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Inverse scattering in acoustics and elasticity using high-order topological derivatives Marc Bonnet 1 1 POems, (ENSTA, CNRS, INRIA, Universit e Paris-Saclay), Palaiseau, France mbonnet@ensta.fr Workshop Analysis and Numerics of Acoustic and

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Inverse scattering in acoustics and elasticity using high-order topological derivatives

Marc Bonnet1

1POems, (ENSTA, CNRS, INRIA, Universit´

e Paris-Saclay), Palaiseau, France mbonnet@ensta.fr

Workshop ”Analysis and Numerics of Acoustic and Electromagnetic Problems”, Oct. 21, 2016

Marc Bonnet1 Inverse scattering with high-order topological derivatives 1 / 42

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Outline

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)
  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 2 / 42

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Model problem: acoustic transmission problem

Scattering of an acoustic wave u by a penetrable obstacle B (ρ⋆, c⋆) embedded in an acoustic medium Ω (ρ, c).

  • ∆ + k2

uB = 0 in Ω\B,

  • β∆ + ηk2

uB = 0 in B, ∂nuB = iρωV D

  • n S,

uB|+ = uB|− and ∂nuB|+ = η∂nuB|−

  • n ∂B.

? u uobs B(ρ⋆, c⋆) Ω(ρ, c)

β := ρ/ρ⋆ (mass density ratio), η := (ρc2)/(ρc2)⋆ (bulk modulus ratio) Problem considered: identification of B by means of small-inclusion asymptotics.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 3 / 42

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Wave-based identification

? u uobs B(ρ⋆, c⋆) Ω(ρ, c)

Standard approach (e.g. Full Waveform Inversion): minimize a cost functional (e.g.

  • utput least-squares)

(B⋆, ρ⋆, c⋆) = arg min

B′,ρ′,c′

  • JLS(B′, ρ′, c′) := 1

2

  • M
  • u[B′, ρ′, c′] − uobs
  • 2

dΓ

  • Entails repeated evaluations of forward solutions u[B′, ρ′, c′] (with varying B′, ρ′, c′)

Impetus for development of alternative identification methods:

◮ Linear sampling method (Colton, Kirsch ’96), factorization method (Kirsch’98)

Mathematically justified; require abundant data, qualitative

◮ Topological derivative (TD):

(very) partial mathematical justification so far; any data, qualitative

◮ High-order topological derivative (this talk): quantitative enhancement of TD, any

data

Marc Bonnet1 Inverse scattering with high-order topological derivatives 4 / 42

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Identification problem and cost functional

Experimental configuration fur true (unknown) inhomogeneity B⋆(ρ⋆, c⋆) Simulation for trial inhomogeneity B′(ρ′, c′)

u u′

B

Ω(ρ, c) B′(ρ′, c′)

The discrepancy between B⋆ and B′ is evaluated by the cost functional J (B′) := J(uB′) = 1 2

  • Γ
  • uB′ − uobs

2 dΓ uB′ := u[B′, ρ′, c′] Usual idea: minimize J (B′) w.r.t. (some of the characteristics of) the trial obstacle B′. Note: quadratic cost functional considered for definiteness in this talk, but more general choices possibble as well.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 5 / 42

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Identification using topological derivative: overview

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)

Numerical example

  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Numerical example

Marc Bonnet1 Inverse scattering with high-order topological derivatives 6 / 42

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Identification using topological derivative: overview

Cost functional for a small trial defect Ba

Experimental configuration fur true (unknown) inhomogeneity B⋆(ρ⋆, c⋆) Simulation for small trial inhomogeneity Ba(ρ′, c′) u ua Ω(ρ, c) a z Ba(ρ′, c′) Ba = z + aB Introducing the scattered field v a := ua −u, we have J (Ba) := J(ua) = 1 2

  • Γ
  • u +v a −uobs

2 dΓ = J(u) + J′(u; v a) + 1 2J′′(u; v a)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 7 / 42

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Identification using topological derivative: overview

Topological derivative T3

The leading-order expansion of J(ua) is already known as: J(ua) = J(u) + a3T3(z, B) + o(a3) T3 is called the topological derivative (or sensitivity, or gradient) of J. [Sokolowski, Zochowski, 1999; Garreau, Guillaume, Masmoudi, 2001; Guzina, B 2004; Amstutz, Takahashi, Wexler 2008] ... Heuristic: locations z where T3(z) takes the most negative values are good candidates for the real defect location. Many empirical validations of this heuristic even for macroscopic defects

◮ using synthetic data (references above and many more) ◮ using experimental data [Tokmashev, Tixier, Guzina 2013].

Heuristic proved in some cases:

◮ β = 0 and “moderate” scatterers [Bellis, B, Cakoni 2013], ◮ small obstacles [Ammari et al. 2012],

High-frequency behavior investigated [Guzina, Pourahmadian 2015] (T3 tends to emphasize the boundaries of the obstacles).

Marc Bonnet1 Inverse scattering with high-order topological derivatives 8 / 42

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Identification using topological derivative: overview

TD as topology optimization tool

20 40 60 80 100

Iterations

4 4.5 5 5.5 6

Compliance

PhD G. Delgado (2014) — Topology optimization combining topological and shape derivatives

Marc Bonnet1 Inverse scattering with high-order topological derivatives 9 / 42

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Identification using topological derivative: overview

TD as imaging functional for qualitative flaw identification

FEM-based computation of T3, transient wave equation, 2D: simultaneous identification

  • f a multiple scatterer

α = 0.5

Bellis, B, Int. J. Solids Struct. (2010)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 10 / 42

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Identification using topological derivative: overview

TD as imaging functional for qualitative flaw identification

Figure 2. Three-dimensional motion sensing via laser Doppler vibrometer (LDV) system.

(a)

9 cm 10 cm 0.6 cm

Bhole Bslit

S

D

S

N

1.5 cm

S

piezo 5

S

piezo 4

S

piezo 3

S

piezo 2

S

piezo 1

1 4 13 14 63 66

(b)

S

  • bs

5

S

  • bs ⊂S

N

Figure 4. Testing configuration: (a) photograph of the damaged plate, and (b) boundary conditions and spatial arrangement of the LDV scan points for five individual source locations (Spiezo

k

, k = 1, 5).

Topological derivative z → T (z) for J(uD) = 1 2 T

  • S

|uD − uobs|2 dS dt

  • R. Tokmashev, A. Tixier, B. Guzina, Inverse Problems (2013)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 11 / 42

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Identification using topological derivative: overview

T3 : Full and partial aperture (time-harmonic elastodynamics)

Full aperture:

Γ u(x) Ωtest Btrue

Partial aperture:

b b b b b b b b b b b b b b b b b b b b b b b b b

Γ = xn u(x) Ωtest Btrue

[PhD thesis R. Cornaggia, 2016]

Marc Bonnet1 Inverse scattering with high-order topological derivatives 12 / 42

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Higher-order topological expansion (acoustics)

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)

Numerical example

  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Numerical example

Marc Bonnet1 Inverse scattering with high-order topological derivatives 13 / 42

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Higher-order topological expansion (acoustics)

Higher-order topological expansion: motivation

Computation of topological derivative of J Non-iterative; Computationally faster than a minimization-based inversion algorithm; (2 forward solutions governed by same linear field equations) Yields qualitative results, since the approximation J (Ba) ≈ J(u) + a3T3(z) (Ω ⊂ RD) cannot be minimized Inaccurate localization when using partial-aperture measurements Higher-order topological expansion − → polynomial (in a) approximation of cost function J : Much faster to compute than J ; Lends itself to minimization w.r.t. defect size a Provide usable results with sparser data

Marc Bonnet1 Inverse scattering with high-order topological derivatives 14 / 42

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Higher-order topological expansion (acoustics)

Expansion of the cost functional

Expansion of the cost functional: J(a) = J(ua) = J(u) + J′(u; v a) + 1 2J′′(u; v a); Known behavior of scattered field: v a(x) = a3W (x; z) + O(a4) for x ∈ Ba, hence: J(ua) = J(u) + J′(u; v a) + a6

1 2J′′(u; W ) + o(1)

  • To retain the contribution of the last term in the expansion, we seek

J(ua) = J(u) + a3T3(z) + a4T4(z) + a5T5(z) + a6T6(z) + o(a6) = ⇒ J′(u; v a) is to be expanded to order O(a6). Identification approach: (aest, zest) = arg min

(a,z)

  • a3T3(z) + a4T4(z) + a5T5(z) + a6T6(z)
  • Quantitative evaluation of size a.

Improved localization when using partial-aperture measurements. Previous work: [B 2008] (3D acoustics, sound-hard obstacles), [B 2010] (2D electrostatics, inclusions), [Silva et al., 2010] (2D elastostatics, holes), in all cases with expansions derived but not justified.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 15 / 42

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Higher-order topological expansion (acoustics)

Adjoint solution

Cost functional expansion (ua = u +v a): J(a) = J(ua) = J(u) + J′(u; v a) + a6

1 2J′′(u; W ) + o(1)

  • Evaluation of leading perturbation J′(u; v a): define adjoint solution ˆ

u by Find ˆ u ∈ H1(Ω),

  • ˆ

u, w

  • Ω − k2

ˆ u, w

  • Ω = J′(u; w) , ∀w ∈ H1(Ω),

(a) Invoke (weak formulations of) background and scattering problems: Find u ∈ H1(Ω),

  • u, w
  • Ω − k2

u, w

  • Ω = iρω
  • V D, w
  • S

∀w ∈ H1(Ω), (b) Find uB ∈ H1(Ω),

  • uB, w

βB

Ω − k2

uB, w ηB

Ω = iρω

  • V D, w
  • S

∀w ∈ H1(Ω), (c) with βB := 1+(β −1)χ(B), ηB := 1+(η −1)χ(B) ∈ L∞(Ω). Combining

  • (a) with w = v a

+

  • (b) with w = ˆ

u

  • (c) with w = ˆ

u

  • yields

J′(u; v a) = Re ua, ˆ u β−

1 Ba

− k2 ua, ˆ u η−

1 Ba

  • Since
  • Ba

() dV = O(a3), inner expansions of ua, ∇ua in Ba needed to order O(a3) These will be established using a volume integral equation (VIE) with support Ba

Marc Bonnet1 Inverse scattering with high-order topological derivatives 16 / 42

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Higher-order topological expansion (acoustics)

Volume integral equation formulation

  • I − La
  • ua = u

in Ω Volume integral operator (VIO) La: La := Ha + k2Ga,

  • Gaw
  • (x) =
  • Ba

(η −1)G(·, x)w dV ,

  • Haw
  • (x) = −
  • Ba

(β −1)∇1G(·, x)·∇w dV G(·, x): Green’s function associated with the domain Ω and Neumann BC on S, defined for a point source x ∈ Ω by −(∆ + k2)G(·, x) = δx in Ω, ∂nG(·, x) = 0

  • n S

(notation: ∇kG means gradient w.r.t. k-th argument)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 17 / 42

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Higher-order topological expansion (acoustics)

Scaling of volume integral equation

Rescaled coordinates: x = z + a¯ x and dVx = a3 d ¯ V¯

x

(x ∈ Ba, ¯ x ∈ B) Decomposition of Green’s function: G(ξ, x) = G∞(ξ −x) +

  • G∞,k(ξ −x) − G∞(ξ −x)
  • + GC(ξ, x)
  • GC,0(x, ξ)

with G∞(r) = 1 4π|r|, G∞,k(r) = eik|r| 4π|r|,

  • GC,0(x, ξ)
  • ≤ C (x, ξ ∈ Ba; a < a0)

Expansions: G(ξ, x) = a−1G∞(¯ x −¯ ξ) + O(1), ∇1G(ξ, x) = a−2∇G∞(¯ x −¯ ξ) + O(1) Rescaling of integral operator (with [Paw](¯ x) := w(z +a¯ x)): I − La = P−1

a (I − HB)Pa − Lc a ,

Lc

a := k2a2P−1 a GBPa + a2HC,0 + k2a3GC,0,

HBW (¯ x) =

  • B

(β −1)∇G∞(¯ x −·)·∇W dV , GBW (¯ x) =

  • B

(η −1)G∞(¯ x −·)W dV Lemma: P−1

a HBPa = O(1),

Lc

a = O(a) (as H1(Ba) → H1(Ba) operators).

HB: VIO for static problems and normalized inclusion. Hence, inner solution leading term(s) (as a → 0) expected to solve rescaled static inclusion problems

Marc Bonnet1 Inverse scattering with high-order topological derivatives 18 / 42

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Higher-order topological expansion (acoustics)

Inner expansion of solution

  • 1. Inner solution ansatz:

ua(x) =

  • U0 + aU1 + a2U2 + a3U3 + a4U4
  • (¯

x) + δa(x) (x ∈ Ba, ¯ x ∈ B),

  • 2. Taylor expansion of background field u about x = z,
  • 3. Rescaling and expansion of G:

G(ξ, x) = G∞(ξ −x) +

  • G∞,k(ξ −x) − G∞(ξ −x)
  • + GC(ξ, x)

G(z +a¯ ξ, z +a¯ x) = a−1G∞(¯ x −¯ ξ) + ik 4π − a k2 8π |¯ x −¯ ξ| + G c

z + a

  • ∇2G c

z ·¯

x + ∇1G c

z ·¯

ξ

  • + O(a2),

∇1G(z +a¯ ξ, z +a¯ x) = a−2∇G∞(¯ ξ − ¯ x) + k2 8π ∇|¯ x −¯ ξ| + a ik3 12π (¯ x −¯ ξ) + ∇1G c

z + a

  • ∇11G c

z ·¯

ξ + ∇12G c

z ·¯

x

  • + O(a2)

(shorthand notations G c

z := GC(z, z), ∇kG c z := ∇kGC(z, z), ∇kℓG c z := ∇kℓGC(z, z))

Use 1, 2, 3 in integral equation

  • I − La
  • ua = u

in Ω

Marc Bonnet1 Inverse scattering with high-order topological derivatives 19 / 42

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Higher-order topological expansion (acoustics)

Inner expansion of solution

Solution ansatz: ua(x) =

  • U0 + aU1 + a2U2 + a3U3 + a4U4
  • (¯

x) + δa(x) (x ∈ Ba, ¯ x ∈ B) We obtain governing VIEs for U0, U1, U2, U3, U4 in B (with shorthand notations uz := u(z), g z := u(z), g (k)

z

:= ∇ku(z)):

  • I − HB
  • U0(¯

x) = uz,

  • I − HB
  • U1(¯

x) = g z ·¯ x,

  • I − HB
  • U2(¯

x) = 1 2g (2)

z :¯

x⊗2 + k2 GBU0

  • (¯

x),

  • I − HB
  • U3(¯

x) = 1 6g (3)

z :¯

x⊗3 + k2 GBU1 + G0

BU0

  • (¯

x) +

  • H0

BU1

  • (¯

x),

  • I − HB
  • U4(¯

x) = 1 24g (4)

z :¯

x⊗4 + k2 GBU2 + G0

BU1 + G1 BU0

  • (¯

x) +

  • H0

BU2 + H1 BU1

  • (¯

x). GB, HB: as defined before; G0

B, G1 B, H0 B, H1 B: known H1(B) → H1(B) VIOs with bounded kernels.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 20 / 42

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Higher-order topological expansion (acoustics)

Inner expansion of solution

Auxiliary VIOs:

  • G0

BU

  • (¯

x) = ik 4π + G c

z B

(η −1)U dV ,

  • G1

BU

  • (¯

x) = −k2UB

  • (η −1)U
  • (¯

x) +

  • B

(η −1)

  • ∇2G c

z ·¯

x + ∇1G c

z ·¯

ξ

  • U dV ,
  • H0

BU

  • (¯

x) = −k2div UB[(β −1)∇U1](¯ x) − ∇1G c

z ·

  • B

(β −1)∇U dV ,

  • H1

BU

  • (¯

x) = −

  • B

(β −1)∇U·

  • ∇12G c

z ·¯

ξ + ∇11G c

z ·¯

x + ik3 12π (¯ x −¯ ξ)

  • dV

in terms of the volume potential U defined for densities g ∈ L2

loc(R3) by

[Ug](x) = 1 8π

  • B

|¯ x −·|g dV .

Marc Bonnet1 Inverse scattering with high-order topological derivatives 21 / 42

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Higher-order topological expansion (acoustics)

Free-space transmission problems, polarization tensors

Free-space transmission problem (FSTP): div (βB∇uB) = div (∇u) in R3, uB(ξ) − u(ξ) = O(|ξ|−2) (|ξ| → ∞). Equivalent VIE form: (I − HB)uB = u in R3. . Lemma Let 0 < β < ∞. I−HB : H1(B) → H1(B) is bounded and invertible with bounded inverse. Polynomial background field: define Up

B = Up B[E p] as the solution of FSTP

(I − HB)Up

B = E p • x⊗ p

in B. When B ellipsoid, Up

B[E p] is polynomial with degree p in B (e.g. Eshelby problem).

Polarization tensors (PTs) Apq: defined by identification through

  • uB[E p] , ϕq[E q]

β−1

B

= E p • Apq • E q Reciprocity property of PTs (implies knowing uB[E m] with m := min(p, q) is sufficient for finding Apq): E p • Apq • E q = E q • Aqp • E p.

(proof: stems from reciprocity identity

  • uB , u′ β−1

B

=

  • u′

B , u

β−1

B

for FSTPs)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 22 / 42

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Higher-order topological expansion (acoustics)

Inner expansion of solution

Recall ansatz ua(x) =

  • U0 + aU1 + a2U2 + a3U3 + a4U4
  • (¯

x) + δa(x) (x ∈ Ba, ¯ x ∈ B) U0 = uz, U1 = U1

B[g z],

U2 = 1 2U2

B[g (2) z ] + k2uzXB,

(¯ x ∈ B) U3 = 1 6U3

B[g (3) z ] + k2Y 1 B + D3 z ,

U4 = 1 24U4

B[g (4) z ] + k2Y 2 B + k4uzZB + D4 z + U1 B[E z].

scalars D3

z , D4 z and vector E z are constant w.r.t. ¯

x and depend on GC, U1 and U2; functions ZB, XB, Y 1

B, Y 2 B are auxiliary FSTP solutions:

  • I − HB
  • ZB =

UB[(η −1)1],

  • I − HB
  • XB = ∆UB[(η −1)1],
  • I − HB
  • Y 1

B = ∆UB[(η −1)U1] − div UB[(β −1)∇U1],

  • I − HB
  • Y 2

B = ∆UB[(η −1)U2] − div UB[(β −1)∇U2].

Marc Bonnet1 Inverse scattering with high-order topological derivatives 23 / 42

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Higher-order topological expansion (acoustics)

Proof of lemmas on VIEs for FSTPs

Only sparse literature on mathematical properties of VIEs, e.g. [Potthast 1999], [Costabel, Darrigrand, Kone 2010], [Kirsch 2008], [Kirsch, Lechleiter 2009] (electromagnetism), [B 2016] (elastodynamics) We have HBU = W[(βB −1)∇U], with the volume potential W defined by W : L2

comp(R3) → H1 loc(R3),

W[h](x) =

  • R3 ∇G∞(x −·)·h dV

Reformulation of FSTP (I − HB)uB = u: set h := (βB −1)∇uB, then: (a)

  • I − (β −1)∇W
  • [h](x) = (βB −1)∇u

in R3, (b) uB = u + W[h] in R3, with A := I − (β −1)∇W. Boundedness of W : L2(B) → H1(B). Use G(ρ) = (4π2|ρ|2)−1 and convolution thm:

  • ∇W[h](ρ) = −(ˆ

ρ⊗ ˆ ρ)· h(ρ) (with ˆ ρ := ρ/|ρ|). Hence | ∇W[h](ρ)| ≤ | h(ρ)| pointwise, so (Plancherel) ∇W[h]L2(R3) ≤ hL2(R3) .

Marc Bonnet1 Inverse scattering with high-order topological derivatives 24 / 42

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Higher-order topological expansion (acoustics)

Proof of lemmas on VIEs for FSTPs9

Bounded invertibility of I − (β −1)∇W : L2

comp(R3) → L2(R3). We have

I − (β −1)∇W = βB +1 2

  • I − K
  • ,

with K := βB −1 βB +1

  • I + 2∇W
  • From previous step, we have F
  • (I + 2∇W)h
  • (ˆ

ρ) =

  • I − 2ˆ

ρ⊗ˆ ρ

  • ·F[h](ρ), implying
  • F
  • (I + 2∇W)h
  • (ˆ

ρ)

  • =
  • h(ρ)
  • pointwise. Therefore:
  • (I + 2∇W)h
  • L2(R3) = hL2(R3),

implying

  • K
  • L2(R3) ≤ sup

R3

  • βB −1

βB +1

  • =
  • β −1

β +1

  • < 1

(0 < β < ∞). I − K is therefore invertible with bounded inverse (Neumann series), and so is I − (β −1)∇W. Consequences: bounded invertibility of (I −HB) : H1(B) → H1(B) P−1

a (I −HB)Pa :

H1(Ba) → H1(Ba) (with a-independent bound for all a0) Fredholm character of (acoustic, elastic) frequency-domain VIEs follows. [Potthast 1999], VIEs for 2D Maxwell with orthotropic inhomogeneities.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 25 / 42

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Higher-order topological expansion (acoustics)

Inner expansion of solution: estimate of remainder

Proposition Let 0 < β < ∞. There exists a1 > 0 and C > 0 such that δaH1(Ba) < Ca11/2 for all a < a1 Sketch of proof: Combining the VIEs governing (i) ua, (ii) U0, . . . U4, δa satisfies a VIE (I − La)δa = γa . γa is found to verify γaH1(Ba) < Ca11/2 for some C > 0; We have I − La = Aa − a2Lc

a

(where Aa := P−1

a [I − HB]Pa)

= Aa

  • I − A−1

a a2Lc a

  • Aa is invertible, and A−1

a ≤ C1 with C1 independent on a.

Lc

a is defined in terms of VIOs with O(|ξ −x|−1) or O(1) kernels (ξ, x ∈ Ba), implying (by

scaling arguments) Lc

a < C2a−1 for any a < a0.

Therefore (Neumann series), I − A−1

a a2Lc a is invertible with bounded inverse for all

a < a1 := min

  • a0, (C1C2)−1

.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 26 / 42

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Higher-order topological expansion (acoustics)

Cost functional expansion: result

Theorem (B, 2016) For a single centrally-symmetric inhomogeneity with support Ba := z +aB and parameters β, η, we have J(a) = J(0) + a3T3(z) + a4T4(z) + a5T5(z) + a6T6(z) + o(a6) with the topological derivatives T3, . . . T6 given by T3(z) = g z ·A11·ˆ g z − k2(η −1)|B|uz ˆ uz T4(z) = 0 T5(z) = 1 6A13 • (g z ⊗ ˆ g (3)

z

+ ˆ g z ⊗g (3)

z ) + 1

4g (2)

z :A22 : ˆ

g (2)

z

− 1 2k2B20 :(uz ˆ g (2)

z

+ ˆ uzg (2)

z )

− k2g z ·(Q11 + N 11)·ˆ g z − k4uz ˆ uz

  • 1, XB

η−1

B

T6(z) = 1 2J′′(u; W ) + E z ·A11·ˆ g z − k2|B|(η −1)D3

z ˆ

uz T3: usual (leading) topological derivative B20, Q11, N 11: additional auxiliary constant tensors defined from FSTP solutions

Marc Bonnet1 Inverse scattering with high-order topological derivatives 27 / 42

slide-28
SLIDE 28

Higher-order topological expansion (acoustics)

Cost functional expansion: result

Sketch of proof: recall expansion J(ua) = J(u) + Re ua, ˆ u β−

1 Ba

− k2 ua, ˆ u η−

1 Ba

  • + a6

1 2J′′(u; W ) + o(1)

  • Main task: expansion of Re

ua, ˆ u β−

1 Ba

− k2 ua, ˆ u η−

1 Ba

  • :

Expansion of Re Ua, ˆ u β−

1 Ba

− k2 Ua, ˆ u η−

1 Ba

  • : use Ua and Taylor expansion about

z of ˆ u, ∇ˆ u, truncate to order O(a6); Estimate remainder by Cauchy-Schwarz inequality:

  • δa, ˆ

u β−

1 Ba

− k2 δa, ˆ u η−

1 Ba

  • ≤ CδaH1(Ba) ˆ

uH1(Ba) ≤ Ca11/2a3/2 = Ca7

Marc Bonnet1 Inverse scattering with high-order topological derivatives 28 / 42

slide-29
SLIDE 29

Higher-order topological expansion (acoustics)

Spherical trial scatterer

All auxiliary solutions, PTs etc are known in closed form: U1

B[E] =

3 β +2E ·x, U2

B[E] =

5 2β +3E :(x ⊗x) + β −1 3β E :I

  • 1− 3|x|2

2β +3

  • ,

∇U1

B[E] =

3 β +2E, ∇U2

B[E] =

10 2β +3E ·x − 2(β −1) β(2β +3)(E :I)x XB = η −1 6β

  • 2β +1 − |x|2

, A11 = 4π β −1 β +2I, B20 = 4π 3 (η −1)(5β −2) 15β I, A13 = 12π 5 β −1 β +2I ⊗I, Q11 = 12π 5 (η −1) (β +2)2 I, A22 = 16π 15 β −1 2β +3

  • 5I − β −1

β I ⊗I

  • ,

N 11 = 8π 5 (β −1)2 (β +2)2 I.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 29 / 42

slide-30
SLIDE 30

Higher-order topological expansion (acoustics) Numerical example

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)

Numerical example

  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 30 / 42

slide-31
SLIDE 31

Higher-order topological expansion (acoustics) Numerical example

Identification of ellipsoidal inhomogeneity in a half-space

z x u uobs Ω(ρ, c)

Ellipsoidal inhomogeneity:

  • center (−2, 0.5, −3.5),
  • axes 41/3(0.2, 0.1, 0.1),
  • β = 2, η = 0.5
  • k = 1, 2, 5

Size estimation and localization procedure J6(a, z) = a3T3(z) + a5T5(z) + a6T6(z) (a) Compute Tj(z) for all z ∈ Ωtest (b) Define amin(z) = arg min

a

J6(a, z) (c) zest = arg min

z∈Ωtest J6(amin(z), z)

(d) aest = amin(zest)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 31 / 42

slide-32
SLIDE 32

Higher-order topological expansion (acoustics) Numerical example

Identification of ellipsoidal inhomogeneity in a half-space

TD with partial aperture (k = 5); (# sources × # sensors) (4 × 25) (25 × 121) (100 × 441) (x, z) (x, y)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 32 / 42

slide-33
SLIDE 33

Higher-order topological expansion (acoustics) Numerical example

Identification of ellipsoidal inhomogeneity in a half-space

TD with full aperture (k = 5); (# sources × # sensors) (20 × 20) (74 × 74) (452 × 452) (x, z) (x, y)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 33 / 42

slide-34
SLIDE 34

Higher-order topological expansion (acoustics) Numerical example

Identification of ellipsoidal inhomogeneity in a half-space

Relative error 4π 3 (aest)3/|B⋆| − 1 1/3

  • n radius:

k 4 × 25 1 0.002 2 0.041 5 0.23 zest − z = 0.1 in all cases (minimum achievable with search grid used; equivalent radius: 0.2) J6(amin(z), z) (x, z) (x, y)

Marc Bonnet1 Inverse scattering with high-order topological derivatives 34 / 42

slide-35
SLIDE 35

Higher-order topological expansion (elastodynamics – with R. Cornaggia)

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)

Numerical example

  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Numerical example

Marc Bonnet1 Inverse scattering with high-order topological derivatives 35 / 42

slide-36
SLIDE 36

Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Identification problem and cost functional

F uobs B(C, ρ) D(C⋆, ρ⋆)

The discrepancy between B⋆ and B′ is evaluated by the cost functional J (B′) := J(uB′) = 1 2

  • Γ
  • uB′ − uobs

2 dΓ uB′ := u[B′, ρ′, C′]

Marc Bonnet1 Inverse scattering with high-order topological derivatives 36 / 42

slide-37
SLIDE 37

Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Cost functional expansion: result (elastodynamics)

Theorem (Cornaggia, B, 2016) For a single spherical inhomogeneity with support Ba := z +aB and parameters ρ⋆, A⋆, embedded in an isotropic elastic background material, we have J(a) = J(0) + a3T3(z) + a4T4(z) + a5T5(z) + a6T6(z) + o(a6) with the topological derivatives T3, . . . T6 given by T3(z) = Re

  • ∇u(z):A11 :∇ˆ

u(z) − |B|ω2∆ρu(z)·ˆ u(z)

  • T3(z) = 0

T5(z) = Re 1 10A11 • ∇(∆u)(z)⊗∇ˆ u(z) + ∇u(z)⊗∇(∆ˆ u)(z)

  • + ∇2u(z) • A22 • ∇2ˆ

u(z) − k2

S∇u(z):Q11 :∇ˆ

u(z) − k2

S∆ρQ20 •

∇2u(z)⊗ ˆ u(z) + u(z)⊗∇2ˆ u(z)

  • − k4

S∆ρ2E5u(z)·ˆ

u(z)

  • T6(z) = 1

2J′′(u; W ) + Re

  • ∇u(z):
  • A + iI H1

1

  • :A:∇ˆ

u(z) − |B|ω2∆ρ

  • [∇u(z):A:∇1G C(z, z)·ˆ

u(z) + ∇ˆ u(z):A:∇1G C(z, z)·u(z)

  • + |B|2ω2∆ρu(z)·G C(z, z)·ˆ

u(z) + ω4∆ρ2E5u(z)·ˆ u(z)

  • where all scalar and tensor-valued constants are known (in closed form if A⋆ is also isotropic)

Similar expansion also established for anisotropic background medium.

Marc Bonnet1 Inverse scattering with high-order topological derivatives 37 / 42

slide-38
SLIDE 38

Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example

  • 1. Identification using topological derivative: overview
  • 2. Higher-order topological expansion (acoustics)
  • 3. Higher-order topological expansion (elastodynamics – with R. Cornaggia)

Numerical example

Marc Bonnet1 Inverse scattering with high-order topological derivatives 38 / 42

slide-39
SLIDE 39

Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example

Identification of spherical inhomogeneity in a full space

b b b b b b b b b b b b b b b b b b b b b b b b b

Γ = xn u(x) Btrue Infinite medium (UC(x, ξ; ω) = 0). Spherical obstacle Dtrue. Discrete measurements, partial aperture. Background field: single plane P-wave u(x) = u0 exp (ikP(e1 · x)) e1. Measurements uobs simulated using the analytical scattering solution known for this configuation. Adjoint field: ˆ u(x) =

N

  • n=1

(u − uobs)(xn) · U(xn − x; ω). [PhD thesis R. Cornaggia, 2016]

Marc Bonnet1 Inverse scattering with high-order topological derivatives 39 / 42

slide-40
SLIDE 40

Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example

Weak scatterer: a = 0.2 = λS/10, ∆C = 0.2C, ∆ρ = 0.2ρ

b b b b b b b b b b b b b b b b b b b b b b b b b

Γ = xn u(x) Ωtest Btrue

Size estimation and localization procedure J6(a, z) = a3T3(z) + a5T5(z) + a6T6(z) (a) Compute Tj(z) for all z ∈ Ωtest (b) Define amin(z) = arg min

a

J6(a, z) (c) zest = arg min

z∈Ωtest J6(amin(z), z)

(d) aest = amin(zest)

T3(z)

1 1.5 2 2.5 3 1 1.5 2 2.5 3

J6(amin(z); z)

1 1.5 2 2.5 3 1 1.5 2 2.5 3 Marc Bonnet1 Inverse scattering with high-order topological derivatives 40 / 42

slide-41
SLIDE 41

Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example

Accuracy of localization and size estimation

u(x) Ωtest

Normalized size a=6S

0.05 0.1 0.15 0.2 0.25 0.3

jzest ! zexactj a

0.1 0.2 0.3 0.4 0.5 0.6

Error in localization

partial aperture full aperture

Normalized size a=6S

0.05 0.1 0.15 0.2 0.25 0.3

a

0.1 0.2 0.3 0.4 0.5 0.6 Exact and estimated obstacle size

aexact aest , partial aperture aest , full aperture Marc Bonnet1 Inverse scattering with high-order topological derivatives 41 / 42

slide-42
SLIDE 42

Higher-order topological expansion (elastodynamics – with R. Cornaggia) Numerical example

Importance of sixth-order term for size estimation

a 0.05 0.1 0.15 0.2 0.25 #10 -6

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 a3T3(zest) a5T5(zest) a6T6(zest) J6(a; zest) real size: atrue = 0:2 estimated size: aest = 0:207

Marc Bonnet1 Inverse scattering with high-order topological derivatives 42 / 42