Inverse scattering for obstacles and cracks with impedance boundary - - PowerPoint PPT Presentation

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Inverse scattering for obstacles and cracks with impedance boundary conditions

Fahmi Ben Hassen

ENIT-LAMSIN, Tunis

PICOF’12, Palaiseau April 2011

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Introduction Obstacle scattering Forward and inverse scattering problems The Point Source Method Inverse scattering of multiple obstacles The Singular Sources Method Inverse scattering for cracks The Linear Sampling Method The Reciprocity Gap-Linear Sampling Method The Data Completion Algorithm and application to RG-LSM List-to-do

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Introduction

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Setting for reconstruction problem

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Forward and inverse

• bstacle scattering problems

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Acoustic scattering

Given an incident field ui, a bounded obstacle D and a surface impedance λ, find a scattered field us governed by

• The time-harmonic Helmholtz equation

∆us + κ2us = 0 in R3 \ D.

• A boundary condition, for example impedance BC

∂u ∂ν + iλu = 0

• n ∂D

for the total field u = ui + us

• The Sommerfeld radiation condition (3d)

lim

r→∞ r(∂us

∂r − iκus) = 0. Usually ui(x, d) = eiκd·x is a plane wave with direction d ∈ S2 or ui(x, z) = Φ(x, z) := 1 4π eiκ|x−z| |x − z| , x = z in R3, a point source with source point z ∈ D.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Solution Techniques

Assume that

• D is a bounded open domain, R3 \ D is connected and ∂D is Lipschitz.
• λ ∈ L∞(∂D) and Re(λ) > 0.
• The wave number κ > 0.

Under these assumptions

• The exterior impedance obstacle problem is well posed.
• Existence can be shown by variational methods, integral equation

methods or several further special techniques. The scattered field has the asymptotic behaviour of an outgoing spherical wave us(x) = eiκ|x| |x| „ u∞(ˆ x) + O „ 1 |x| «« , as |x| → ∞ uniformly in all directions of observation ˆ x =

x |x|.

The function u∞ is the far field pattern of us.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Inverse Problem

Measured data are either the scattered field us on some surface Γ or the far field pattern u∞ on S2. Task: reconstruct the shape and properties

• f the

unknown scatterer!!!!

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Reconstruction Schemes

• Iterative methods
• Least Squares Method
• Newton’s type Method
• Level Set Methods
• Evolutionary (Newton or Gradient) Methods
• Successive Iteration Schemes (Kress-Rundell)
• Decomposition methods
• Series Expansion Methods: Spherical Waves, Bessel functions
• Fourier - Plane Wave Expansion Methods
• Kirsch-Kress Method or Potential Method
• Point Source Method : Green representation and Point Source Approx.
• Sampling and probe methods
• Linear sampling method (Colton-Kirsch)
• Factorization method (Kirsch)
• Singular sources method (Potthast) - Probe method (Ikehata)
• No response test (Luke-Potthast)
• Orthogonality Sampling (Potthsat-Nakamura)
• PSM uses measurements for one wave, but needs to know the boundary

condition for shape reconstruction

• SSM uses measurements for many incident waves, does not need to

know the boundary condition

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

PSM : acoustics 3d, field reconstruction

Point Source Method : Reconstruct an approximation to us from u∞ for one incident wave. Definition Consider a set of sampling domains Gx parameterized by x ∈ R3, such that x / ∈ Gx. We define the illuminated area E := {x ∈ R3 : D ⊂ Gx}. Theorem (FBH-Erhard-Potthast 2006) The back-projection operator (Aεu∞)(x) := 4π Z

S2 u∞(−ˆ

x)gx,ε(ˆ x) ds(ˆ x), x ∈ R3 \ D, converges uniformly to us on the illuminated area E lim

ε→0 sup x∈E

|us(x) − Aεu∞(x)| = 0.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Acceleration of the PSM algorithm

(1) Use fixed reference approximation domain G0 and solve (αI + H∗H)g0,α = H∗Φ(·, 0)

• n G0

with Herglotz operator (Hg)(y) = R

Λ eiκy·dg(d) ds(d) ,

y ∈ ∂G0. (2) For Gx = x + G0 compute approximating density gx,α(d) = e−iκx·dg0,α(d). (3) Compute the reconstructed field Aεu∞, with density gx,α. (4) Repeat (1)-(3) with a different parameter α2, calculate an approximation Ec = {x ∈ R3 : |Ienl(x)| < c} of the illuminated area from Ienl = urec,1 − urec,2. (5) Determine the ball Br(x0) given by x0 =

R

Ec x dx

R

Ec dx , r = q

„

3 R

Ec dx

4π

« 1

3

, 0 < q < 1. (6) Repeat the steps for different reference domains G(j)

0 ⇒ A(j) ε u∞ and Brj (x(j) 0 )

(7) A complete reconstruction of the field is given as us

rec(x) =

Pn

j=1 us,(j) rec (x) χBrj (x(j)

0 )(x)

Pn

j=1 χBrj (x(j)

0 )(x)

, x ∈

n

[

j=1

Brj (x(j)

0 ).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

PSM: acoustics 2d, field reconstruction

Scattering by an obstacle (λ = 1 + i, κ = 2, d = (−1, 0)T)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

PSM: acoustics 3d, field reconstruction

Scattering by a sound-soft ball (κ = 2)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

PSM: acoustics 3d, shape reconstruction

Reconstructions of a sound-soft torus (κ = 2)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Reconstruction of a sound-hard or a complex obstacle by one incident wave : Kress R and Serranho P : A hybrid method for sound-hard obstacle reconstruction, J. Comp. Appl.

• Math. 2007

Serranho P : A hybrid method for inverse scattering for shape and impedance, IP 2006

• 1. Use the Potential Method to reconstruct the scattered wave from the far

field pattern.

• 2. Use a regularized Newton method applied to a nonlinear operator

equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem. G : (z, λ) → z′⊥ |z′| · (∇u ◦ z) + i(λu) ◦ z in [0, 2π] Find z such that G(z) = 0 where ∂D = {z(s) : s ∈ [0, 2π]}.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Limitation of the standard formulation of the PSM

(a) Simulation of the scattered field (b) reconstruction of the the scattered field via PSM without taking into account the non-convexity of the scatterer. The non-convex part of the fields and domains cannot be reconstructed since it is outside of the illuminated area.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Scattered Wave Splitting

Assume that D = D1 ∪ D2, D1 ∩ D2 = ∅ and Dj is connected with C2-smooth boundary. Definition Consider two domains G1, G2 with G1 ∩ G2 = ∅ and C2-smooth boundary are given s. t. ¯ D1 ⊂ G1, ¯ D2 ⊂ G2. Set G := G1 ∪ G2. Theorem : Uniqueness of source splitting (FBH-Liu-Potthast) Assume that we are given a decomposition us = us

1 + us 2, s. t. :

• 1. us

j satisfies the radiation condition for j = 1, 2;

• 2. us

j solves the Helmholtz equation in Rn\Gj for j = 1, 2;

• 3. (us

j )+ :=

lim

x∈Gj ,x→∂Gj

us

j (x) exists and is continuous on ∂G.

Then the splitting of us is unique in Rn\Gj, j = 1, 2.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Far field splitting via single-layer potentials

Let us express the scattered wave by single-layer approach us(x) := (Sϕ)(x), x ∈ Rn\G The density ϕ lives on ∂G = ∂G1 ∪ ∂G2. We have Sϕ = S1ϕ1 + S2ϕ2, where ϕj(y) := ϕ(y) for y ∈ ∂Gj and (Sjϕj)(x) := Z

∂Gj

Φ(x, y)ϕj(y)ds(y), x ∈ Rn, The splitting of the far field of a scatterer D = D1 ∪ D2 is obtained from the following three steps.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Far field splitting via single-layer potentials 2

Algorithm (FBH-Liu-Potthast 2007)

• 1. Solve S∞ϕ = u∞ to generate ϕ on ∂G.
• 2. Define us

j (x) := (Sjϕj)(x), x ∈ Gj, which can be considered as a

scattered wave outside Gj.

• 3. Compute the far field patterns of us

j defined by

u∞

j

:= S∞

j ϕj,

j = 1, 2.

• If G is chosen s. t. κ2 is not the interior Dirichlet eigenvalue of −∆ in Gj

then the far field equation is uniquely solvable.

• The single-layer approach is a constructive method for the unique

decomposition of the scattered field us = us

1 + us 2

in Rn\G

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Far field splitting via single-layer potentials 3

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Application of Wave Splitting for Shape Reconstruction

Algorithm : The reconstruction of Dj by wave splitting

• 1. Specify two domains G1 and G2 s. t. G1 ∩ G2 = ∅, ¯

Dj ⊂ Gj (this choice depends on a priori informations about Dj).

• 2. Use the splitting procedure to determine densities ϕj on ∂Gj to generate

us = us

1 + us 2 outside G and calculate u∞ j .

• 3. Use PSM to reconstruct (an approximation of) us

2 from u∞ 2

in G2\D2.

• 4. Calculate an approximation to the total field u = ui + S1ϕ1 + us

2 in G2\D2.

• 5. Search for the zero curve of u to calculate an approximation to ∂D2 (for

sound soft obstacle).

• 6. ∂D1 can be reconstructed analogously.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Application of Wave Splitting for Shape Reconstruction

Original total field via simulation (a) and total field via splitting procedure and point source method after reconstruction (b).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Main Idea of Sampling and Probe Methods

Define an indicator function I(z) based on the knowledge of the far field patterns u∞ for all incident plane waves such that I blows up when z tends to ∂D knowledge of the type of the boundary condition on ∂D is not necessary. Singular Sources Method (Potthast-2000) Based on the singular behaviour of Φs(z, z) (point source scattering)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Main Idea of Sampling and Probe Methods

Define an indicator function I(z) based on the knowledge of the far field patterns u∞ for all incident plane waves such that I blows up when z tends to ∂D knowledge of the type of the boundary condition on ∂D is not necessary. Singular Sources Method (Potthast-2000) Based on the singular behaviour of Φs(z, z) (point source scattering)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Properties of fields for singular excitations

Theorem (FBH-Erhard-Potthast 2010) Consider the scattering of a point source Φ(·, z) by a Robin absorbing scatterer D ⊂ R3. Then, ˛ ˛Φs(z, z) ˛ ˛ ≥ c |d(z, D)| holds in the tube 0 < d(z, D) < τ and ˛ ˛Φs(z, z) ˛ ˛ ≤ C |d(z, D)| for all z ∈ B(0, R) \ D, d(z, D) denotes the Hausdorff distance.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The asymptotic expansion of the indicator function

I0(z) := |Φs(z, z)| , z ∈ B(0, R) \ D may serve as an indicator function for the reconstruction of D (Use the blow-up of I0 to find the unknown shape as level curves of |Φs(z, z)|). The precise behaviour of I0(z) for z near ∂D is given by Theorem (FBH-Ivanyshyn-Sini 2010) I0(z) = 1 8π(z − a) · ν(a) − iλ(a) + 1

4∆fa(a′)

2π ln |(z − a) · ν(a)| + O(1), for every z := a + hν(a), h > 0 small enough. ∆fa(a′) is the Mean curvature of the surface ∂D at the point a ∈ ∂D where fa is the local parametrization of ∂D near a and a′ := (a1, a2).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The asymptotic expansion of the indicator function

I0(z) := |Φs(z, z)| , z ∈ B(0, R) \ D may serve as an indicator function for the reconstruction of D (Use the blow-up of I0 to find the unknown shape as level curves of |Φs(z, z)|). The precise behaviour of I0(z) for z near ∂D is given by Theorem (FBH-Ivanyshyn-Sini 2010) I0(z) = 1 8π(z − a) · ν(a) − iλ(a) + 1

4∆fa(a′)

2π ln |(z − a) · ν(a)| + O(1), for every z := a + hν(a), h > 0 small enough. ∆fa(a′) is the Mean curvature of the surface ∂D at the point a ∈ ∂D where fa is the local parametrization of ∂D near a and a′ := (a1, a2).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

PSM in multi-wave setting

Scattered field Φs(·, z) of incident point source Φ(·, z) can be reconstructed with PSM Φs(x, z) ≈ 4π Z

S2 Φ∞(−d, z)gx,ε(d) ds(d)

The mixed reciprocity relation Φ∞(d, z) =

1 4π us(z, −d) yields

Φs(x, z) ≈ Z

S2 us(z, d)gx,ε(d) ds(d) .

Applying the PSM once more Φs(x, z) ≈ 4π Z

S2

Z

S2 u∞(−ˆ

x, d)gz,˜

ε(ˆ

x) ds(ˆ x) gx,ε(d) ds(d) . The indicator function I0(z) ≈ 4π Z

S2

Z

S2 u∞(−ˆ

x, d)gz,˜

ε(ˆ

x) ds(ˆ x) gz,ε(d) ds(d) z ∈ E

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

λ = 1 λ = 0.5 + 10i λ = 5 + 4z3 λ = 5 + 4cos(10z1) Reconstruction of the ellipsoid

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

λ = 1 λ = 0.5 + 10i λ = 5 + 4z3 λ = 5 + 4cos(10z1) Reconstruction of the cushion

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Reconstruction of the surface impedance

• M. Sini : On uniqueness and reconstruction of rough and complex
• bstacles from acoustic scattering data, Comp. Meth. Appl. Math., 2011.

λ(z) − Z

∂D

λ(x) ∂G ∂νz (x, z)ds(x) = −i Z

∂D

∂G ∂νz (x, z) »∂w ∂ν (x) – ds(x) where the Robin Green function (∆ + κ2)G(x, z) = −δ(x, z) in Rn\D ∂νG + iλG = 0

• n ∂D

G(·, z) satisfies the Sommerfeld radiation condition and w is solution of the Dirichlet exterior problem (∆ + κ2)w = 0 in Rn\D w = 1

• n ∂D

w satisfies the Sommerfeld radiation condition

• F. Cakoni and D. Colton : The determination of boundary coefficients

from far field measrements, J. Int. Eq. Appl., 2010.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Reconstruction of the surface impedance

• M. Sini : On uniqueness and reconstruction of rough and complex
• bstacles from acoustic scattering data, Comp. Meth. Appl. Math., 2011.

λ(z) − Z

∂D

λ(x) ∂G ∂νz (x, z)ds(x) = −i Z

∂D

∂G ∂νz (x, z) »∂w ∂ν (x) – ds(x)

• F. Cakoni and D. Colton : The determination of boundary coefficients

from far field measrements, J. Int. Eq. Appl., 2010. For a partially coated obstacle Z

∂D

λ(x)|vg|2ds = −κFg2 + √ 8πκIm “ eiπ/4(Fg, g) ” where the far field operator F : L2(S1) → L2(S1), Fg(x) = Z 2π u∞(x, d)g(y)ds(d) and vg is the total field associated to v i

g(x) =

Z

S1 eiκx·dg(d)ds(d).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Inverse scattering for cracks with impedance boundary conditions

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The impedance crack problem

Given a smooth nonintersecting open arc σ, a wave number κ > 0, an incident plane wave ui(x, d) = eiκd·x and impedance functions λ± ∈ L∞(σ) with Im(λ±) ≥ 0, λ+ + λ− = 0 a.e : Find the scattered field us unique solution of ∆us + κ2us = 0 in R2 \ σ, satisfying the Sommerfeld radiation condition lim

r→∞

√ r(∂rus − iκus) = 0, and the impedance boundary condition ∂νu± ± λ±u± = 0

• n σ

for the total field u = ui + us where u±(x) = limh→0+ u(x ± hν).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method 1

Inverse scattering by an impedance crack : Given the far field pattern u∞(·, ·) on S1 × S1, Determine the crack σ and the impedances λ±.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method 1

Inverse scattering by an impedance crack : Given the far field pattern u∞(·, ·) on S1 × S1, Determine the crack σ and the impedances λ±.

• D. Colton and A. Kirsch :

A simple method for solving inverse scattering problem in the resonance region, Inverse Problems 1996.

• F. Cakoni and D. Colton :

The linear sampling method for cracks, Inverse Problems, 2003.

• F. Cakoni and D. Colton :

The determination of the surface impedance of a partially coated

• bstacle from far field data, SIAM J. Appl. Math., 2003.
• J. Liu and M. Sini :

Reconstruction of cracks of different types from far field measurements,

• Math. Meth. Appl. Sci., 2010.
• N. Zeev and F. Cakoni :

The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 2009.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method 1

Inverse scattering by an impedance crack : Given the far field pattern u∞(·, ·) on S1 × S1, Determine the crack σ and the impedances λ±.

• Represent the search domain as a grid.

σ

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method 1

Inverse scattering by an impedance crack : Given the far field pattern u∞(·, ·) on S1 × S1, Determine the crack σ and the impedances λ±.

• Represent the search domain as a grid.
• Solve the linear equation

F(gL)(ˆ x) = Φ∞

L (ˆ

x)

L σ

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method 1

Inverse scattering by an impedance crack : Given the far field pattern u∞(·, ·) on S1 × S1, Determine the crack σ and the impedances λ±.

• Represent the search domain as a grid.
• Solve the linear equation

F(gL)(ˆ x) = Φ∞

L (ˆ

x)

• The far field operator

F : L2([0, 2π]) → L2([0, 2π]) F(g)(x) = Z 2π u∞(x, y)g(y)ds(y)

L σ

• Φ∞

L (θ) = eiπ/4

√ 8κπ Z

L

„ αL(y)e−iκθy + βL(y)∂e−iκθy ∂ν(y) « ds(y)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Linear Sampling Method

Theorem (FBH-Boukari-Haddar 2011) Let L be a non-intersecting smooth open arc.

• 1. If L ⊂ σ; there exists a sequence (gn)n∈N on L2(S1) such that

lim

n→∞ F(gn) − Φ∞ L L2(S1) = 0

and lim

n→∞ vgn∗ < ∞,

where vgn∗ := ∂νvgn + λ+vgn

H− 1

2 (σ) + ∂νvgn − λ−vgn

H− 1

2 (σ)

• 2. Otherwise, for any sequence (gn)n∈N ⊂ L2(S1) that satisfies

lim

n→∞ F(gn) − Φ∞ L L2(S1) = 0

we have lim

n→∞ vgn∗ = ∞.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical scheme

ν

L << 1

z

• If z ∈ σ and ν is the normal to σ then g is small
• Otherwise g is large.

Consider two type of solutions : gz corresponding to αL(z) = 1, βL(z) = 0 in Φ∞

L and gz,ν associated to αL(z) = 0, βL(z) = 1.

Let ν1 = (0, 1)t, ν2 = (1, 0)t and ν := ζν1 + p 1 − ζ2ν2, 0 ≤ ζ ≤ 1. By linearity of the far field equation gz,ν = ζgz,ν1 + p 1 − ζ2gz,ν2. Determine on each point z →

1 gz + 1 gz,ν

where gz,ν corresponds with ζ that minimizes gz,ν2 = ζ2gz,ν12 + (1 − ζ2)gz,ν22 + 2ζ p 1 − ζ2 gz,ν1, gz,ν2.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical results

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140

λ± = 0.01(1 + i) (left), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (right).

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 5 10 15 20 25 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 50 100 150 200 250 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40

λ± = 0.01(1 + i) (left), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (right).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Determination of the impedance values

• The natural approach :
• Solve (ζI + F∗F)([u], [∂νu]) = F∗u∞(ˆ

x), where F : ˜ H1/2(σ) × H−1/2(σ) − → L2(S1) defined by F([u], [∂νu])(ˆ x) := Z

σ

“ [u](y)∂νeiκˆ

x·y − [∂νu](y)eiκˆ x·y”

ds(y).

• us

± = Kσ([u]) − Sσ([∂νu]) ± [u]

2 , ∂νus

± = Tσ([u]) − K ′ σ([∂νu]) ± [∂νu]

2 .

• λ± = ∓ ∂νu±

u± .

• For constant impedances λ± = ∓

Z

σ

u±∂νu±

|σ|

Z

σ

|u±|2 .

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Determination of the impedance values

• The natural approach :
• Solve (ζI + F∗F)([u], [∂νu]) = F∗u∞(ˆ

x),

• us

± = Kσ([u]) − Sσ([∂νu]) ± [u]

2 , ∂νus

± = Tσ([u]) − K ′ σ([∂νu]) ± [∂νu]

2 .

• λ± = ∓ ∂νu±

u± .

• The approach inspired by the LSM algorithm :
• if L ⊂ σ then ∂νΦ±

L ± λ±Φ± L ≃ −(∂νvg ± λ±vg), on σ.

• Hence, for constant impedances,

λ± ≃ ∓ Z

L

(∂νΦ±

L + ∂νvg)(Φ± L + vg)

|L| Z

L

|Φ±

L + vg|2

.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Impedance crack within inhomogeneous media

Assume that the index of refraction n is piecewise-constant such that ℑ(n) > 0 and n(x) =  n0 for x ∈ Ω 1 for |x| ≥ R For a source point x0 ∈ C, the outgoing Green function satisfies ∆G(·, x0) + κ2nG(·, x0) = −δx0, in R2. Find the total field u(·, x0) = G(·, x0) + us(·, x0) unique solution ∆us(., x0) + nκ2us(., x0) = 0 in R2 \ σ, satisfying the Sommerfeld radiation condition lim

r=|x|→+∞

√ r(∂rus(·, x0) − inκus(·, x0)) = 0

σ1 σ2 n0 n1 Ω3 n2 n3 Ω4 n4 C Ω Ω1 Γ

and the impedance boundary condition ∂νu± ± λ±u± = 0 on σ, with λ± ∈ L∞(σ), ℑ(λ±) > 0

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The Reciprocity Gap-Linear Sampling Method 1

Inverse scattering by a crack in an inhomogeneous medium : Given {(u(., x0), ∂νu(., x0))|∂Ω, for x0 ∈ C}, determine the crack σ.

• D. Colton and H. Haddar : An application of the reciprocity gap functional to

inverse scattering theory, Inverse Problems, 2005.

• For the crack

F . Delbary : Identifications de fissures par des ondes acoustiques, Université de Paris 6, 2006.

• A crack characterization is given by solving:

R(u(., x0), vg) = R(u(., x0), ΦL)

with R(u, v) = Z

∂Ω

(u∂νv − v∂νu) is the reciprocity gap functional introduced by

• S. Andrieux and A. Ben abda : Identification of planar cracks by complete
• ver-determined data: inversion formulae, IP

, 1991.

• In the vacuum R(u, vg) = R(u, ΦL) ⇐

⇒ F(g) = Φ∞

L .

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

RG-LSM

Solve for any smooth arc L: R(u, Sg) = R(u, ΦL) with R(u, v) = Z

∂Ω

(u∂νv − v∂νu) Sg is the incident wave with density g ∈ L2(Σ), Sg(x) = Z

Σ

g(y)Φ(x, y)ds(y) ΦL(x) = Z

L

(ϕL(y)Φ(x, y) + βL(y)∂νΦ(x, y)) ds(y) ∆Φ(., x0) + k 2n0Φ(., x0) = −δx0

• n R2

σ1 σ2 n0 n1 Ω3 n2 n3 Ω4 n4 Ω Ω1 Γ Σ C

N : L2(Σ) − → L2(C) g − → R(u(., x0), Sg) ΛL(x0) = R(ΦL, u(., x0)) for x0 ∈ C.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

RG-LSM

Solve for any smooth arc L: Ng = ΛL with R(u, v) = Z

∂Ω

(u∂νv − v∂νu) Sg is the incident wave with density g ∈ L2(Σ), Sg(x) = Z

Σ

g(y)Φ(x, y)ds(y) ΦL(x) = Z

L

(ϕL(y)Φ(x, y) + βL(y)∂νΦ(x, y)) ds(y) ∆Φ(., x0) + k 2n0Φ(., x0) = −δx0

• n R2

σ1 σ2 n0 n1 Ω3 n2 n3 Ω4 n4 Ω Ω1 Γ Σ C

N : L2(Σ) − → L2(C) g − → R(u(., x0), Sg) ΛL(x0) = R(ΦL, u(., x0)) for x0 ∈ C.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Theoretical results associated to the RG-LSM

• N is a compact operator

N(g)(x0) = Z

∂Ω

(u(x, x0)∂νSg(x) − ∂νu(x, x0)Sg(x))ds(x) = Z

Σ

g(y)A(x0, y)ds(y) with A(x0, y) := Z

∂Ω

` u(x, x0)∂ν(x)Φ(x, y) − ∂ν(x)u(x, x0)Φ(x, y) ´ ds(x) for (x0, y) ∈ C × Σ

• Factorisation of N = PH

H : L2(S1) − → ˜ H1/2(σ) × H−1/2(σ) g − → (Sg, ∂νSg)|σ P : ˜ H1/2(σ) × H−1/2(σ) − → L2(C) (α , β) − → − Z

σ

(β(x)[u](x, x0) − α(x)[∂νu](x, x0)) ds(x)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Key properties of the operators

Theorem (Boukari-Haddar-FBH 2011) Assume that (λ+ + λ−)−1 ∈ L∞(σ). Then, the operator N is one to one and has a dense range. Theorem (Boukari-Haddar-FBH 2011) Let (λ+ + λ−)−1 ∈ L∞(σ) and L be a smooth and non intersecting arc such that ¯ L ⊂ Ω, for (ϕL, βL) = (0, 0) with ϕL ∈ H−1/2(L) and βL ∈ H1/2(L). Then L ⊂ σ if and only if ΛL ∈ Range(P) Theorem (Boukari-Haddar-FBH 2011) The operator H : L2(S1) − → ˜ H1/2(σ) × H−1/2(σ) is one to one and has a dense range.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Main Theorem of the RG-LSM

Theorem (Boukari-Haddar-FBH 2011) Let L be a non intersecting smooth open arc, ϕL ∈ ˜ H− 1

2 (L) and βL ∈ ˜

H

1 2 (L)

such that (ϕL, βL) = (0, 0).

• 1. If L ⊂ σ, there exists a sequence (gn)n∈N ∈ L2(Σ) such that

(1) lim

n→+∞ N(gn) − ΛLL2(C) = 0.

Furthermore, we have lim

n→+∞ ||Sgn||∗ < +∞

where Sgn∗ = ∂νSgn

H− 1

2 (σ) + Sgn

H

1 2 (σ).

• 2. Otherwise, for any sequence (gn)n∈N ∈ L2(Σ) satisfying (1)

lim

n→+∞ ||Sgn||∗ = ∞.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical results

Geometry of the problem

n =1 Medium 1 n =1 + 0.5 i Medium 2 n =1.5 + 2 i Medium 3 n =2 + 0.5 i Medium 4 The 80 point sources location λ0

The measurements are taken on the boundary of the Medium 1.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical results

−1 1 −1 −0.5 0.5 1 z1 z2

λ n =1

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 −1 1 −1 −0.5 0.5 1 z1 z2

λ n =1

0.005 0.01 0.015 0.02 0.025 0.03 −1 1 −1 −0.5 0.5 1 z1 z2

λ n =1

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 −1 1 −1 −0.5 0.5 1 z1 z2

λ n =2

0.02 0.04 0.06 0.08 0.1 0.12 −1 −0.5 0.5 1 −1 −0.5 0.5 1 z1 z2

λ n =2

0.02 0.04 0.06 0.08 0.1 −1 1 −1 −0.5 0.5 1 z1 z2

λ n =2

0.02 0.04 0.06 0.08 0.1 0.12

Reconstruction of different cracks for λ− = λ+ = 10−2(1 + i) (top) and λ− = 1.2 + 2i, λ+ = 2 + 1.2i (bottom).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical results

Absorbing medium

−1 1 −1 −0.5 0.5 1 z1 z2

λ n =1 + 0.5 i

1 2 3 4 x 10

−4

Figure: Reconstruction of an L-shaped crack for λ− = λ+ = 0.1(1 + i) and a medium index n = 1 + 0.5i.

Multiple cracks

Figure: Reconstruction of multiple cracks for λ±

1 = λ± 4 = .01 and

λ±

2 = λ± 3 = 10(1 + i) for n = 2.

• Does not require to compute the Green function of the background.
• Requires to know the index of the medium containing the crack.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Extension of the application of the RG-LSM

Retrieve the crack σ in the following case

n1 Ω2 n2 (f, g) = (u, ∂νu) Γ1 Ω1 (u, ∂νu) ∆u + k2n1u = 0

knowing (u(., x0), ∂νu(., x0))|∂Ω1, compute (u(., x0), ∂νu(., x0))|∂Ω2 by using a data completion algorithm.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

On the Cauchy problem

There is huge literature, just to name a few methods For the Laplace equation The quasi-reversibility: Lattes and Lions (1967), Bourgeois, Klibanov, Pereverzev Using Tikhonov regularization: Cimetiere, Delavre, Jaoua, Pons. Kozlov algorithm: Kozlov 1991, Ben Abda, Ben Belgacem, El Fekih, Jelassi. Best constrained harmonic or analytical approxiamtion (BEP) : Leblond, Partington, Baratchart, Chaabane, Mahjoub For the Helmholtz equation Ben Abda, Ben Fatma, et al (Decomposition domain, Steklov Poincarré

• perator, Richardson algorithm).

The boundary element method (BEM): Marina et al

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Data completion algorithm

n1 Ω2 n2 (f, g) = (u, ∂νu) Γ1 Ω1 (u, ∂νu) ∆u + k2n1u = 0

u(x) = −DLΓ1(f) + SLΓ1(g) + DLΓ2(u|Γ2) − SLΓ2(∂νu|Γ2), ∀x ∈ Ω1 \ Ω2. where SLΓi ψ(x) := Z

Γi

ψ(y)Φ(x, y)ds(y), DLΓi ϕ(x) := Z

Γi

ϕ(y)∂Φ(x, y) ∂ν(y) ds(y),

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Data completion algorithm 2

Using the trace relationship on Γ1 we have 2 4 DLΓ2→Γ1 −SLΓ2→Γ1 TLΓ2→Γ1 −DL′

Γ2→Γ1

3 5 2 4 u|Γ2 ∂νu|Γ2 3 5 = − 2 4 −KΓ1 − I

2

SΓ1 −TΓ1 K ′

Γ1 − I 2

3 5 2 4 f g 3 5 and on Γ2 2 4 −KΓ2 + I

2

SΓ2 −TΓ2 K ′

Γ2 + I 2

3 5 2 4 u|Γ2 ∂νu|Γ2 3 5 = − 2 4 DLΓ1→Γ2 −SLΓ1→Γ2 TLΓ1→Γ2 −DL′

Γ1→Γ2

3 5 2 4 f g 3 5 SLΓi →Γj ψ(x) := Z

Γi

ψ(y)Φ(x, y)ds(y), DLΓi →Γj ϕ(x) := Z

Γi

ϕ(y)∂Φ(x, y) ∂ν(y) ds(y), DL′

Γi →Γj ψ(x) :=

Z

Γi

ψ(y)∂Φ(x, y) ∂ν(x) ds(y), TLΓi →Γj ϕ(x) := ∂ ∂ν(x) Z

Γi

ϕ(y)∂Φ(x, y) ∂ν(y) ds(y) for x ∈ Γj.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Data completion algorithm 3

We define the Calderon operator PΓi : H1/2(Γi) × H−1/2(Γi) − → H1/2(Γi) × H−1/2(Γi) by PΓi 2 4 ϕ ψ 3 5 = 2 4 −KΓi + I

2

SΓi −TΓi K ′

Γi + I 2

3 5 2 4 ϕ ψ 3 5 , for i = 1 : 2, and TΓi →Γj : H1/2(Γi) × H−1/2(Γi) − → H1/2(Γ1) × H−1/2(Γj) by TΓi →Γj 2 4 ϕ ψ 3 5 := − 2 4 DLΓi →Γj −SLΓi →Γj TLΓi →Γj −DL′

Γi →Γj

3 5 2 4 ϕ ψ 3 5 , for i = j; i, j = 1 : 2. Hence, we have to solve 2 4 TΓ2→Γ1 PΓ2 3 5 2 4 u|Γ2(., x0) ∂νu|Γ2(., x0) 3 5 = 2 4 (PΓ1 − I) TΓ1→Γ2 3 5 2 4 f g 3 5

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Properties of the operators

• PΓi is a projector, satisfies P2

Γi = PΓi

• the operator TΓi →Γj is compact and not injective.
• Ker(TΓ2→Γ1) = Ker(PΓ2 − I)
• Range(TΓ2→Γ1) = Ker(PΓ1) = Range(PΓ1 − I)
• Range(TΓ1→Γ2) = Ker(PΓ2 − I) = Ker(PΓ2)
• Denote

A 2 4 ϕ ψ 3 5 2 4 ϕ ψ 3 5 = 2 4 TΓ2→Γ1 PΓ2 3 5 B 2 4 ϕ ψ 3 5 = 2 4 PΓ1 − I TΓ1→Γ2 3 5 2 4 ϕ ψ 3 5 Main results (Boukari-Haddar 2012) The operator A : X → Y is one to one and has a dense range, where X = H1/2(Γ2) × H−1/2(Γ2) and Y = Ker(PΓ1) × Ker(PΓ2 − I). Moreover Range(A) = Range(B).

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

The main Theorem of the data completion algorithm

Corollary (Boukari-Haddar 2012) Let (f, g) ∈ H1/2(Γ1) × H−1/2(Γ1) and (u|Γ2, ∂νu|Γ2) ∈ H1/2(Γ2) × H−1/2(Γ2) such that A „ u|Γ2 ∂νu|Γ2 « = B „ f g « . (f δ, gδ) ∈ H1/2(Γ1) × H−1/2(Γ1) such that ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ „ f δ gδ « − „ f g «˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ≤ δ Let the Tikhonov solution (uδ, ∂νuδ) satisfying (α(δ) + A∗A) „ uδ ∂νuδ « = A∗B „ f δ gδ « where α(δ) is determined using the Morozov discrepancy principle, then ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ „ uδ ∂νuδ « − „ u ∂νu «˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ − → 0 as δ − → 0

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical tests

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 Exact geometry: test (a)

σ Γ2

n2=2 Γ1 n1=4

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical tests

50 100 −0.5 0.5 representation of the real part of ∂νu 50 100 −0.4 −0.2 0.2 0.4 representation of the imaginary part of ∂νu 50 100 −0.1 −0.05 0.05 0.1 representation of the real part of u 50 100 −0.05 0.05 representation of the imaginary part of u

Figure: Reconstruction of the Cauchy data on Γ2 using the data completion algorithm (red) the exact data (blue) without added noise.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Numerical tests

50 100 −0.4 −0.2 0.2 0.4 0.6 representation of the real part of ∂νu 50 100 −0.4 −0.2 0.2 0.4 representation of the imaginary part of ∂νu 50 100 −0.1 −0.05 0.05 0.1 representation of the real part of u 50 100 −0.05 0.05 0.1 0.15 representation of the imaginary part of u

exact ζ=10% ζ=1%

Figure: Reconstruction of the Cauchy data on Γ2 with added noise.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Application of the data completion algorithm to the RG-LSM

−0.5 0.5 −0.5 0.5 z1 z2

λ n =2

0.005 0.01 0.015 0.02 −0.5 0.5 −0.5 0.5 z1 z2

λ n =2

0.01 0.02 0.03 0.04 0.05 0.06 −0.5 0.5 −0.5 0.5 z1 z2

λ n =2

0.01 0.02 0.03 0.04

Figure: Reconstruction of a crack using 100 discretization points on Γ1, Γ2 and σ, for λ± = 10−3 (left), λ± = 10 (middel), λ± = 103 (right).

−0.5 0.5 −0.5 0.5 z1 z2

λ n =2

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 −0.5 0.5 −0.5 0.5 z1 z2

λ n =2

0.01 0.02 0.03 0.04

Figure: Reconstruction of a crack using 100 discretization points on Γ1, Γ2 and σ, for λ± = 103, without completion (left) and using the completion algorithm (right)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Work in progress and Prospects

• Extend the application of the data completion algorithm to the following

case.

n1 Ω2 Ω1 (u, ∂νu) Γ1 (f, g) = (u, ∂νu)

• Asymptotic analysis of the LSM indicator function in terms of the

geometry and the impedances

• Formulate the Factorization for the RG-LSM configuration.
• Extend the results for the Maxwell equations and the numerics to 3D

setting.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Work in progress and Prospects 2

• Reconstruction of an unkown number of multiple obstacles.
• Identification of a generalized impedance boundary condition (GIBCs)

from the far fields created by one or several incident plane waves at a fixed frequency. More specifically, we shall considered ∂u ∂ν + div∂D(µ∇∂Du) + λu = 0 The wider class of GIBCs is commonly used to model thin coatings or gratings as well as more accurate models for imperfectly conducting

• bstacles.
• Inverse obstacle scattering from measurements of the far field pattern at

multiple frequencies. The ill-posedness of the inverse problem decreases as the frequency increases.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do

Work in progress and Prospects 3

Inverse elastic wave scattering Arens T : Linear sampling methods for 2D inverse elastic wave, IP , 2001 Gintides D and Sini M : Identification of obstacles using only the pressure parts (or only the sheare parts) of the elastic waves. Submitted Navier system µ∆u + (λ + µ)grad(div u) + ρω2u = 0, Kupradze’s radiation condition a boundary condition λ, µ : Lamé constants, ρ : the density, ω the circular frequency.