-N.(np ); p ← p + pEnd ; until N > Nmax or each Zi contains less than 16 mesh elements; Output: npEnd Algorithm 2: Adaptive refinement
Yann Grisel
Defects localization applied to the inverse medium problem
14/ 25
Application 2: adaptive refinement
0.6 0.8
0.4
0.6
0.75 0.6
0.58
0.4
0.7
0.4
0.56
0.2
0.54
0
0.52
0.2
0.75 0.2
0
0
0.7
−0.2 −0.4 −0.6 0
0.5
0.6
−0.2
−0.2
0.65−0.4
0.55−0.4
−0.6
0.5 −0.6
0.6 −0.5
0.65
1
−0.5
0
0.5
1
0.5 0.48 0.46 −0.5
0
0.5
1
(g) Loop 2 - zone to be splitted (h) Loop 3 - zone to be splitted (i) Loop 4 - zone to be splitted (maxZi S{n7 ,n? } (x))
(maxZi S{n11 ,n? } (x)) 2
0.6 0.4
0
2
0.6
1.8
0.2
(maxZi S{n15 ,n? } (x))
0.4 0.2
0.4
1.8
0.2 0
1.6
−0.4
−0.2 1.4 −0.4
−0.2 1.4 −0.4
1.4
−0.6
1.2−0.6
1.2−0.6
1.6
−0.2
−0.5
0
0.5
1
(j) Loop 2 - reconstruction with 7 parameters
0
2
0.6
1.8 1.6
−0.5
0
0.5
1
1.2 −0.5
0
0.5
1
(k) Loop 3 - reconstruction
(l) Loop 4 - reconstruction with
with 10 parameters
13 parameters
Figure: Adaptive refinement Yann Grisel
Defects localization applied to the inverse medium problem
15/ 25
Application 2: adaptive refinement 2
0.6 0.4
0.6
0
1.6
0.4
1.8
0.2
0.2
1.4
0
1.6
−0.2
−0.2 1.4
−0.4 −0.6 0
0.5
1
−0.6
1.2 −0.5
1.2
−0.4
1
−0.5
0
0.5
1
(b) Unknown index n?
(a) Loop 8 - reconstruction with 25 parameters
Relative error (e p ) in %
Figure: Adaptive refinement
13
N>10
12
N>20
11
10
9 0
Yann Grisel
Selective reconstruction error Standard reconstruction error
10
20 Iterations (p)
30
Figure: Evolution of the relative error Defects localization applied to the inverse medium problem
16/ 25
Selective reconstruction and adaptive refinement combined
1 2 3
Input: n0 ∈ L2 (O) = known initial state Si ← maxZi S{n0 ,n? } (x); OT ← the set of zones for which Si > T max(Si ); npEnd ← Adaptive refinement(n0 OT ) (indices are extended by n0 outside OT ); Output: npEnd Algorithm 3: Selective reconstruction followed by adaptive refinement
Yann Grisel
Defects localization applied to the inverse medium problem
17/ 25
Selective reconstruction and adaptive refinement combined 0.6 1.6
0.4
0.6 Selected
0.4
0.2
1.4
0.2
1.2
−0.2
0
0
−0.2 −0.4
−0.4
1
−0.6 −0.5
0
0.5
Rejected
−0.6
1
−0.5
(a) Initial guess
0
0.5
1
(b) Thresholded selection T = 10%
0.6
0.6 1.6
0.4 0.2
1.4
0
1.6
0.4 0.2
1.4
0
−0.2
1.2
−0.4
−0.2
1.2
−0.4 1
−0.6 −0.5
0
0.5
1
(c) Selective reconstruction with 1 parameter
1
−0.6 −0.5
0
0.5
1
(d) Selective reconstruction with 14 adaptive refinements
Figure: Adaptive reconstruction of a perturbed index (real part) Yann Grisel
Defects localization applied to the inverse medium problem
18/ 25
Selective reconstruction and adaptive refinement combined
T = 10% T = 20% T = 30%
Relative error (e p ) in %
11 10 9 8 7 6 0
10
20 30 Iterations (p)
40
Figure: Evolution of the relative error for 3 different selection thresholds
Yann Grisel
Defects localization applied to the inverse medium problem
19/ 25
3 - A new approach to the inverse medium problem
Yann Grisel
Defects localization applied to the inverse medium problem
20/ 25
Uniqueness of the solution Theorem If n(x), n? (x) ∈ R (n − n? )(x) > 0 or < 0 Incoming and measurement directions covering the whole unit sphere Then un∞ = un∞? on (S d−1 × S d−1 ) ⇐⇒ S{n,n? } = 0 on R ⇐⇒ n = n? on R Usual (regularized) cost-function: J(n) := kF(n) − uε k2L2 (Γe ×Γm ) + λ kn − n0 k2L2 (O) . Defects (regularized) cost-function:
2 JS (n) := S{n, n? } L2 (O) + λ kn − n0 k2L2 (O) . Observation space: L2 (O) =⇒ Straightforward domain decomposition Yann Grisel
Defects localization applied to the inverse medium problem
21/ 25
A new fidelity term Derivative of the defects localization (eigen-system of F# version) ∂n S{n,n? } (x) dn = 2 X −2 S{n,n? } (x) σi |hun , ψi i|2 ∂n σi (n)dn o n 2 X −1 − S{n,n? } (x) σi 2Re hun , ψi i (hun , ∂n ψi (n)dni + hψi , ∂n un dni) ∂n un dn = v , (∆ + k 2 n)v = −k 2 dn un and Sommerfeld radiation condition ∂n ψi (n)dn = −(F# − σi id)† (∂n F# (n)dn)ψi ∂n σi (n)dn = h(∂n F# (n)dn)ψi , ψi i ∂n F# (n)dn = X + Y , Re (Fn − Fn? )Sn? X + X Re (Fn − Fn? )Sn? = Re (Fn − Fn? )Sn? Re ∂n Fn dn Sn? + 2ik |γ| (Fn − Fn? )∂n Fn dn Im (Fn − Fn? )Sn? Y + Y Im (Fn − Fn? )Sn? = Im (Fn − Fn? )Sn? Im ∂n Fn dn Sn? + 2ik |γ|2 (Fn − Fn? )∂n Fn dn D E ∂n Fn dn = un dn, un (−·, ·) Yann Grisel
L2 (O) Defects localization applied to the inverse medium problem
22/ 25
A new fidelity term Derivative of the defects localization (singular-system of Fn version) ∂n S{n,n? } (x) dn = 2 X −2 S{n,n? } (x) σi |hun , ψi i|2 ∂n σi (n)dn o n 2 X −1 − S{n,n? } (x) σi 2Re hun , ψi i (hun , ∂n ψi (n)dni + hψi , ∂n un dni) ∂n un dn = v , (∆ + k 2 n)v = −k 2 dn un and Sommerfeld radiation condition ∂n ψi (n)dn = (Fn? Fn − σi2 id)† Fn? (∂n Fn (n)dn)ψi + σi (Fn? Fn − σi2 id)† (∂n Fn (n)dn)? φi ∂n σi (n)dn = h(∂n Fn (n)dn)ψi , φi i D E ∂n Fn (n)dn = un dn, un (−·, ·)
L2 (O)
0.6
2.2
0.4
2.1
0.2 2
0
1.9
−0.2 −0.4
1.8
−0.6 Yann Grisel
Defects localization applied to the inverse medium problem
−0.5
0
0.5
1
23/ 25
Conclusion and perspectives
Yann Grisel
Defects localization applied to the inverse medium problem
24/ 25
Achievements Localization of defects Selective reconstruction Adaptive refinement New reconstruction approach Perspectives Extension of the localization to limited aperture data and absorbing media → S{n,n? } requires n(x) ∈ R, n? (x) ∈ R and Γe = Γm = S d−1 Motion detection in inhomogeneous media → S{n,n? } requires un∞? , un∞ and un Free domain decomposition through the new reconstruction approach → S{n,n? } (x) = 0 on a subdomain =⇒ n = n? on that subdomain L1 -norm minimisation → Seems better adapted to ”0/1” functions Thank you for your attention
Yann Grisel
Defects localization applied to the inverse medium problem
25/ 25
