29 o Col oquio Brasileiro de Matem atica A New Method for the - - PowerPoint PPT Presentation

29o Col´

• quio Brasileiro de Matem´

atica A New Method for the Inverse Potential Problem Based on the Topological Derivative Concept Antonio Andr´ e Novotny1, Alfredo Canelas2 & Antoine Laurain3

1Laborat´

• rio Nacional de Computa¸

c˜ ao Cient´ ıfica, LNCC/MCTI

• Av. Get´

ulio Vargas 333, 25651-075 Petr´

• polis - RJ, Brasil

2Instituto de Estructuras y Transporte, Facultad de Ingenier´

ıa,

• Av. Julio Herrera y Reissig 565, C.P. 11.300, Montevideo, Uruguay

3Institut f¨

ur Mathematik, Technical University Berlin, Street 36, A-8010 Berlin, Germany

IMPA - 31th July, 2013

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 1 / 34

Outline

1

Motivation

2

Topological Derivative Concept

3

Applications of the Topological Derivative

4

Second-Order Topological Derivative

5

Inverse Potential Problem Problem Formulation Topological Derivative Calculation Numerical Results

6

Conclusions

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 2 / 34

Motivation inf

Ω∈E ψ(Ω) ψ(Ω): shape functional Ω: geometrical domain E: set of admissible domains

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 3 / 34

Motivation inf

Ω∈E ψ(Ω) ψ(Ω): shape functional Ω: geometrical domain E: set of admissible domains

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 3 / 34

Topological Derivative Concept

Sokolowski & Zochowski, 1999

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 4 / 34

Topological Derivative Concept

Sokolowski & Zochowski, 1999 ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) , where Ωε( x) = Ω \ ωε( x) and f (ε) → 0, when ε → 0.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 4 / 34

Topological Derivative Concept

Sokolowski & Zochowski, 1999 ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) , where Ωε( x) = Ω \ ωε( x) and f (ε) → 0, when ε → 0. T ( x) = lim

ε→0

ψ(Ωε( x)) − ψ(Ω) f (ε) .

In general, f (ε) = |ωε|. It depends on the boundary condition on ∂ωε.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 4 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as:

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures Image Processing: segmentation, restoration, denoising

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Applications of the Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymp- totic expansion of a shape functional with respect to a singu- lar domain perturbation, like the insertion of holes, inclusions

• r cracks.

The first term of this expansion, called topologi- cal derivative, is now of common use for resolution of several problems, such as: Topology Optimization Inverse Problems: EIT, gravimetry, etc. Multi-Scale Modeling: optimal design of micro-structures Image Processing: segmentation, restoration, denoising Mechanical Modeling: fracture and damage mechanics

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 5 / 34

Topology Optimization

Energy-Based Topological Derivative in Linear Elasticity

Figure : unperturbed problem defined in the domain Ω.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 6 / 34

Topology Optimization

ψ(Ω) := JΩ(u) = 1 2

• Ω

σ(u) · ∇us −

• ΓN

q · u ,            Find u, such that −divσ(u) = in Ω , σ(u) = C∇us u = u

• n

ΓD , σ(u)n = q

• n

ΓN . C = E 1 + ν

• I +

ν 1 − 2ν I ⊗ I

• ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 7 / 34

Topology Optimization

Topological Derivative Calculation

Figure : perturbed problem defined in the domain Ωε.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 8 / 34

Topology Optimization

Topological Asymptotic Expansion ψ(Ωε( x)) = ψ(Ω) − πε3Pσ(u( x)) · ∇us( x) + o(ε3) , P = 3 4 1 − ν 7 − 5ν

• 10I − 1 − 5ν

1 − 2ν I ⊗ I

• A.A. Novotny et al.

Inverse Gravimetry Problem IMPA - 31th July, 2013 9 / 34

Topology Optimization

A benchmark example in 3D ΨΩ(u) := −JΩ(u) + β |Ω| , T = Pσ(u) · ∇us − β .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 10 / 34

Topology Optimization

(a) iteration 13 (b) iteration 35 (c) iteration 52 (d) iteration 76

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 11 / 34

Topology Optimization

(a) top (b) bottom (c) lateral

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 12 / 34

Topology Optimization

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 13 / 34

Second-Order Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + f2(ε)T 2( x) + R(f2(ε)) ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 14 / 34

Second-Order Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + f2(ε)T 2( x) + R(f2(ε)) , where f (ε) → 0 and f2(ε) → 0 with ε → 0, and lim

ε→0

f2(ε) f (ε) = 0 , lim

ε→0

R(f2(ε)) f2(ε) = 0 .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 14 / 34

Second-Order Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + f2(ε)T 2( x) + R(f2(ε)) , where f (ε) → 0 and f2(ε) → 0 with ε → 0, and lim

ε→0

f2(ε) f (ε) = 0 , lim

ε→0

R(f2(ε)) f2(ε) = 0 . (first order) topological derivative T ( x) := lim

ε→0

ψ(Ωε( x)) − ψ(Ω) f (ε) .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 14 / 34

Second-Order Topological Derivative

ψ(Ωε( x)) = ψ(Ω) + f (ε)T ( x) + f2(ε)T 2( x) + R(f2(ε)) , where f (ε) → 0 and f2(ε) → 0 with ε → 0, and lim

ε→0

f2(ε) f (ε) = 0 , lim

ε→0

R(f2(ε)) f2(ε) = 0 . (first order) topological derivative T ( x) := lim

ε→0

ψ(Ωε( x)) − ψ(Ω) f (ε) . second order topological derivative T 2( x) := lim

ε→0

ψ(Ωε( x)) − ψ(Ω) − f (ε)T ( x) f2(ε) .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 14 / 34

Inverse Potential Problem

Problem Formulation: Gravimetry Inverse Problem        Find b∗, such that −∆u = b∗ in Ω , u −∂nu = = u∗ q∗

• n

ΓM . b∗ = γχω∗ ∈ PCγ(Ω) , PCγ(Ω) := {b ∈ L∞(Ω) : b = γχω, ω ⊂ Ω is measurable} ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 15 / 34

Inverse Potential Problem

u[b∗](x) =

• Ω

K(x, y)b∗(y) dy , K(x, y) =      1 4π|x − y| for n = 3 , − 1 2π ln |x − y| for n = 2 .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 16 / 34

Inverse Potential Problem

u[b∗](x) =

• Ω

K(x, y)b∗(y) dy , K(x, y) =      1 4π|x − y| for n = 3 , − 1 2π ln |x − y| for n = 2 . u∗ := u[b∗]|ΓM and q∗ := −∂nu[b∗]|ΓM .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 16 / 34

Inverse Potential Problem

Difficulties The problem is over determined and highly ill-posed; Additional measurements do not provide extra information; Lack of uniqueness if the intensity γ and the region ω∗ are unknown.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 17 / 34

Inverse Potential Problem

Difficulties The problem is over determined and highly ill-posed; Additional measurements do not provide extra information; Lack of uniqueness if the intensity γ and the region ω∗ are unknown. ⊲ We assume that the intensity γ is known

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 17 / 34

Inverse Potential Problem

Theorem (Uniqueness Result)

Let ω1 and ω2 be two star-shaped domains with respect to their centers of gravity. If u1 = u2 and ∂nu1 = ∂nu2 on ΓM, with |ΓM| = 0, then ω1 = ω2.

• V. Isakov. Inverse Source Problems. American Mathematical

Society, Providence, Rhode Island, 1990.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 18 / 34

Inverse Potential Problem

Kohn-Vogelius Criterion min

b∈PCγ(Ω) J(b) := 1

2

• Ω
• uD[b] − uN[b]

2 ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 19 / 34

Inverse Potential Problem

Kohn-Vogelius Criterion min

b∈PCγ(Ω) J(b) := 1

2

• Ω
• uD[b] − uN[b]

2 ,    −∆uD = b in Ω , uD = u∗

• n

ΓM , uD = uT

• n

Γ ,    −∆uN = b in Ω , −∂nuN = q∗

• n

ΓM , uN = uT

• n

Γ ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 19 / 34

Inverse Potential Problem

Kohn-Vogelius Criterion min

b∈PCγ(Ω) J(b) := 1

2

• Ω
• uD[b] − uN[b]

2 ,    −∆uD = b in Ω , uD = u∗

• n

ΓM , uD = uT

• n

Γ ,    −∆uN = b in Ω , −∂nuN = q∗

• n

ΓM , uN = uT

• n

Γ , uT[b] =

• Ω

K(x, y)b(y) dy .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 19 / 34

Inverse Potential Problem

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 20 / 34

Inverse Potential Problem

be,ˆ

x = b + γ

• i∈I

χB(εi ,

xi) .

̟e,ˆ

x = ∪i∈IB(εi,

xi) , with I = {1, ..., m} e := {εi}i∈I ˆ x := {ˆ xi}i∈I , with εi > 0 ,

• xi ∈ Ω

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 20 / 34

Inverse Potential Problem

J(be,ˆ

x) = J(b) −

• Ω

(uD[b] − uN[b])

• i∈I

aihi + 1 2

• Ω
• i∈I

aihi 2 where ai := |B(εi, xi)| and    −∆hi = in Ω , −∂nhi = gi

• n

ΓM , hi =

• n

Γ , with gi = ∂nvi on ΓM and    −∆vi = γδ(x − xi) in Ω , vi =

• n

ΓM . vi = γK(x, xi)

• n

Γ .

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 21 / 34

Inverse Potential Problem

J(be,ˆ

x) = J(b) −

• Ω

(uD[b] − uN[b])

• i∈I

aihi + 1 2

• Ω
• i∈I

aihi 2 Minimization with respect to ai yields Hijaj = fi Hij =

• Ω

hihj and fi =

• Ω

(uD − uN)hi ,

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 22 / 34

Inverse Potential Problem

J(be,ˆ

x) = J(b) −

• Ω

(uD[b] − uN[b])

• i∈I

aihi + 1 2

• Ω
• i∈I

aihi 2 Minimization with respect to ai yields Hijaj = fi Hij =

• Ω

hihj and fi =

• Ω

(uD − uN)hi , J(be,ˆ

x) = J(b) − 1

2aifi

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 22 / 34

Inverse Potential Problem

Example 1: Looking for three anomalies target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 23 / 34

Inverse Potential Problem

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 24 / 34

Inverse Potential Problem

(a) one ball

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 24 / 34

Inverse Potential Problem

(a) one ball (b) two balls

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 24 / 34

Inverse Potential Problem

(a) one ball (b) two balls (c) three balls

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 24 / 34

Inverse Potential Problem

(a) one ball (b) two balls (c) three balls (d) four balls

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 24 / 34

Inverse Potential Problem

Example 2: Partial boundary measurement target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 25 / 34

Inverse Potential Problem

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 26 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 26 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 26 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4 (c) |ΓM| = 0.2

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 26 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4 (c) |ΓM| = 0.2 (d) |ΓM| = 0.1

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 26 / 34

Inverse Potential Problem

Example 3: Noisy data target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 27 / 34

Inverse Potential Problem

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 28 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 28 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 28 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4 (c) |ΓM| = 0.2

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 28 / 34

Inverse Potential Problem

(a) |ΓM| = 1.0 (b) |ΓM| = 0.4 (c) |ΓM| = 0.2 (d) |ΓM| = 0.1

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 28 / 34

Inverse Potential Problem

Example 4: Shape and topology reconstruction

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 29 / 34

Inverse Potential Problem

Example 4: Shape and topology reconstruction

(a) target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 29 / 34

Inverse Potential Problem

Example 4: Shape and topology reconstruction

(a) target (b) result

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 29 / 34

Inverse Potential Problem

Example 5: Two anomalies far from each other

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 30 / 34

Inverse Potential Problem

Example 5: Two anomalies far from each other

(a) target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 30 / 34

Inverse Potential Problem

Example 5: Two anomalies far from each other

(a) target (b) result

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 30 / 34

Inverse Potential Problem

Example 6: Two anomalies close to each other

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 31 / 34

Inverse Potential Problem

Example 6: Two anomalies close to each other

(a) target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 31 / 34

Inverse Potential Problem

Example 6: Two anomalies close to each other

(a) target (b) result

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 31 / 34

Inverse Potential Problem

Example 7: Hidden object

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 32 / 34

Inverse Potential Problem

Example 7: Hidden object

(a) target

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 32 / 34

Inverse Potential Problem

Example 7: Hidden object

(a) target (b) result

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 32 / 34

Inverse Potential Problem

Conclusions

The number of unknown anomalies can be found after some trials. Due to the combinatorial nature of the search procedure, the problem is tractable only in the case of small number of unknown measures. Completely hidden anomalies can be detected from very few information (single partial boundary measurement). Corrupted measurements with a high level of noise can be reconstructed with acceptable precision. The characterization of the biggest set PCγ(Ω) seems to be an open problem.

A.A. Novotny et al. Inverse Gravimetry Problem IMPA - 31th July, 2013 33 / 34

Muito Obrigado!

A.A. Novotny (joint with A. Canelas and A. Laurain). A Non-Iterative Method for the Inverse Potential Problem Based on the Topological

• Derivative. Oberwolfach Seminars vol. 47. Geometries, Shapes and

Topologies in PDE-based Applications. Edited by M. Hinterm¨ uller, G. Leugering, J. Sokolowski. To appear.

• M. Hinterm¨

uller, A. Laurain, A.A. Novotny, Second-order topological expansion for electrical impedance tomography, Advances in Computational Mathematics 36(2):235–265, 2012. A.A. Novotny & J. Soko

• lowski. Topological Derivatives in Shape
• Optimization. Mechanics and Mathematics Iteraction Series. 432p.

Springer, 2013.