the inverse problem of obstacle detection via …mgodoy/presentaciones/examenes/examen...jury:...

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.

Mat́ıas GODOY (DIM UChile - IMT UPS) Santiago - 8th July 2016 1 / 52

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?


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


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).


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).

• . . .


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

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!


Example of reconstruction

An example of reconstruction with noisy data.


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.


Outline






6 Going Further


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).


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


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).


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 ) = +∞.


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 .


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)

).


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)


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).


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.


Reconstructions - easy case


Reconstructions - hard case


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?


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


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.


Outline






6 Going Further


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



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.



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



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.


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).


Outline






6 Going Further


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}



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



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.


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.



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 .



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!.


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.


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



We continue with an example where things go (expectedly) wrong:

Figure: Bad Detection for a ‘very big sized’ object



Let us see an animated example of how our algorithm works, where theobject to be detected has a different geometry to our default shape:


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).


Outline






6 Going Further


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.


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.


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.


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


Outline






6 Going Further


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).


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).


Gracias por su atención.Merci pour votre attention.

Thank you for your attention.


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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