Logarithmic stability in determination of a 3D viscoelasticity system coefficientMaya de Buhan(1,2) Axel Osses(2)

(1) Universite Pierre et Marie Curie, LJLL, Paris, France. (2) Universidad de Chile, CMM, Santiago, Chile.

Introduction

Let Ω be an open bounded domain of R3 with a Lipschitz continuous boundary ∂Ω.We consider the following system endowed with initial and boundary conditions:∣∣∣∣∣∣∣∣∣

Pu(x, t) = f (x, t), ∀(x, t) ∈ Ω× (0,+∞),u(x, 0) = u0(x), ∀x ∈ Ω,∂tu(x, 0) = u1(x), ∀x ∈ Ω,u(x, t) = 0, ∀(x, t) ∈ ∂Ω× (0,+∞),

(1)

where P is the integro-differential hyperbolic operator:

Pu(x, t) = ∂2tu(x, t)−∇ ·

(µ(x)(∇u(x, t) +∇u(x, t)T ) + λ(x)(∇ · u)(x, t)I

)+∫ t

0∇ ·

(µ(x, s)(∇u(x, t− s) +∇u(x, t− s)T ) + λ(x, s)(∇ · u)(x, t− s)I

)ds.

This system models the dynamics of a viscoelastic material subjected to a loadf , u being the displacement vector, (λ, µ) the Lame coefficients and (λ, µ) theviscosity coefficients. We assume that the coefficient µ can be decomposed as

µ(x, t) = p(x)h(t), ∀(x, t) ∈ Ω× (0,+∞). (2)

Hence, knowing (λ, µ, λ, h, u0, u1, f ), we would like to recover p(x), ∀x ∈ Ω, frommeasurements of

u(x, t), ∀(x, t) ∈ Γ× (0, T ),

where Γ is a part of ∂Ω and T > 0.We proved a Carleman estimate and a sharp unique continuation result for thesystem (1). We used these results to obtain a logarithmic stability estimate forthe inverse problem of recovering the coefficient p from a unique measurementon an arbitrary part of the boundary. We proposed a method to solve the problemnumerically and illustrated the theoretical result by a numerical example.

Logarithmic stability result

Condition 1. The scalar function q satisfies Condition 1 if• there exists K > 0 such that ∀x ∈ Ω, q(x) ≥ K,

• there exists x0 ∈ R3 \ Ω such that ∀x ∈ Ω,1

2q(x) +∇q(x) · (x− x0) ≥ 0.

Theorem 1. Let u (resp. u) be the solution of the system (1)-(2), associated tothe coefficient p (resp. p). We assume that• (λ, µ) ∈ C2(Ω)2 and (λ, µ) ∈ C2(Ω× (0,+∞))2 are such that the solutions u

and u ∈ W 8,∞(Ω× (0,+∞))3,•µ and λ + 2µ satisfy Condition 1,• p = p is known in a neighborhood ω of ∂Ω,•h(0) 6= 0, h′(0) = 0,

• there exists M > 0 such that, ∀x ∈ Ω \ ω, |(∇u0 +∇uT0 )(x) · (x− x0)| ≥M .Then, for Γ ⊂ ∂Ω arbitrarly small, there exist κ ∈ (0, 1) and T0 > 0 such that, forall T > T0, the following estimate holds true:

‖p− p‖H2(Ω) ≤ C

log

2 +C∑

1≤|α|≤2

‖∂αx (u− u)‖2L2(Γ×(0,T ))

−κ

(3)

where C depends on the C2(Ω)-norm of p and p and on the W 8,∞(Ω × (0, T ))3-norm of u and u.

Idea of the proof

It consists in :•writing a Carleman estimate for the system (1) by

– decoupling the system in equations for u, ∇ ∧ u and ∇ · u as in Imanuvilovand Yamamoto (2005),

– using the change of variable of Cavaterra, Lorenzi and Yamamoto (2006) totreat the integral term of the operator P ,

– applying a pointwise Carleman estimate of Klibanov and Timonov (2004) fora scalar hyperbolic equation,

• applying the method of Bukhgeim and Klibanov to obtain the Holder stability

‖p− p‖H2(Ω) ≤ C‖u− u‖κH2(ω×(0,T )), (4)

• proving a sharp unique continuation for the system (1) by– applying the method of Bellassoued (2001) which uses some results of Rob-

biano (1995),– introducing a new transformation inspired from the Fourier-Bros-Iagolnitzer

transform but which is able to treat the integral term of P .

Application

The system (1) is a simplified model for the mechanical behavior of the brainstructures. We consider it now in 2D, with the following coefficients:•µ(x) = λ(x) = 1200, λ(x, t) = 400h(t) and h(t) = e−t/τ with τ = 1,• p(x) = 400, in the healthy tissue and p(x) > 400, in the tumor as in Fig 1,• f (x, t) = 0, u1(x) = 0 and u0 as in Fig 1, T = 50 and ω as in Fig 1.

Fig 1. Coefficient p, initial data u0, observation zone ω, computational meshWe solve the direct problem (1) by discretizing the equations• in space using P1 Lagrange Finite Elements in the mesh shown on Fig 1,• in time using a θ-scheme with θ = 0.5 (implicit centered scheme) and δt = 1,• by using the trapezium formula for the integral term.

A uniform experimental error of δ = 2% is added to the observed solution uobs.We are looking for the minimizer p∗ of the non quadratic functional

J(p) =1

2

∫ T

0

∫ω

(|u(p)− uobs|2 + |∇(u(p)− uobs)|2

)dx dt,

We solve the minimization problem by a BFGS algorithm.

Fig 2. Initial mesh and basis functions n1, 2 and 4

As regularization method, we look for p∗ in the finite dimensional space PK of theK first eigenfunctions of the mesh (see Fig 2), that is

PK =

p ∈ P, p = p|∂Ω +K∑i=1

piϕi

,where p|∂Ω is a lifting of the trace of p on ∂Ω and∣∣∣∣∣ −∆ϕi(x) = σiϕi(x), ∀x ∈ Ω,

ϕi(x) = 0, ∀x ∈ ∂Ω.

We propose an adaptive method to solve the problem accurately. After computinga first solution p∗0 on the initial mesh, we use it to refine the mesh and to adaptthe spectral basis as follows:∣∣∣∣∣ −∇ · (a(x)∇ϕi(x)) = σiϕi(x), ∀x ∈ Ω,

ϕi(x) = 0, ∀x ∈ ∂Ω,with a(x) =

1

∇p∗0(x).

Fig 3. Adapted mesh and basis functions n1, 2 and 4 at step 2

Numerical results

10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

number of eigenvectors in the basis

% o

f e

rro

r —— step 0

—— step 1 with K0 = 40

—— step 1 with K0 = 80

Fig 4. Relative error for the coefficient p in L2-norm with respect to K

Fig 5. Exact coefficient, recovered coefficient at step 0 for K = 100 and K = 50,recovered coefficient at step 2 for K = 50