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The Inverse Problem of Obstacle Detection viaOptimization
Methods
Examen de grado - Soutenance de thèseMat́ıas GODOY CAMPBELL
Departamento de Ingenieŕıa Matemática, Universidad de
ChileInstitut de Mathématiques de Toulouse, Université Paul
Sabatier
Advisors: Fabien CAUBET - Carlos CONCA.Jury: Grégoire ALLAIRE,
Franck BOYER, Marc DAMBRINE, Axel OSSES.
Referees: Grégoire ALLAIRE, Marc BONNET.
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The inverse problem of obstacle detection: Motivation
Figure: Can we measure ‘something’ in Γobs ⊂ ∂Ω in order to
precise the locationof one or several objects ω? What awesome
things could be inside the bottle? Atreasure map? Riemann
hypothesis proof?
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Inverse obstacle problem
• The data:
- Ω open set of Rd (d ≥ 2), with Lipschitz boundary- Γobs ⊂ ∂Ω,
|Γobs | > 0, Γi := ∂Ω \ Γobs- (gD, gN): (possibly noisy) Cauchy
data in H1/2(Γobs)× H−1/2(Γobs).
• Problem:Find an inclusion ω∗, ω∗ ⊂ Ω, Ω \ ω∗ connected, and u
∈ H1(Ω \ ω∗), s.t.
(P)
∆u = 0 in Ω \ ω∗u = gD on Γobs∂nu = gN on Γobsu = 0 on ∂ω∗
PropertyThere exists at most one couple (ω∗, u) solution of
(P).
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Reconstruction methods (non-exhaustive list)
• 2d problem: method based on conformal mappings:
Conformalmappings and inverse boundary value problem, H. Haddar and
R. Kress,Inverse Problems 21 (2005).
• Exterior approach, based on the Quasi-reversibility method:
Aquasi-reversibility approach to solve the inverse obstacle
problem, L.Bourgeois and J.D., Inverse problems and Imaging
(2010).
• Shape optimization methods: Detecting perfectly insulated
obstacles byshape optimization techniques of order two, L.
Afraites, M. Dambrine, K.Eppler, D. Kateb, Discrete Contin. Dyn.
Syst. Ser. B (2007).
• . . .
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Shape optimization for inverse obstacle problems: general
strategy
1 Initial situation2 choose an arbitrary open set ω b Ω
3 compute u solving the direct problem
∆uω = 0 in Ω \ ω∂νuω = gN on ∂Ωuω = 0 on ∂ω
4 compute J(ω) =∫∂D(gD − uω)2ds → if zero, ω = ω∗, if not,
compute
the shape derivative of J w.r.t. ω → gradient algorithm to
update ω.Mat́ıas GODOY (DIM UChile - IMT UPS) Santiago - 8th July
2016 6 / 52
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Shape optimization for inverse obstacle problems: general
strategy
1 Initial situation2 choose an arbitrary open set ω b Ω
3 compute u solving the direct problem
∆uω = 0 in Ω \ ω∂νuω = gN on ∂Ωuω = 0 on ∂ω
4 compute J(ω) =∫∂D(gD − uω)2ds → if zero, ω = ω∗, if not,
compute
the shape derivative of J w.r.t. ω → gradient algorithm to
update ω.Mat́ıas GODOY (DIM UChile - IMT UPS) Santiago - 8th July
2016 6 / 52
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Shape optimization for inverse obstacle problems: general
strategy
1 Initial situation2 choose an arbitrary open set ω b Ω
3 compute u solving the direct problem
∆uω = 0 in Ω \ ω∂νuω = gN on ∂Ωuω = 0 on ∂ω
4 compute J(ω) =∫∂D(gD − uω)2ds → if zero, ω = ω∗, if not,
compute
the shape derivative of J w.r.t. ω → gradient algorithm to
update ω.Mat́ıas GODOY (DIM UChile - IMT UPS) Santiago - 8th July
2016 6 / 52
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Shape optimization for inverse obstacle problems: general
strategy
1 Initial situation2 choose an arbitrary open set ω b Ω
3 compute u solving the direct problem
∆uω = 0 in Ω \ ω∂νuω = gN on ∂Ωuω = 0 on ∂ω
4 compute J(ω) =∫∂D(gD − uω)2ds → if zero, ω = ω∗, if not,
compute
the shape derivative of J w.r.t. ω → gradient algorithm to
update ω.Mat́ıas GODOY (DIM UChile - IMT UPS) Santiago - 8th July
2016 6 / 52
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Kohn-Vogelius functional
• In our computation, we will minimize a Kohn-Vogelius
functional:
minωK(ω) :=
∫Ω\ω|∇(uDω − uNω )|2 dx
where uDω , uNω solve∆uDω = 0 in Ω \ ωuDω = gD on ∂ΩuDω = 0 on
∂ω
,
∆uNω = 0 in Ω \ ω∂νuNω = gN on ∂ΩuNω = 0 on ∂ω
.
• Advantages:(gD, gN) are treated symmetricallyonly voluminic
quantitiesnumerically: better reconstructions.
• Still a severely ill-posed problem!
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Example of reconstruction
An example of reconstruction with noisy data.
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Incomplete data
• The whole strategy is possible only if the data (gD, gN) are
available onthe whole boundary of the domain Ω (at least one
actually of them).
• But in lots of practical applications, some parts of the
boundary areunaccessible → no measurements on them (particularly
true for fluid problems).
• ⇒ the whole strategy fails.
• Main objective: propose a shape optimization strategy to
reconstruct theunknown inclusion when only Cauchy data are
available only on a subpartof the boundary of the domain of
study.
• Clearly, we have to reconstruct both ω and the missing data −→
datacompletion problem.
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Incomplete data
• The whole strategy is possible only if the data (gD, gN) are
available onthe whole boundary of the domain Ω (at least one
actually of them).
• But in lots of practical applications, some parts of the
boundary areunaccessible → no measurements on them (particularly
true for fluid problems).
• ⇒ the whole strategy fails.
• Main objective: propose a shape optimization strategy to
reconstruct theunknown inclusion when only Cauchy data are
available only on a subpartof the boundary of the domain of
study.
• Clearly, we have to reconstruct both ω and the missing data −→
datacompletion problem.
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Incomplete data
• The whole strategy is possible only if the data (gD, gN) are
available onthe whole boundary of the domain Ω (at least one
actually of them).
• But in lots of practical applications, some parts of the
boundary areunaccessible → no measurements on them (particularly
true for fluid problems).
• ⇒ the whole strategy fails.
• Main objective: propose a shape optimization strategy to
reconstruct theunknown inclusion when only Cauchy data are
available only on a subpartof the boundary of the domain of
study.
• Clearly, we have to reconstruct both ω and the missing data −→
datacompletion problem.
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Data completion problem
• The data:
- Ω open set of Rd (d ≥ 2), with Lipschitz boundary- Γobs ⊂ ∂Ω,
|Γobs | > 0. Γi := ∂Ω \ Γobs .- (gD, gN): (possibly noisy)
Cauchy data in H1/2(Γobs)× H−1/2(Γobs).
• Problem: find u ∈ H1(Ω), s.t. (Pc)
∆u = 0 in Ωu = gD on Γobs∂nu = gN on Γobs
• This problem is severely ill-posed (exponentially ill-posed),
it has at mostone solution that does not depend continuously on the
data.In particular, the set of data for which the problem has no
solution isdense in H1/2(Γobs)×H−1/2(Γobs) ⇒ high instability ⇒ it
is mandatory topropose a regularization method to solve the problem
numerically.
• In the sequel, we denote by uex the exact solution
corresponding toexact data (gD, gN).
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Data completion problem
• The data:
- Ω open set of Rd (d ≥ 2), with Lipschitz boundary- Γobs ⊂ ∂Ω,
|Γobs | > 0. Γi := ∂Ω \ Γobs .- (gD, gN): (possibly noisy)
Cauchy data in H1/2(Γobs)× H−1/2(Γobs).
• Problem: find u ∈ H1(Ω), s.t. (Pc)
∆u = 0 in Ωu = gD on Γobs∂nu = gN on Γobs
• This problem is severely ill-posed (exponentially ill-posed),
it has at mostone solution that does not depend continuously on the
data.In particular, the set of data for which the problem has no
solution isdense in H1/2(Γobs)×H−1/2(Γobs) ⇒ high instability ⇒ it
is mandatory topropose a regularization method to solve the problem
numerically.
• In the sequel, we denote by uex the exact solution
corresponding toexact data (gD, gN).
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Kohn-Vogelius minimization strategy
• Introduced in Solving Cauchy problems by minimizing an
energy-likefunctional, S. Andrieux, T.N. Baranger and A. Ben Abda,
InverseProblems 22, (2006).• Main idea: minimize the energy
functional
K(ϕ,ψ) := 12
∫Ω|∇(uϕ − uψ)|2 dx
over all (ϕ,ψ) ∈ H−1/2(Γi )× H1/2(Γi ), where uϕ and uψ
verify∆uϕ = 0 in Ωuϕ = gD on Γobs∂νuϕ = ϕ on Γi
,
∆uψ = 0 in Ω∂νuψ = gN on Γobsuψ = ψ on Γi .
• Easy remark: K(ϕ,ψ) = 0⇔ uϕ = uex = uψ + cte.
Property
infH−1/2(Γi )×H1/2(Γi )
K(ϕ,ψ) = 0
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Regularization of the K-V functional
• Regularized Kohn-Vogelius functional: for ε > 0, for(ϕ,ψ) ∈
H−1/2(Γi )× H1/2(Γi ),
Kε(ϕ,ψ) = K(ϕ,ψ) +ε
2(‖vϕ‖2H1(Ω) + ‖vψ‖
2H1(Ω)
).
with ∆vϕ = 0 in Ωvϕ = 0 on Γobs∂νvϕ = ϕ on Γi
,
∆vψ = 0 in Ω∂νvψ = 0 on Γobsvψ = ψ on Γi .
PropertyThere exists a unique (ϕε, ψε) ∈ H−1/2(Γi )× H1/2(Γi )
s.t.
Kε(ϕε, ψε) = argmin(ϕ,ψ)∈H−1/2(Γi )×H1/2(Γi )
Kε(ϕ,ψ).
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Convergence results
PropertyThe sequence (ϕε, ψε)ε (ε→ 0) is a minimizing sequence
for K.
TheoremSuppose (Pc) admits a (necessarily unique) solution uex .
Then (ϕε, ψε)converges to (∂νuex , uex + cte) ⇔ uϕε
ε→0−−−−→H1(Ω)
uex .
Furthermore, the convergence is monotonic: the mapε 7→ ‖uϕε −
uex , uψε − uex‖H1(Ω)×H1(Ω) is strictly increasing.
Suppose (Pc) does not admit a solution. Thenlimε→0 ‖ϕε,
ψε‖H−1/2(Γi )×H1/2(Γi ) = +∞.
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Derivatives of Kε
• We define wN , wD ∈ H1(Ω) solutions of∆wN = εvψ in Ω∂νwN =
∂νuϕ − gN on ΓobswN = 0 on Γi
,
∆wD = εvϕ in ΩwD = uψ − gD on Γobs∂νwD = 0 on Γi
.
Property
For all (ϕ,ψ), (ϕ̃, ψ̃) in H−1/2(Γi )× H1/2(Γi ), we have
∂Kε∂ϕ
(ϕ,ψ)[ϕ̃] = 〈ϕ̃, uϕ + εvϕ + wD − ψ〉Γi
and∂Kε∂ψ
(ϕ,ψ)[ψ̃] = 〈∂νuψ + ε∂νvψ + ∂νwN − ϕ, ψ̃〉Γi .
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Inverse obstacle problem with partial Cauchy data
• Problem:Find an inclusion ω∗, ω∗ ⊂ Ω, Ω \ ω∗ connected, and u
∈ H1(Ω \ ω∗), s.t.
(P)
∆u = 0 in Ω \ ω∗u = gD on Γobs∂nu = gN on Γobsu = 0 on ∂ω∗.
• Kohn-Vogelius strategy: minimization of the regularized
Kohn-Vogeliusfunctional w.r.t to ω, ϕ and ψ.
Kε(ω, ϕ, ψ) :=∫
Ω\ω|∇(uϕ − uψ)|2 dx +
ε
2(‖vϕ‖2H1(Ω\ω) + ‖vψ‖
2H1(Ω\ω)
).
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Computation of the shape derivative
• As usual, for V ∈W 2,∞(Rd ), compactly supported in Ω, we
note
DKε(ω) := limt→0Kε((I + tV)ω)−Kε(ω)
t .
PropertyWe have
DKε(ω) · V = −∫∂ω
(∂νρuN ∂νuϕ + ∂νρvN ∂νvϕ)(V · ν)
−∫∂ω
(∂νρuD ∂νuϕ + ∂νρvD ∂νvψ)(V · ν)
+ 12
∫∂ω|∇(uϕ − uψ)|2(V · ν)
+ ε2
∫∂ω
(|∇vϕ|2 +∇vψ|2 + |vϕ|2 + |vψ|2
)(V · ν)
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Computation of the shape derivative
We parametrize the boundary of the object ω in polar
coordinates:
∂ω ={( x0y0 )+ r(ϕ)( cosϕsinϕ ) , ϕ ∈ [0, 2π)} ,
And taking into account of the ill-posedness of the problem, we
regularize withthe approximation of the polar radius r by its
truncated Fourier series
rN(ϕ) := aN0 +N∑
k=1aNk cos(kϕ) + bNk sin(kϕ),
Then, the unknown shape is entirely defined by the coefficients
(ai , bi ). Hence,for k = 1, . . . ,N, the corresponding
deformation directions are respectively,
V1 := Vx0 :=( 1
0), V2 := Vy0 :=
( 01), V3(ϕ) := Va0 (ϕ) :=
( cosϕsinϕ
),
V2k+2(ϕ) :=Vak (ϕ) :=cos(kϕ)( cosϕ
sinϕ),V2k+3(ϕ) := Vbk (ϕ) := sin(kϕ)
( cosϕsinϕ
),
ϕ ∈ [0, 2π).
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Global algorithm
- choose an initial guess (ω0, ϕ0, ψ0)
- at step n,1 solve 10 (!) elliptic problems in Ω \ ωn to obtain
uϕn , uψn , vϕn , vψn ,
wN , wD, ρuD, ρuN , ρvD and ρvN2 compute the descent directions
ϕ̃ and ψ̃3 compute the ∇Kε(ωn)4 update ϕn, ψn, ωn (line search) →
ϕn+1, ψn+1, ωn+1.
- repeat until stopping criterion is reached.
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Reconstructions - easy case
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Reconstructions - hard case
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Reconstructions - hard case
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Noisy data case
Up to now, we have explored the problem when the Cauchy data
(gD, gN)is perfect. What can we expect if we can only obtain
contaminatedCauchy data?• Exact data: (gD, gN) ∈ H1/2(Γ)× H−1/2(Γ)→
K, uex .
• Noisy data: (gδD, gδN) ∈ H1/2(Γ)× H−1/2(Γ)→ Kδ.
• The amplitude of noise is known:
‖gD − gδD‖H1/2(Γ) + ‖gN − gδN‖H−1/2(Γ) ≤ δ.
Is it possible to retrieve uex in the limit δ → 0?
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Noisy data case
We have to propose a strategy to set the parameter of
regularization εw.r.t. noise amplitude
PropertyWe have
‖(ϕ∗ε, ψ∗ε)− (ϕ∗ε,δ, ψ∗ε,δ)‖H−1/2(Γi )×H1/2(Γi ) ≤ Cδ√ε.
CorollaryIf we consider ε = ε(δ) such that
limδ→0
ε(δ) = 0 and limδ→0
δ√ε
= 0
we have‖(ϕ∗ε,δ, ψ∗ε,δ)− (ϕ∗, ψ∗)‖H−1/2(Γi )×H1/2(Γi ) → 0
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Publications
This work has generated the following publication:
[1] F. Caubet, J. Dardé and M. Godoy.A Kohn-Vogelius approach
to study the data completion problem andthe inverse obstacle
problem with partial Cauchy data for Laplace’sequation.To be
submitted.
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Mathematical Model: Posing the Detection Problem
Let us consider the non-compressible and stationary fluid
case:
Stokes system −ν∆u +∇p = 0 in Ω \ ωdiv u = 0 in Ω \ ωu = f on
∂Ωu = 0 on ∂ω
Our aim now is to reconstruct an obstacle ω∗ based in a
measurement onO ⊂ ∂Ω:
σ(u, p)n = g on O ⊂ ∂Ω
where σ(u, p) = νD(u)− pI = ν(∇u + t∇u)− pI
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Mathematical Model: Posing the Detection Problem
The following result implies the well definition of the
problem:
Identifiability Result [Alvarez et al. 2005]Let (ui , pi ), i =
1, 2 the solutions for the problems defined in Ω \ ωi foreach
correspondant i :
−ν∆ui +∇pi = 0 in Ω \ ωidiv ui = 0 in Ω \ ωiui = f on ∂Ωui = 0
on ∂ωiσ(ui , pi )n = gi on O ⊂ ∂Ω
If g1 = g2 then ω1 = ω2.
Important hypothesis: The measurements gi are perfect (this
means,without error). Also, we only need ∂Ω to be Lipschitz.
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Mathematical Model: Posing the Detection Problem
In virtue of the previous result, we consider the following
strategy:
StrategyLet us consider the pairs (uN , pN) ∈ H1(Ω \ ω)× L2(Ω \
ω),(uD, pD) ∈ H1(Ω \ ω)× L20(Ω \ ω) who solves:
(PN)
−ν∆uN +∇pN = 0 in Ω \ ωdiv uN = 0 in Ω \ ωσ(uN , pN)n = g on OuN
= f on ∂Ω \ OuN = 0 on ∂ω
, (PD)
−ν∆uD +∇pD = 0 in Ω \ ωdiv uD = 0 in Ω \ ωuD = f on ∂ΩuD = 0 on
∂ω
.
Minimize (over all the admissible ‘objects’ ω) the Kohn-Vogelius
functional, givenby:
JKV (ω) = FKV (uN(ω),uD(ω)) =ν
2
∫Ω\ω|D(uN(ω)− uD(ω))|2dx
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Mathematical Model: Posing the Detection Problem
Notice that we have the following equivalence:
Inverse Problem as a Minimization ProblemGiven the data g, f on
O and ∂Ω respectively, to determine the position ofthe desired
object, we have to solve:
(O){
Find ω∗ ∈ Dad , such that :JKV (ω∗) = minω∈Dad JKV (ω)
We will use two different criterias to perform this
minimization, each onewill respond to a different objective and
therefore will consider a differenttechnique.
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Mathematical Model: Posing the Detection Problem
Notice that we have the following equivalence:
Inverse Problem as a Minimization ProblemGiven the data g, f on
O and ∂Ω respectively, to determine the position ofthe desired
object, we have to solve:
(O){
Find ω∗ ∈ Dad , such that :JKV (ω∗) = minω∈Dad JKV (ω)
We will use two different criterias to perform this
minimization, each onewill respond to a different objective and
therefore will consider a differenttechnique.
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References (non-exhaustive list)
[1] C. Álvarez, C. Conca, I. Fritz, O. Kavian and J.
Ortega.Identification of immersed obstacles via boundary
measurements.Inverse Problems. 21 (2005), 1531–1552.
[2] A. Ben Abda, M. Hassine, M. Jaoua and M.
Masmoudi.Topological sensitivity analysis for the location of small
cavities inStokes flow.SIAM J. Control Optim. 48 (2010).
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Topological Optimization
In our first approach the minimization will be based under the
criteria thatwe want to detect several objects without knowing the
number of them apriori. For this, we will use a tool called
‘Topological derivative’, whichdefines the way we will perform this
(topological) optimization:
AssumptionOur admissible domain space will be restricted to
domains with fixedshape and small size:
Dad = {ωz,ε = z + εω, ω is open with 0 ∈ ω ⊂ B(0, 1), 0 < ε�
1, ωz,ε ⊂⊂ Ω}
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Topological Optimization
In our first approach the minimization will be based under the
criteria thatwe want to detect several objects without knowing the
number of them apriori. For this, we will use a tool called
‘Topological derivative’, whichdefines the way we will perform this
(topological) optimization:
AssumptionOur admissible domain space will be restricted to
domains with fixedshape and small size:
Dad = {ωz,ε = z + εω, ω is open with 0 ∈ ω ⊂ B(0, 1), 0 < ε�
1, ωz,ε ⊂⊂ Ω}
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Topological Optimization
Topological Optimization basically explores how changes a cost
functionalwhen we add an object (with prescribed shape) in the
original domain.The objective is to obtain an expression like:
Topological asymptotic expansion
JKV (Ω \ ωz,ε) = JKV (Ω) + ξ(ε) · δJKV (z) + o(ξ(ε)), (1)
where ξ(ε) is a positive function of ε which goes to 0 when ε→
0. Theterm δJKV (z) is called the ‘topological derivative’ of JKV
in z ∈ Ω.
The topological gradient δJKV (z) basically measures the cost of
add theobject ωz,ε on the domain Ω. Noticing that ξ(ε) is positive,
the strategythen should be:
Add ωz,ε where δJKV (z) is the most negative possible
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Topological Optimization
Topological Optimization basically explores how changes a cost
functionalwhen we add an object (with prescribed shape) in the
original domain.The objective is to obtain an expression like:
Topological asymptotic expansion
JKV (Ω \ ωz,ε) = JKV (Ω) + ξ(ε) · δJKV (z) + o(ξ(ε)), (1)
where ξ(ε) is a positive function of ε which goes to 0 when ε→
0. Theterm δJKV (z) is called the ‘topological derivative’ of JKV
in z ∈ Ω.
The topological gradient δJKV (z) basically measures the cost of
add theobject ωz,ε on the domain Ω. Noticing that ξ(ε) is positive,
the strategythen should be:
Add ωz,ε where δJKV (z) is the most negative possible
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Topological Optimization
Let us consider the following notation: Ωz,ε = Ω \ ωz,ε.From now
we focus in the 2D case, we consider the problems (PD), (PN)with ω
= ωz,ε, so the solutions will depend of ε and we will use
thenotation (uεN , pεN) and (uεD, pεD).With this in mind, we get
our main result (Caubet, Conca and Godoy,2015):
Theorem: Topological asymptotic expansion for our functional in
2D
JKV (Ωz,ε) = JKV (Ω) +4πν− log ε(|u
0D(z)|2 − |u0N(z)|2) + o(ξ(ε)), (2)
or equivalently: ξ(ε) = 1− log ε , δJKV (z) = 4πν(|u0D(z)|2 −
|u0N(z)|2
).
Where superscript 0 denotes the solutions of problems (PD), (PN)
in Ω.
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Topological Optimization
Let us consider the following notation: Ωz,ε = Ω \ ωz,ε.From now
we focus in the 2D case, we consider the problems (PD), (PN)with ω
= ωz,ε, so the solutions will depend of ε and we will use
thenotation (uεN , pεN) and (uεD, pεD).With this in mind, we get
our main result (Caubet, Conca and Godoy,2015):
Theorem: Topological asymptotic expansion for our functional in
2D
JKV (Ωz,ε) = JKV (Ω) +4πν− log ε(|u
0D(z)|2 − |u0N(z)|2) + o(ξ(ε)), (2)
or equivalently: ξ(ε) = 1− log ε , δJKV (z) = 4πν(|u0D(z)|2 −
|u0N(z)|2
).
Where superscript 0 denotes the solutions of problems (PD), (PN)
in Ω.
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Topological asymptotic expansion proof
The proof is based in two ‘big’ parts:1 Obtain an asymptotic
expansion of uεD/N2 Estimate JKV (Ωε)− JKV (Ω)
A detailed presentation of each step (and the rest of this
presentation!)has been published:
[1] F. Caubet, C. Conca and M. Godoy.On the detection of several
obstacles in 2D Stokes flow: topologicalsensitivity and combination
with shape derivatives.Inverse Problems and Imaging. 10(2) (2016),
327–367.
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Topological asymptotic expansion proof
Theorem: Asymptotic expansion of uεD/N in 2D
The respective solutions uεD ∈ H1(Ωz,ε) and uεN ∈ H
1(Ωz,ε) of Problems (PD)and (PN) admit the following asymptotic
expansion (with the subscript \ = Dand \ = N respectively):
uε\(x) = u0\(x) +1
− log ε (C \(x)−U\(x)) + OH1(Ωz,ε)
(1
− log ε
), (3)
where (U\,P\) ∈ H1(Ω)× L20(Ω) solves the following Stokes
problem defined inthe whole domain Ω{
−ν∆U\ +∇P\ = 0 in Ωdiv U\ = 0 in Ω
U\ = C \ on ∂Ω,(4)
with C \(x) := −4πνE (x − z)u0\(z), where E is the fundamental
solution of theStokes equations in R2 given byE (x) = 14πν
(− log ‖x‖I + er ter
), P(x) = x2π‖x‖2 .
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Topological asymptotic expansion proof
Estimate JKV (Ωε)− JKV (Ω)We have:
JKV (Ωε)− JKV (Ω) = AD + AN ,where
AD :=12ν∫
ΩεD(uεD − u0D) :D(uεD − u0D)
+ ν∫
ΩεD(uεD − u0D) :D(u0D)−
12ν∫
ωε
|D(u0D)|2
andAN :=
∫∂ωε
[σ(uεN − u0N , pεN − p0N)n
]· u0N −
12ν∫
ωε
|D(u0N)|2.
Proof: (A little bit tricky) Integration by parts.Remark: Now we
have a decoupled expression!.
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Numerical Algorithm
Algorithm1 fix an initial shape ω0 = ∅, a maximum number of
iterations M and
set i = 1 and k = 0,2 solve Problems (PD) and (PN) in Ω\
(⋃kj=0 ωj
),
3 compute the topological gradient δJKV using the formula from
(1),
δJKV (P) = 4πν(|u0D(P)|2 − |u0N(P)|2
)∀P ∈ Ω\
k⋃j=0
ωj
,4 seek P∗k+1 := argmin
(δJKV (P), P ∈ Ω\
(⋃kj=0 ωj
)),
5 if ‖P∗k+1−Pj0‖ < rk+1 + rj0 + 0.01 for j0 ∈ {1, . . . , k},
where rj0 is theradius of ωj0 and rk+1 is defined such that there
are no intersections,then rj0 = 1.1 ∗ rj0 , get back to the step 2.
and i ← i + 1 while i ≤ M,
6 set ωk+1 = B(P∗k+1, rk+1), where rk+1 is defined such that
there areno intersections,
7 while i ≤ M, get back to the step 2, i ← i + 1 and k ← k +
1.
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Numerical Algorithm: Some tests
Framework of the examples:The domain Ω is the rectangle [−0.5,
0.5]× [−0.25, 0.25], g is measuredon all faces except on the one
given by y = 0.25, and we considerf = (1, 1)t .We start with a
simple example:
Figure: Detection of ω∗1 , ω∗2 and ω∗3
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Numerical Algorithm: Some tests
We continue with an example where things go (expectedly)
wrong:
Figure: Bad Detection for a ‘very big sized’ object
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Numerical Algorithm: Some tests
Let us see an animated example of how our algorithm works, where
theobject to be detected has a different geometry to our default
shape:
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References (non-exhaustive list)
[1] J. Soko lowski and A. Żochowski.On the topological
derivative in shape optimization.SIAM J. Control Optim., 37(4)
(1999), 1251–1272.
[2] V. Bonnaillie-Noel and M. Dambrine.Interactions between
moderately close circular inclusions: TheDirichlet-Laplace equation
on the plane.Asymptotic Analysis. 84 (2013).
[3] F. Caubet, M. Dambrine, D. Kateb, C. Timimoun.Localization
of small obstacles in Stokes flow.Inverse Problems. 28 (2012).
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Geometrical Optimization: A complementary task
Up to now, we’ve been able to detect (small) objects, precise
their numberand their relative location.We would like to improve
the quality of our algorithm in the followingsenses:
1 The numerical simulations suggest the need to improve
thecomputation of the relative location of the objects.
2 The shape of the objects: Our topological approach works with
afixed shape for the objects, naturally we cannot expect to find
alwayscircular objects.
To this end, we will consider a useful and classical tool from
geometricaloptimization: the shape gradient.
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Geometrical Optimization: A complementary task
Up to now, we’ve been able to detect (small) objects, precise
their numberand their relative location.We would like to improve
the quality of our algorithm in the followingsenses:
1 The numerical simulations suggest the need to improve
thecomputation of the relative location of the objects.
2 The shape of the objects: Our topological approach works with
afixed shape for the objects, naturally we cannot expect to find
alwayscircular objects.
To this end, we will consider a useful and classical tool from
geometricaloptimization: the shape gradient.
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Geometrical Optimization: A complementary task
For our functional, we have the following result:
Shape Gradient for our functional (Caubet et al. 2011)For θ ∈ U,
the Kohn-Vogelius cost functional JKV is differentiable at Ω \ ω in
thedirection θ with
DJKV (Ω\ω) · θ = −∫
∂ω
(σ(w , q) n) · ∂nuD(θ · n) +12ν∫
∂ω
|D(w)|2 (θ · n), (5)
where (w , q) is defined by w := uD − uN and q := pD − pN .
As in the first part we parametrize the boundary of the object ω
in polarcoordinates and we consider the truncated fourier series
for the polarradius.
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Geometrical Optimization: A complementary task
For our functional, we have the following result:
Shape Gradient for our functional (Caubet et al. 2011)For θ ∈ U,
the Kohn-Vogelius cost functional JKV is differentiable at Ω \ ω in
thedirection θ with
DJKV (Ω\ω) · θ = −∫
∂ω
(σ(w , q) n) · ∂nuD(θ · n) +12ν∫
∂ω
|D(w)|2 (θ · n), (5)
where (w , q) is defined by w := uD − uN and q := pD − pN .
As in the first part we parametrize the boundary of the object ω
in polarcoordinates and we consider the truncated fourier series
for the polarradius.
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Mixed Approach Algorithm
Algorithm1 fix a number of iterations M and take the initial
shape ω0 (which can
have several connected components) given by the previous
topologicalalgorithm,
2 solve problems (PD) and (PN) with ωε = ωi ,3 compute ∇JKV (Ω \
ωi ) using formula (5),4 move the coefficients associated to the
shape:ωi+1 = ωi − αi∇JKV (ωi ),
5 get back to the step 2. while i < M.Note: The number of
parameters increases gradually during the algorithm.
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Mixed Approach Algorithm: Results
For Ω the unit circle, f = (n2,−n1)t and g measured in the whole
disk ∂Ωexcept the lower right quadrant, we have:
Figure: Detection of squares ω∗1 and ω∗2 with the combined
approach (the initialshape is the one obtained after the
“topological step”) and zoom on theimprovement with the geometrical
step for ω∗2
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Outline
1 Introduction: First part
2 Data completion and obstacle detectionProblemKohn-Vogelius
minimization strategyNoisy data
3 Introduction: Second Part
4 Topological OptimizationMain ResultNumerical Results
5 A Combination with Geometrical Optimization
6 Going Further
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Going Further: Obstacle detection with incomplete data.
Future (and present) things which seems a natural continuation
of ourwork:
1 Data completion for other systems: Obstacle detection for
fluids withincomplete data (Stokes, Navier-Stokes): Ben Abda
(numerical trialsonly).
2 Data completion via KV minimization in an abstract setting:
Dardé(Abstract data completion via QR).
3 Computation of regularization parameter ε: Propose
concretestrategies to choose ε in noisy cases: Dardé (Using
abstract setting +QR method).
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Going Further: Topological gradient techniques.
1 How about considering Navier-Stokes (at least stationary):
Amstutz,Chetboun (in an applied context).
2 Final Challenge: What if the obstacles can move?: Conca
(Complexvariable), Conca, Schwindt, Takahashi (rigid convex body,
Stokes).
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Gracias por su atención.Merci pour votre attention.
Thank you for your attention.
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Introduction: First partData completion and obstacle
detectionProblemKohn-Vogelius minimization strategyNoisy data
Introduction: Second PartTopological OptimizationMain
ResultNumerical Results
A Combination with Geometrical OptimizationGoing Further
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