Some inverse problems with applications in …...Some inverse problems with applications in...

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

A. Doubova Geometric inverse problems

Outline

1 IntroductionInverse problems governed by PDEs (wave, Lamé)Motivation: Elastography

2 Uniqueness results

3 Method 1 of reconstruction (FEM, FreeFem++. . . )

4 Numerical results with Method 1Case 1 (Lamé): N = 2, D is a circleCase 2 (Lamé): N = 2, D is the interior of an ellipseCase 3 (Lamé): N = 2, D is a star-shaped domainCase 4 (Lamé): N = 3, D is a sphere

5 Method 2 of reconstruction (meshless, MFS. . . )

6 Numerical results with Method IICase 5 (Poisson): N = 2, D is a circleCase 6 (Poisson): N = 3, D is a sphere

7 Work in progress


IntroductionInverse 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)

. . .


IntroductionGeometric inverse problem

Goal

Identification of a shape of a domain

(a) Direct problem:Data: Ω, D, T > 0, ϕ = ϕ(x , t) and γ ⊂ ∂ΩResult: the solution u

(1)

utt −∆u = 0 in (Ω \ D)× (0,T )u = ϕ on ∂Ω× (0,T )u = 0 on ∂D × (0,T )u(x , 0) = u0, ut (x , 0) = u1 in Ω

Information:(2)

∂u∂n

:= α on γ × (0,T )

(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. . .A. Doubova Geometric inverse problems

MotivationWhat 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 stiffmatter.For example, tumor tissue is 5-28 times stiffer than normal tissue, thenthe resulting deformation after a mechanical action is smaller and thedifference can be captured by images:

It is used in various fields of Medicine (detection and description ofbreast, liver, prostate and other cancers, fibrosis, . . . )

First works: [Ophir-et-al 1991], [Muthupillai-et-al 1995], [Sinkus-et-al 2000],[McKnight-et-al 2002], . . .


Elasticity systems: isotropic case, constant Lamé coefficientsSimilar inverse problem

Similarly: elasticity system

(b) Inverse problem: given α = α(x , t), ϕ = ϕ(x , t), µ, λ > 0, find D such thatutt − 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) = u0, ut (0) = u1 in Ω \ D

satisfies

σ(u) · n =(µ(∇u +∇ut ) + λ(∇ · u)Id.

)· n := α(x , t) on γ × (0,T )

Explanations:

u = (u1, u2, u3) 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


Inverse problems: main questions

3 main questions:1 Uniqueness: If u0 and u1 are two solutions corresponding to

u itt −∆u i = 0 in (Ω \ Di )× (0,T )

u i = ϕ on ∂Ω× (0,T )

u i = 0 on ∂Di × (0,T )

u i (x , 0) = u0, ut (x , 0) = u1 in Ω \ Di

such that∂u0

∂n=∂u1

∂non γ × (0,T ) =⇒ do we have D0 = D1?

2 Stability: Find an estimate of the “size” of (D0 \ D1) ∪ (D1 \ D0) in termsof the “size” of α0 − α1:

Size(

(D0 \ D1) ∪ (D1 \ D0))≤ CF

(‖α0 − α1‖A(γ×(0,T ))

)for all D1 “close” to D0, for some F : R+ 7→ R+ with F (s)→ 0 as s → 0,some suitable space A(γ × (0,T )) and C = C(D0,Ω, γ,T , ϕ0)

3 Reconstruction: Devise iterative algorithms to compute D from α


Outline







7 Work in progress


Uniqueness results IN-dimensional wave equation

u i

tt −∆u i = 0 in Ω \ Di × (0,T ), i = 0, 1u i = ϕ on ∂Ω× (0,T )

u i = 0 on ∂Di × (0,T )

u i (x , 0) = 0, u it (x , 0) = 0 in Ω \ Di

Theorem

T > T∗(Ω, γ), Di convex, ϕ 6= 0

∂u0

∂n=∂u1

∂non γ × (0,T )

=⇒ D0 = D1

Main steps of the proof:

1 Consider G := Ω \ D0 ∪ D1 and u := u0 − u1 in G × (0,T ):utt −∆u = 0 in G × (0,T )u = 0 on ∂Ω× (0,T )∂u∂n

= 0 on ∂γ × (0,T )

Unique continuation ⇒ u ≡ 0 in G × (0,T )


Uniqueness results IIN-dimensional wave equation

2 Assume D0 6⊂ D1, i.e. D0 \ D1 6= ∅. In (D0 \ D1)× (0,T ), consider u1:u1

tt −∆u1 = 0 in (D0 \ D1)× (0,T )

u1 = 0 on ∂(D0 \ D1)× (0,T )

u1 = 0, u1t = 0 in (D0 \ D1)

Uniqueness of zero solution⇒ u1 = 0 in D0 \ D1

3 In Ω \ D1 × (0,T ) consider u1.Unique continuation implies that u1 ≡ 0, but u1 = ϕ 6= 0, therefore→←and then D0 ⊂ D1.

4 In a similar way, we can prove that D1 ⊂ D0, therefore D0 = D1.

Here we need only Unique Continuation, then NO geometrical condition on γ

Also: N-dimensional isotropic Lamé system with constant coefficients: OK


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


Outline







7 Work in progress


Method 1: Optimization problem, FEM, FreeFem++. . . IAugmented Lagrangian method (ff-NLopt - AUGLAG)

Assume: N = 2, D = B(x0, y0; r)

Inverse problem: given α = α(x , t), find x0, y0, r such that D ⊂ Ω and thesolution u to the Lamé system satisfies

σ[x0, y0; r ] :=(µ(x)(∇u +∇ut ) +λ(x)(∇ · u)Id.

)· n = α(x , t) on γ× (0,T )

Constrained optimization problem (case of a circle)

Find x0, y0 and r such that (x0, y0, r) ∈ Xb and

J(x0, y0, r) ≤ J(x ′0, y′0, r′) ∀ (x ′0, y

′0, r′) ∈ Xb

the function J : Xb 7→ R is defined by

J(x0, y0, r) :=12

∫ T

0‖σ[x0, y0, r ]− α‖2

H−1/2(γ) dt

Xb := (x0, y0, r) ∈ R3 : B(x0, y0; r) ⊂ Ω


Method 1: Optimization problem, FEM, FreeFem++. . . IIAugmented Lagrangian method (ff-NLopt - AUGLAG)

The problem formulation contains inequality constraintsMinimize f (x)

Subject to x ∈ X0; ci (x) ≥ 0, 1 ≤ i ≤ I

X0 = x ∈ Rm : x j ≤ xj ≤ x j , 1 ≤ j ≤ m

Introducing slack variables si :

ci (x) ≥ 0 rewired as ci (x)− si = 0, si ≥ 0, 1 ≤ i ≤ I

Augmented Lagrangian is defined in therms of the constraints ci (x)− si = 0

Optimization problem (for Augmented Lagrangian)

Minimize LA(x , λk ;µk ) := f (x)−I∑

i=1

λki (ci (x)− si ) +

12µk

I∑i=1

(ci (x)− si )2

Subject to x ∈ X0; si ≥ 0, 1 ≤ i ≤ I

λki : multipliers, µk : penalty parameters


Method 1: Optimization problem, FEM, FreeFem++. . . IIIAugmented Lagrangian method (ff-NLopt - AUGLAG)

Algorithm (Augmented Lagrangian: inequality constraints)

(a) Fix µ1 and starting points x0 and λ1;(b) Then, for given k ≥ 1, µk , xk−1, λk :

(b.1) Unconstrained optimization: Find an approximate minimizer xk of LA( · , λk ;µk ),starting at xk−1:

Minimize LA(x, λk ;µk )

Subject to x ∈ X0

(b.2) Update the Lagrange multipliers:

λk+1i = max

(λ

ki −

ci (xk )

µk, 0), 1 ≤ i ≤ I

(b.3) Choose a new parameter and check whether a stopping convergence test is satisfied:

µk+1 ∈ (0, µk )

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 forglobal optimization


Outline







7 Work in progress


Numerical results: 2-D Lamé system, D is a circle ICase 1: D is a circle

Assume: N = 2, D is the interior of an circle: 3 variables (x0, y0; r)

Test 1: N = 2, Ω = B(0; 10), D = B(x0, y0; r), T = 5

u01 = 10x , u02 = 10y , u11 = 0, u12 = 0, ϕ1 = 10x , ϕ2 = 10y

x0ini = 0, y0ini = 0, rini = 0.6

x0des = -3, y0des = 0, rdes = 0.4

NLopt (AUGLAG + DIRECTNoScal), FreeFem++:

x0cal = -3.000224338y0cal = -0.0005268693985rcal = 0.4000228624

Figure: Test 1 – Initial mesh and the target D. Number oftriangles: 992; number of vertices: 526.


Numerical results: 2-D Lamé system, D is a circle IICase 1: D is a circle

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5x 10

5

Iterations

Current Function Values

Function v

alu

e

Figure: Test 1 – The evolution of the cost of DIRECTNoScal (left) and computed centerand radius (right)


Numerical results: 2-D Lamé system, D is an ellipse ICase 2: D is the interior of an ellipse

Assume: N = 2, D is the interior of an ellipse: 5 variables (x0, y0, θ, a, b)

Test 2: Ω = B(0; 10), T = 5, u01 = 10x , u02 = 10y , u11 = 0,u12 = 0, ϕ1 = 10x , ϕ2 = 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


x0cal =-3.002591068y0cal =-3.001574963sin(thetacal)=0.00548696845acal =0.8036351166bcal =0.400617284

Figure: Test 2 – Initial mesh and the target D.Number of triangles: 1206; number of vertices: 633.


Numerical results: 2-D Lamé system, D is a star-shaped domain ICase 3: D is a star-shaped domain

Assume: N = 2, D is a star-shaped domain: 7 variables (x0, y0, ri ), 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


x0cal = -2.962962963y0cal = -0.009144947417r1cal = 0.5932098765r2cal = 0.8660493827r3cal = 1.288683128r4cal = 0.9462962963r5cal = 0.9997942387

Figure: Test 3 – Initial mesh and the target D.


Numerical results: 3-D Lamé system ICase 4: D is a sphere

Test 4: N = 3, Ω is a sphere centered at (0, 0, 0) and radius R = 10, T = 5,

u01 = 10x , u02 = 10y u03 = 10z, u11 = 0, u12 = 0 u13 = 0ϕ1 = 10x , ϕ2 = 10y ϕ3 = 10z

x0des = -2, y0des = -2, z0des = -2, rdes = 1

x0ini = 0, y0ini = 0, z0ini = 0, rini = 0.6


x0cal = -1.981405274y0cal = -2.225232904z0cal = -2.148084171rcal = 0.9504115226


Numerical results: 3-D Lamé system IICase 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: 8406.

Figure: Test 4 – The computed first componentof the observation at final time and the finalmesh.

Similar results for the wave equation. . .

AD, E. Fernández-Cara, Some geometric inverse problems for the linearwave equation, Inverse Problems and Imaging, 9 (2015), no. 2, 371–393

AD, E. Fernández-Cara, Some Geometric Inverse Problems for theLamé System with Applications in Elastography, Appl Math Optim(2018), 1-21


Outline







7 Work in progress


Method 2 of reconstructionMethod of fundamental solutions (MFS), meshless,. . .

MFS is meshless method developed for solving N dimensional waveequations (direct problem), based on:

1 Wave equation is considered as Poisson equation with time-dependentsource term: −∆u = −utt

2 Houbolt finite difference scheme, then Poisson problem3 Method of particular solutions (MPS) - fundamental solutions (MFS):

u(x) = uP(x) + uH(x) =Nf∑j=1

βjF (|x − ηj |) +Nb∑

k=1

αk G(|x − ξk |), where

uP is particular solution of nonnhomogeneous equationuH is homogeneous solution of Laplace equationF is integrated radial basis function: ∆F (r) = f (r), f (r) is radial basis func.βj coefficients of the basis function, αk intensity of the source pointsG is fundamental solution of the Laplace equationNf is number of the field points: ηjNb is number of the source points: ξk

4 PDE+BC+IC⇒ resolution of linear system: M(βα

)= Z for βj , αk

For Inverse Problem: for simplicity, we consider Poisson type equation. . .A. Doubova Geometric inverse problems

Outline







7 Work in progress


Method 2 of reconstruction: Poisson type equationCase 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)

(3)

−∆u + au = h in Ω \ D (PDE)u = ϕ on ∂Ω (BC on ∂Ω )u = 0 on ∂D (BC on ∂D )

(4)∂u∂n

= α on γ (BC on γ )

We take

u(x) = uP(x) + uH(x) =Nf∑j=1


k=1

αk G(|x − ξk |)

Assume: Ω = B(0, 10), D is a circle: D = B(x0, y0; rho)


-10 -5 0 5 10

-10

-5

0

5

10

Boundary of Omega

Boundary of D

Points on gamma

Boundary points

Source points

Field points

Figure: Initial configuration for MFS

Method 2 of reconstruction: Poisson type 2D-equation ICase 5 (Poisson): N = 2, D is a circle

Ω = B(0, 10)

a = 0.2√

x2 + y2, h = 0.3x , ϕ = 10x

Nb0 = 60 : Nb of boundary points on ∂Ω

Nb00 = 10 : Nb. of boundary points on γ

Nd = 12 : Nb. of boundary points on ∂D

Nb = Nb0 + Nd : Nb. of source points

x0ini = 0, y0ini = 0, rho0ini = 1.5

x0des = 2, y0des = 4, rhodes = 1-10 -5 0 5 10

-10

-5

0

5

10

Boundary of Omega

Boundary of D

Points on gamma

Boundary points

Source points

Field points

u(x) = uP(x) + uH(x) =Nf∑j=1


k=1

αk G(|x − ξk |),

PDE + BC’s on ∂Ω, ∂D, γ yield to nonlinear system of equations

M(x0, y0, rho)

(βα

)= Z ⇒

Least square formulation fmincon, MATLAB

Nf + Nb + 3 unknowns at each iteration

x0cal = 2.000274, y0cal = 4.000057, rhocal = 0.999658


Method 2 of reconstruction: Poisson type 2D-equation IICase 5 (Poisson): N = 2, D is a circle

x0cal = 2.000274, y0cal = 4.000057, rhocal = 0.999658

x0des = 2, y0des = 4, rhodes = 1

-10 -5 0 5 10

-10

-5

0

5

10

Outer boundary

Initail configuration

Inner (computed) boundary

Desired boundary

Figure: Desired and computed configuration

-10 -5 0 5 10

-10

-5

0

5

10

Outer boundary

Inner (computed) boundaries

Figure: Computed domain and iterations


Method 2 of reconstruction: Poisson type 2D-equation IIICase 5 (Poisson): N = 2, D is a circle

0 20 40 60 80 100 120 140 160

Iterations

-20

-15

-10

-5

0

5

10

15

log

(Fu

n.

va

lue

s)

Function Values

Figure: The evolution of the cost along 146 iterations of fmincon


Method 2 of reconstruction: Poisson type 2D-equation IVCase 5 (Poisson): N = 2, D is a circle

% random noise Cost in the active-set N. iterations1% 1.e-02 162

0.1% 1.e-05 920.01% 1.e-05 880.001% 1.e-08 127

0% 1.e-09 146

Table: The evolution of the cost at the (best) stopping (β, α, x0, r) with random noisesin the target.


Method 2 of reconstruction: 3D-Poisson equation ICase 6 (Poisson): N = 3, D is a sphere

Ω = S(0, 10),

a = 0.2√

x2 + y2 + z2, h = 0.3x , ϕ = 10x

(x0ini,y0ini,z0ini)=(0,0,0), rhoini = 1.5

(x0des,y0des,z0des)=(-6,0,0), rhodes = 1

u(x) = uP(x) + uH(x)

=Nf∑j=1


k=1

αk G(|x − ξk |)

many variables . . .

Minimize J(x0, y0, z0, rho) :=12

∫γ

∣∣∣∂u∂n

(x0, y0, z0, rho)− α∣∣∣2 ds

(x0, y0, z0, rho) 7→ (α, β) 7→ ∂u∂n

(x0, y0, z0, rho) : 4 unknows . . .

x0cal = 6.000111, y0cal = -0.001903, z0cal = 0.000217

rhocal = 0.998924


Method 2 of reconstruction: 3D-Poisson equation IICase 6 (Poisson): N = 3, D is a sphere

10

-8

-6

8

-4

6

-2

0

48

2

2 6

4

40

6

2

8

-20

-4 -2

-4-6

-6-8

-8

-10

Figure: Desired configuration Figure: Computed configuration and iterations


Method 2 of reconstruction: 3D-Poisson equation IIICase 6 (Poisson): N = 3, D is a sphere

0 50 100 150 200 250 300 350 400

Iterations

-25

-20

-15

-10

-5

0

log

(Fu

n.

va

lue

s)

Function Values

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


References

AD, E. Fernández-Cara, Some geometric inverse problems for the linearwave equation, Inverse Problems and Imaging, 9 (2015), no. 2, 371–393

AD, E. Fernández-Cara, Some Geometric Inverse Problems for theLamé System with Applications in Elastography, Appl Math Optim(2018), 1-21

M.-H. Gu, C.-M. Fan, and D.-L. Young, The method of fundamentalsolutions for multidimensional wave equations, Journal of MarineScience and Technology, Vol. 19, No. 6, pp. 586-595 (2011)

F. Hecht, New development in freefem++, J. Numer. Math., 20, 251–265(2012)

J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 1999

J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi X. Li, Elastography: Aquantitative method for imaging the elasticity of biological tissues,Ultrasonic Imaging, 13 (1991), 111 –134

W. L. Price, A controlled random search procedure for globaloptimisation, The Computer Journal, 20 (1977), 367–370.

D. E. Finkel, Direct optimization algorithm user guide, Center forResearch in Scientific Computation, North Carolina State University, 2.


Outline







7 Work in progress


Work in progress IOpen problems

1 A semilinear elliptic equation

−∆u + f (u) = 0, x ∈ Ω \ D,

where f : R 7→ 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: OKnow M will depend on x0 and r , but also on β and α:

J(β, α, x0, r) :=12

∥∥∥∥M(β, α, x0, r)

(βα

)− Z

∥∥∥∥2


Work in progress IIOpen problems

2 General elasticity system:utt −∇ · σ(u) = 0 in Ω \ D × (0,T )u = ϕ on ∂Ω× (0,T )u = 0 on ∂D × (0,T )

u(0) = u0, ut (0) = u1 in Ω \ D

σkl (u) =N∑

i,j,k,l=1

aijklεij (u), εij (u) =12

(∂iuj + ∂jui )

aijkl = aklij = aijlk ∈ L∞(Ω) 1 ≤ i, j, k , l ≤ 3N∑

i,j,k,l=1

aijklξijξkl ≥ αN∑

i,j=1

|ξij |2, ∀ ξij ∈ RN×Nsym

Uniqueness, stability and numerical reconstruction of D?


Work in progress IIIOpen problems

3 Similar inverse problems for the Navier-Stokes and related systems withapplications to fluid mechanics. For example, in the stationary case, theidea is to find D such that a solution to −ν∆u + (u · ∇)u +∇p = 0, ∇ · u = 0 in Ω \ D,

u = ϕ on ∂Ω,u = 0 on ∂D

satisfies

(−p Id + ν(∇u +∇ut )) · n = α on γ.

Uniqueness (OK), stability and numerical reconstruction ?

4 Ellipsoids, other more complicated geometries ?

5 Internal observations ?


Some inverse problems with applications in …...Some inverse problems with applications in...

Documents

Transcript of Some inverse problems with applications in …...Some inverse problems with applications in...