1. An Inverse Geometrical Problem in Fluid Mechanics
Carlos Conca
Depto. Ingeniería Matemática
Centro de Modelamiento Matemático
UMR 2071 CNRS – U. de Chile
(with C. Alvarez, L. Friz, O. Kavian and J. H. Ortega)
2/23/2019
WONAPDE 2004

2. This lecture is organized as follows:
Setting of the Problem
Introduction to Inverse Problems
Identifiability and Stability Results
Algorithm and Numerical Results
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3. Setting of the Problem
Recover geometrical information (position and shape) about an a-priori unknown body D immersed in an incompressible viscous liquid.
To this end, we perform measurements (on velocity and stress forces) along the boundary of the cavity Ω fulfilled by the liquid.
Non-steady incompressible Stokes equations for the liquid.
Non-homogeneous Dirichlet boundary condition on Ω.
Non-slip condition on D. D Ω
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4. What are Inverse Problems?
In an inverse boundary problem one seeks to determine the internal properties of a medium by performing measurements along the boundary of the medium.
The appropriate mathematical model of the physical situation is usually given by a PDE inside the medium.
The boundary measurements are encoded in a certain boundary map. The inverse problem is to determine the coefficients of the PDE from knowledge of this boundary map.
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5. A Propotypical Example: Calderon´s Problem
Electrical Impedance Tomography (I)
Let  be a bounded smooth domain in Rn. Let (x) be the unknown electrical conductivity of the medium fulfilling Ω.
(x) is assume to be in L1 () and strictly positive. The potential u in Ω with voltage f on Ω satisfies
div (  (x) r u) = 0 in 
u = f on 
In this case the boundary map is the so-called voltage to current map
f   (f ) =  u on  n
Calderon´s inverse problem is to recover  from the boundary map  :   
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WONAPDE 2004

6. A Prototypical Example: Calderon´s Problem Electrical Impedance Tomography (II)
More precisely, we can divide Calderon’s Inverse problem in several parts
Injectivity of the mapping  (Identifiability)
Continuity of  and its inverse -1 if it exists (Stability)
What's the range of ? (Characterization)
Formula to recover  form  (Reconstruction)
Give a numerical algorithm to find and approximation of  (Numerical Reconstruction)
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7. Mathematical Framework for our Inverse Problem in Fluid Mechanic (I)
Let  be a smooth bounded set in Rn and let D   be an unknown rigid body immersed in the liquid.
Let  2 H1Á2(  ) be a non homogeneous Dirichlet boundary data and let (v,p) be the solution of Stokes equations in ¤ : =  n D
-   v +  p = 0 in ¤
div v =0 in ¤ ….(P)
v =  on 
v = 0 on D
Let  (v, p) be the linear stress tensor, defined by
 (v,p) = -pI + 2  e(v)
and
e(v) = (r v + (r v)T) 2
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8. Mathematical Framework for our Inverse Problem in Fluid Mechanic (II)
The set of admissible bodies D is
Uad = f D   : D is a smooth, open and simply connected region in  g
Let  be the following boundary map, velocity to stress tensor
: D ! D
defined as follows
D ( ) =  (v,p)n on  
(v,p) being the solution of the Stokes system (P).
Our Inverse Problem is to recover D from the above boundary map velocity to stress tensor  : D ! D
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9. Pioneering works Inverse Conductivity Problems A. Calderón (1980)
R. Kohn, M. Vogelius (1984)
J. Sylvester, G. Ulhmann (1986)
Inverse Geometrical Problems
S. Andrieux, A. Ben Abda y M. Jaona (1993)
E. Beretta y S. Vesella (1999)
G. Alessandrini, A. Morassi y E. Rosset (2002)

10. D1  D2 )  (v1, p1)n   (v2, p2)n on , 8
Main Issues
Identifiability result, that is, the injectivity of the velocity to stress tensor map :
D1  D2 )  (v1, p1)n   (v2, p2)n on , 8
Stability result, that is, the continuity of the inverse of the velocity to stress tensor map  (if two measures are close each other, then the rigid bodies are also close).
Algorithm and numerical results allow us to recover the volume and position of the unknown rigid body.
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WONAPDE 2004

11. Identifiability Result
Theorem (C. C., L. Friz, O. Kavian, J. Ortega)
Let  be a smooth open bounded domain in ℝn and D0 , D 1  Uad . Let  be an open non empty subset of  and  2 H1Á2( ). Let (v0, p0) and (v1, p1) be solutions of
-  v0 + p0 = 0 in  \ D0
div v0 =0 in \D0
v0 =  on 
v0 = 0 on D0
and
-   v1 + p1 = 0 in  \ D1
div v1 =0 in \D1
v1 =  on 
v1 = 0 on D1
such that
(v1,p1)n =(v0,p0)n on ,
then
D 1=D0 .
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12. v = 0 and p = constant in  n D.
Sketch of the proof (I)
Let us define D=D0 [ D1 . On  n D we set
v = v1 – v and p = p1 – p0
The couple (v,p) solves
-   v + p = 0 in  \ D
div v = 0 in  \ D
v = 0 on 
and it satisfies
 (v,p) n = 0 on .
From the unique continuation property (C. Fabre & G. Lebeau, 1990), we conclude that
v = 0 and p = constant in  n D.
Then v0 =v1 in  n D. D0 D1
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13. Sketch of the proof (II)D1 D0
If A0 = D1n D0 is non empty, we can write
-   v0 + p0 = 0 in A0
div v0 = 0 in A0
v0 = 0 on A0 = ( D1 Å (D0)c) [ ( D0Å D1)
Multiplication by v0 yields
sA0 |r v0 |2 = 0 ) v0 =0 in A0
Applying the unique continuation property we conclude
v0 =0 in nD0
This is a contradiction because
v0 =   0 2  
Thus A0 = ;. Analogously, one can prove D0n D1 = ; , and hence
D0=D1.
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14. Stability Question D D +u
We want to prove that if the external boundary measurements are close, then the rigid bodies are close.
The main tool we use to tackle stability is the so-called shape differentiation which involves small perturbations of the domain.
We consider a reference domain D and a deformed domain D+u.
The deformation u is assumed to be smooth and such that u = 0 in a neighborhood of . D
D +u 
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15.   vu +  pu = 0 in ¤+u =  n (D-u)
Smoothness of the velocity and pressure with respect to deformations of the rigid body
For each given smooth deformation u and for any Dirichlet boundary condition  2 H1Á2()n, such that s ¢ n =0, we consider the problem:
  vu +  pu = 0 in ¤+u =  n (D-u)
div vu = 0 in ¤+u ……(Pu)
vu =  on ,
vu = 0 on  (D – u)
There exists a unique solution (vu , pu) 2 H1() n £ L2(), of the above problem (Pu).
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16. Regularity of the solutions
Lemma (C. C., L. Friz, O. Kavian, J. Ortega)
There exists a neighborhood W around u=0 such that the mapping u ! ( vu , pu ) ± ( I + u ), which is defined in W and takes values in H1( )n £ L2( ), is analytic in W. Here, (vu , pu) is the unique solution of the problem (Pu).
The proof is based on the implicit function theorem.
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17. Stability Issue
Theorem (C.C., L. Friz, O. Kavian, J. Ortega)
Let u0 2 W3,1(; Rn) be given. Assume that u=tu0. Then there exists a strictly positive constant C=C(uo,¤,D,) and an integer m = m(uo,¤,D,) 2 N, such that
|| D ( ) -  D-tuo () ||H-1Á2( ) ¸ C | t |m
where
D () = (vo , po)n on 
and
D-tuo () = (vtuo, ptuo)n on .
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WONAPDE 2004

18. Sketch of the proof (I)
Assume that the deformed domain has the form ¤+u = nD-u.
Let  2 H1Á2() be given and let (,q) be solution of
-   +  q = 0 in 
div  = 0 in 
 =  on ,
Multiplying (P0) and (Ptuo) by  and integrating by parts we have
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19. Sketch of the proof (II)
Subtracting both identities before we obtain
Therefore
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20. Sketch of the proof (III)
Recall that u = t u0.. Thanks to regularity results, one can expand the right hand side of the above inequality as follows (Murat-Simon´s approach to shape differentiation (1974))
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21. Sketch of the proof (IV)
The previous identity yields the existence of a non negative constant C=C(¤,D, u0,  ) such that for  =  we can write
If C>0, this completes the stability result with m=1.
If C=0, we need to go further and come back to the previous asymptotic expansion analyzing the second order term. A similar calculation to the done above yields the existence of a nonnegative constant C=C(¤,D,u0,) such that
… and so on ….
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22. Algorithm and Numerical Experiments
Test Problem
vu=(0,0) 0 2
 (vu,pu)n =0
vu=(0,0) 1
-  vu + pu = 0 in ¤ + u
div vu = 0 in ¤ + u
vu =  on  = 0[ 1 [ 2
 (vu , pu) n =0 on n 
vu = 0 on D-u.
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23. Algorithm and Numerical Experiments A Suitable Objective Function
Let D be the stress linear tensor on the external boundary Ω measured for the unknown body D.
For numerical purposes, a suitable formulation of our inverse problem is to write it down as the following minimization problem
which has a unique global minimum u=0, due to the identifiability result.
2/23/2019
WONAPDE 2004

24. Explicit Formula for the Gradient
A classical formula of shape differentiation shows that the gradient of our objective function is
(,q) being the unique solution of the adjoint problem
-   + q = 0 in ¤ + u
div  = 0 in ¤ + u
 = 2[σ(vu, pu) – σD]n on 
 ( , q) n = on n
 = 0 on D-u.
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25. Numerical Method Main Features
Non convex and nonlinear objective function.
Steepest descent algorithm (SD) and non linear conjugate gradient (NLCG) method (an explicit formula for the gradient is available).
Existence of multiple local minimum (Heuristic initialization by means of a Simulated Annealing type algorithm).
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26. Velocity Fields
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27. Vorticity
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28. Difference of Vorticities
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29. Iterations
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