Some inverse problems with applications in Elastography: theoretical and numerical aspects

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1 Some inverse problems with applications in Elastography: theoretical and numerical aspects Anna DOUBOVA Dpto. E.D.A.N. - Univ. of Sevilla joint work with E. Fernández-Cara (Universidad de Sevilla) J. Rocha de Faria (Universidade Federal da Paraiba) P. de Carvalho (Universidad Estadual da Piauí) Doc-Course "Partial Differential Equations: Analysis Numerics and Control" Sevilla-Málaga-Granada-Cádiz, April 2nd to June 5th, 2018

2 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

3 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

4 Introduction Inverse problems governed by PDEs Some words on inverse problems: General settings of a direct problem: Data (Ω, D) = Results (R) = Observation (additional information) (I) Well posed problems (in the sense of Hadamard): existence, uniqueness, continuous dependence on the data) A related inverse problem: Some data (Ω) + Information (I) = The other data (D) Known as ill posed problems Here: geometric inverse problems (the unknown is a part of the domain). Other possible inverse problems: unknown coefficients (parameter identification): (γ u) = 0 in Ω unknown the source term (source identification): (γ u) = f in Ω unknown the initial dada (data identification, data assimilation)...

5 Introduction Geometric inverse problem Goal Identification of a shape of a domain (a) Direct problem: Data: Ω, D, T > 0, ϕ = ϕ(x, t) and γ Ω Result: the solution u (1) (2) Information: u tt u = 0 in (Ω \ D) (0, T ) u = ϕ on Ω (0, T ) u = 0 on D (0, T ) u(x, 0) = u 0, u t(x, 0) = u 1 in Ω u := α on γ (0, T ) n (b) Inverse problem: Some data: Ω, T, ϕ and γ Ω (Additional) information: α = α(x, t) Goal: Find D such that the solution to (1) satisfies (2) Lamé vibrations are much greater in one direction than the other, then neglecting small terms, one component of the displacement field approximately satisfies a wave equation...

6 Motivation What is Elastography? Elastrography is noninvasive technique of imaging by ultrasound or MRI, allowing to detect elastic properties of a tissue in real time. Based on the fact that soft tissues are more deformable than stiff matter. For example, tumor tissue is 5-28 times stiffer than normal tissue, then the resulting deformation after a mechanical action is smaller and the difference can be captured by images: It is used in various fields of Medicine (detection and description of breast, liver, prostate and other cancers, fibrosis,... ) First works: [Ophir-et-al 1991], [Muthupillai-et-al 1995], [Sinkus-et-al 2000], [McKnight-et-al 2002],...

7 Elasticity systems: isotropic case, constant Lamé coefficients Similar inverse problem Similarly: elasticity system (b) Inverse problem: given α = α(x, t), ϕ = ϕ(x, t), µ, λ > 0, find D such that satisfies σ(u) n = u tt µ u + (µ + λ) ( u) = 0 in Ω \ D (0, T ) u = ϕ on Ω (0, T ) u = 0 on D (0, T ) u(0) = u 0, u t(0) = u 1 in Ω \ D ( ) µ( u + u t ) + λ( u)id. n := α(x, t) on γ (0, T ) Explanations: u = (u 1, u 2, u 3 ) is the displacement vector σ(u) n is normal stress Small displacements, hence linear elasticity The tissue is described by the Lamé coefficients λ and µ

8 Inverse problems: main questions 3 main questions: 1 Uniqueness: If u 0 and u 1 are two solutions corresponding to utt i u i = 0 in (Ω \ D i ) (0, T ) u i = ϕ on Ω (0, T ) u i = 0 on D i (0, T ) u i (x, 0) = u 0, u t(x, 0) = u 1 in Ω \ D i such that u0 n = u1 n on γ (0, T ) = do we have D0 = D 1? 2 Stability: Find an estimate of the size of (D 0 \ D 1 ) (D 1 \ D 0 ) in terms of the size of α 0 α 1 : ( ) ) Size (D 0 \ D 1 ) (D 1 \ D 0 ) CF ( α 0 α 1 A(γ (0,T )) for all D 1 close to D 0, for some F : R + R + with F (s) 0 as s 0, some suitable space A(γ (0, T )) and C = C(D 0, Ω, γ, T, ϕ 0 ) 3 Reconstruction: Devise iterative algorithms to compute D from α

9 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

10 Uniqueness results I N-dimensional wave equation utt i u i = 0 in Ω \ D i (0, T ), i = 0, 1 u i = ϕ on Ω (0, T ) u i = 0 on D i (0, T ) u i (x, 0) = 0, ut(x, i 0) = 0 in Ω \ D i Theorem T > T (Ω, γ), D i convex, ϕ 0 u 0 n = u1 n on γ (0, T ) = D0 = D 1 Main steps of the proof: 1 Consider G := Ω \ D 0 D 1 and u := u 0 u 1 in G (0, T ): u tt u = 0 in G (0, T ) u = 0 on Ω (0, T ) u = 0 on γ (0, T ) n Unique continuation u 0 in G (0, T )

11 Uniqueness results II N-dimensional wave equation 2 Assume D 0 D 1, i.e. D 0 \ D 1. In (D 0 \ D 1 ) (0, T ), consider u 1 : utt 1 u 1 = 0 in (D 0 \ D 1 ) (0, T ) u 1 = 0 on (D 0 \ D 1 ) (0, T ) u 1 = 0, ut 1 = 0 in (D 0 \ D 1 ) Uniqueness of zero solution u 1 = 0 in D 0 \ D 1 3 In Ω \ D 1 (0, T ) consider u 1. Unique continuation implies that u 1 0, but u 1 = ϕ 0, therefore and then D 0 D 1. 4 In a similar way, we can prove that D 1 D 0, therefore D 0 = D 1. Here we need only Unique Continuation, then NO geometrical condition on γ Also: N-dimensional isotropic Lamé system with constant coefficients: OK

12 Reconstruction and numerical algorithms Reconstruction: Iterative algorithms to compute D from α 1 Method 1: Optimization Problem, FEM, FreeFem++ 2 Method 2: Mesh-less method, MFS, Optimization Problem, MATLAB

13 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

14 Method 1: Optimization problem, FEM, FreeFem++... I Augmented Lagrangian method (ff-nlopt - AUGLAG) Assume: N = 2, D = B(x 0, y 0 ; r) Inverse problem: given α = α(x, t), find x 0, y 0, r such that D Ω and the solution u to the Lamé system satisfies ( ) σ[x 0, y 0 ; r] := µ(x)( u + u t ) + λ(x)( u)id. n = α(x, t) on γ (0, T ) Constrained optimization problem (case of a circle) Find x 0, y 0 and r such that (x 0, y 0, r) X b and J(x 0, y 0, r) J(x 0, y 0, r ) the function J : X b R is defined by (x 0, y 0, r ) X b J(x 0, y 0, r) := 1 2 T 0 σ[x 0, y 0, r] α 2 H 1/2 (γ) dt X b := { (x 0, y 0, r) R 3 : B(x 0, y 0 ; r) Ω }

15 Method 1: Optimization problem, FEM, FreeFem++... II Augmented Lagrangian method (ff-nlopt - AUGLAG) The problem formulation contains inequality constraints { Minimize f (x) Subject to x X 0 ; c i (x) 0, 1 i I Introducing slack variables s i : X 0 = {x R m : x j x j x j, 1 j m} c i (x) 0 rewired as c i (x) s i = 0, s i 0, 1 i I Augmented Lagrangian is defined in therms of the constraints c i (x) s i = 0 Optimization problem (for Augmented Lagrangian) Minimize L A (x, λ k ; µ k ) := f (x) Subject to x X 0 ; s i 0, 1 i I I i=1 λ k i (c i (x) s i ) + 1 2µ k I (c i (x) s i ) 2 i=1 λ k i : multipliers, µ k : penalty parameters

16 Method 1: Optimization problem, FEM, FreeFem++... III Augmented Lagrangian method (ff-nlopt - AUGLAG) Algorithm (Augmented Lagrangian: inequality constraints) (a) Fix µ 1 and starting points x 0 and λ 1 ; (b) Then, for given k 1, µ k, x k 1, λ k : (b.1) Unconstrained optimization: Find an approximate minimizer x k of L A (, λ k ; µ k ), starting at x k 1 : { Minimize LA (x, λ k ; µ k ) (b.2) Update the Lagrange multipliers: λ k+1 i = max Subject to x X 0 ( λ k i c ) i (x k ), 0, 1 i I µ k (b.3) Choose a new parameter and check whether a stopping convergence test is satisfied: µ k+1 (0, µ k ) In this Algorithm, we have to solve unconstrained optimization problems (point (b.1)), the we must specify a new (subsidiary) optimization algorithm. Among others: DIRECTNoScal is variant of the DIviding RECTangles algorithm for global optimization

17 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

18 Numerical results: 2-D Lamé system, D is a circle I Case 1: D is a circle Assume: N = 2, D is the interior of an circle: 3 variables (x 0, y 0 ; r) Test 1: N = 2, Ω = B(0; 10), D = B(x0, y0; r), T = 5 u 01 = 10x, u 02 = 10y, u 11 = 0, u 12 = 0, ϕ 1 = 10x, ϕ 2 = 10y x0ini = 0, y0ini = 0, rini = 0.6 x0des = -3, y0des = 0, rdes = 0.4 NLopt (AUGLAG + DIRECTNoScal), FreeFem++: x0cal = y0cal = rcal = Figure: Test 1 Initial mesh and the target D. Number of triangles: 992; number of vertices: 526.

19 Numerical results: 2-D Lamé system, D is a circle II Case 1: D is a circle 2.5 x 105 Current Function Values 2 Function value Iterations Figure: Test 1 The evolution of the cost of DIRECTNoScal (left) and computed center and radius (right)

20 Numerical results: 2-D Lamé system, D is an ellipse I Case 2: D is the interior of an ellipse Assume: N = 2, D is the interior of an ellipse: 5 variables (x 0, y 0, θ, a, b) Test 2: Ω = B(0; 10), T = 5, u 01 = 10x, u 02 = 10y, u 11 = 0, u 12 = 0, ϕ 1 = 10x, ϕ 2 = 10y x0des=-3, y0des=-3, sin(thetades)=0, ades=0.8, bdes=0.4 x0ini=-1, y0ini=-1, sin(thetaini)=0, aini=0.5, bini=0.5 NLopt (AUGLAG + DIRECTNoScal), FreeFem++: x0cal = y0cal = sin(thetacal)= acal = bcal = Figure: Test 2 Initial mesh and the target D. Number of triangles: 1206; number of vertices: 633.

21 Numerical results: 2-D Lamé system, D is a star-shaped domain I Case 3: D is a star-shaped domain Assume: N = 2, D is a star-shaped domain: 7 variables (x 0, y 0, r i ), i = 1,... 5 Test 3: Ω = B(0; 10), the same initial and boundary conditions... x0des = -3, y0des = 0, rides = {0.6, 0.9, 1.2, 1.0, 1.0} x0ini = 0, y0ini = 0, riini = {0.5, 0.7, 0.9, 0.9, 0.8} NLopt (AUGLAG + DIRECTNoScal), FreeFem++: x0cal = y0cal = r1cal = r2cal = r3cal = r4cal = r5cal = Figure: Test 3 Initial mesh and the target D.

22 Numerical results: 3-D Lamé system I Case 4: D is a sphere Test 4: N = 3, Ω is a sphere centered at (0, 0, 0) and radius R = 10, T = 5, u 01 = 10x, u 02 = 10y u 03 = 10z, u 11 = 0, u 12 = 0 u 13 = 0 ϕ 1 = 10x, ϕ 2 = 10y ϕ 3 = 10z x0des = -2, y0des = -2, z0des = -2, rdes = 1 x0ini = 0, y0ini = 0, z0ini = 0, rini = 0.6 NLopt (AUGLAG + DIRECTNoScal), FreeFem++: x0cal = y0cal = z0cal = rcal =

23 Numerical results: 3-D Lamé system II Case 4: D is a sphere Figure: Test 4 Initial mesh and the target D. Number of tetrahedra: 4023; number of vertices: 829; number of faces: Similar results for the wave equation... Figure: Test 4 The computed first component of the observation at final time and the final mesh. AD, E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Problems and Imaging, 9 (2015), no. 2, AD, E. Fernández-Cara, Some Geometric Inverse Problems for the Lamé System with Applications in Elastography, Appl Math Optim (2018), 1-21

24 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

25 Method 2 of reconstruction Method of fundamental solutions (MFS), meshless,... MFS is meshless method developed for solving N dimensional wave equations (direct problem), based on: 1 Wave equation is considered as Poisson equation with time-dependent source term: u = u tt 2 Houbolt finite difference scheme, then Poisson problem 3 Method of particular solutions (MPS) - fundamental solutions (MFS): Nf Nb u(x) = u P (x) + u H (x) = β j F( x η j ) + α k G( x ξ k ), j=1 k=1 where u P is particular solution of nonnhomogeneous equation u H is homogeneous solution of Laplace equation F is integrated radial basis function: F (r) = f (r), f (r) is radial basis func. β j coefficients of the basis function, α k intensity of the source points G is fundamental solution of the Laplace equation Nf is number of the field points: η j Nb is number of the source points: ξ k ( ) 4 β PDE+BC+IC resolution of linear system: M = Z for β α j, α k For Inverse Problem: for simplicity, we consider Poisson type equation...

26 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

27 Method 2 of reconstruction: Poisson type equation Case 5 (Poisson): N = 2, D is a circle Inverse problem: (Partial) data: Ω, T, ϕ, h and γ Ω (Additional) information: α = α(x) Goal: Find D such that the solution u to (3) satisfies (4) u + au = h in Ω \ D (PDE) (3) u = ϕ on Ω (BC on Ω ) u = 0 on D (BC on D ) (4) u = α on γ (BC on γ ) n We take Nf Nb u(x) = u P (x) + u H (x) = β j F( x η j ) + α k G( x ξ k ) j=1 k=1 Assume: Ω = B(0, 10), D is a circle: D = B(x 0, y 0 ; rho)

28 10 Boundary of Omega Boundary of D Points on gamma Boundary points Source points Field points Figure: Initial configuration for MFS

29 Method 2 of reconstruction: Poisson type 2D-equation I Case 5 (Poisson): N = 2, D is a circle Boundary of Omega Boundary of D Points on gamma Boundary points Source points Field points Ω = B(0, 10) a = 0.2 x 2 + y 2, h = 0.3x, ϕ = 10x 10 Nb0 = 60 : Nb00 = 10 : Nd = 12 : Nb = Nb0 + Nd : Nb of boundary points on Ω Nb. of boundary points on γ Nb. of boundary points on D Nb. of source points x0ini = 0, y0ini = 0, rho0ini = 1.5 x0des = 2, y0des = 4, rhodes = 1-10 Nf Nb u(x) = u P (x) + u H (x) = β j F( x η j ) + α k G( x ξ k ), j=1 k= PDE + BC s on Ω, D, γ yield to nonlinear system of equations ( ) { β Least square formulation fmincon, MATLAB M(x0, y0, rho) = Z α Nf + Nb + 3 unknowns at each iteration x0cal = , y0cal = , rhocal =

30 Method 2 of reconstruction: Poisson type 2D-equation II Case 5 (Poisson): N = 2, D is a circle x0cal = , y0cal = , rhocal = x0des = 2, y0des = 4, rhodes = 1 Outer boundary Initail configuration Inner (computed) boundary Desired boundary Outer boundary Inner (computed) boundaries Figure: Desired and computed configuration Figure: Computed domain and iterations

31 Method 2 of reconstruction: Poisson type 2D-equation III Case 5 (Poisson): N = 2, D is a circle 15 Function Values 10 5 log(fun. values) Iterations Figure: The evolution of the cost along 146 iterations of fmincon

32 Method 2 of reconstruction: Poisson type 2D-equation IV Case 5 (Poisson): N = 2, D is a circle % random noise Cost in the active-set N. iterations 1% 1.e % 1.e % 1.e % 1.e % 1.e Table: The evolution of the cost at the (best) stopping (β, α, x 0, r) with random noises in the target.

33 Method 2 of reconstruction: 3D-Poisson equation I Case 6 (Poisson): N = 3, D is a sphere Ω = S(0, 10), a = 0.2 x 2 + y 2 + z 2, h = 0.3x, ϕ = 10x (x0ini,y0ini,z0ini)=(0,0,0), rhoini = 1.5 (x0des,y0des,z0des)=(-6,0,0), rhodes = 1 u(x) = u P (x) + u H (x) = Nf Nb β j F( x η j ) + α k G( x ξ k ) j=1 many variables... k=1 Minimize J(x 0, y 0, z0, rho) := 1 2 γ u n (x 0, y 0, z 0, rho) α 2 ds (x 0, y 0, z 0, rho) (α, β) u n (x 0, y 0, z 0, rho) : 4 unknows... x0cal = , y0cal = , z0cal = rhocal =

34 Method 2 of reconstruction: 3D-Poisson equation II Case 6 (Poisson): N = 3, D is a sphere Figure: Desired configuration Figure: Computed configuration and iterations A. Doubova Geometric inverse problems

35 Method 2 of reconstruction: 3D-Poisson equation III Case 6 (Poisson): N = 3, D is a sphere 0 Function Values -5 log(fun. values) Iterations Figure: The evolution of the cost along 399 iterations of fmincon Joint work with: AD, E. Fernández-Cara, J. Rocha de Faria, P. de Carvalho, submited

36 References AD, E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Problems and Imaging, 9 (2015), no. 2, AD, E. Fernández-Cara, Some Geometric Inverse Problems for the Lamé System with Applications in Elastography, Appl Math Optim (2018), 1-21 M.-H. Gu, C.-M. Fan, and D.-L. Young, The method of fundamental solutions for multidimensional wave equations, Journal of Marine Science and Technology, Vol. 19, No. 6, pp (2011) F. Hecht, New development in freefem++, J. Numer. Math., 20, (2012) J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 1999 J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi X. Li, Elastography: A quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imaging, 13 (1991), W. L. Price, A controlled random search procedure for global optimisation, The Computer Journal, 20 (1977), D. E. Finkel, Direct optimization algorithm user guide, Center for Research in Scientific Computation, North Carolina State University, 2.

37 Outline 1 Introduction Inverse problems governed by PDEs (wave, Lamé) Motivation: Elastography 2 Uniqueness results 3 Method 1 of reconstruction (FEM, FreeFem++... ) 4 Numerical results with Method 1 Case 1 (Lamé): N = 2, D is a circle Case 2 (Lamé): N = 2, D is the interior of an ellipse Case 3 (Lamé): N = 2, D is a star-shaped domain Case 4 (Lamé): N = 3, D is a sphere 5 Method 2 of reconstruction (meshless, MFS... ) 6 Numerical results with Method II Case 5 (Poisson): N = 2, D is a circle Case 6 (Poisson): N = 3, D is a sphere 7 Work in progress

38 Work in progress I Open problems 1 A semilinear elliptic equation u + f (u) = 0, x Ω \ D, where f : R R is a locally Lipschitz-continuous given function. The analysis of uniqueness and stability is not trivial.... Reconstruction of the unknown domain D with MFS-MPS: OK now M will depend on x 0 and r, but also on β and α: J(β, α, x 0, r) := 1 ( β 2 M(β, α, x 0, r) α ) Z 2

39 Work in progress II Open problems 2 General elasticity system: u tt σ(u) = 0 in Ω \ D (0, T ) u = ϕ on Ω (0, T ) u = 0 on D (0, T ) u(0) = u 0, u t(0) = u 1 in Ω \ D σ kl (u) = N i,j,k,l=1 a ijkl ε ij (u), ε ij (u) = 1 2 ( iu j + j u i ) a ijkl = a klij = a ijlk L (Ω) 1 i, j, k, l 3 N N a ijkl ξ ij ξ kl α ξ ij 2, {ξ ij } R N N sym i,j,k,l=1 i,j=1 Uniqueness, stability and numerical reconstruction of D?

40 Work in progress III Open problems 3 Similar inverse problems for the Navier-Stokes and related systems with applications to fluid mechanics. For example, in the stationary case, the idea is to find D such that a solution to satisfies ν u + (u )u + p = 0, u = 0 in Ω \ D, u = ϕ on Ω, u = 0 on D ( p Id + ν( u + u t )) n = α on γ. Uniqueness (OK), stability and numerical reconstruction? 4 Ellipsoids, other more complicated geometries? 5 Internal observations?