An inverse problem in viscoelasticity:
stability result and numerical example
Maya de Buhan
Pascal Frey
Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie
Paris, France
Bourse CNRS
Axel Osses
Departamento de Ingenierı́a Matemática
Facultad de Ciencias Fı́sicas y Matemáticas
Universidad de Chile, Santiago, Chile
Beca CONICYT
CIMPA Summer School on Inverse Problems
Student’s Seminar
January 12th 2010
Direct problem: the 3D linear viscoelasticity system
Let Ω be an open bounded domain of R3 with a Lipschitz boundary ∂Ω.
The displacement vector u : Ω × (0, +∞) −→ R3 satisfies
∂t2 u − ∇ · σ(u) = f
in Ω × (0, +∞)
u(0) = ū0
in Ω
∂t u(0) = ū1
in Ω
u=0
(1)
on ∂Ω × (0, +∞)
where
σ(u)(x, t) = µ(x)(∇u(x, t) + ∇u(x, t)T ) + λ(x)(∇ · u)(x, t)I
Z t
µ̃(x, s)(∇u(x, t − s) + ∇u(x, t − s)T ) + λ̃(x, s)(∇ · u)(x, t − s)I ds
−
0
with (λ, µ) the Lamé coefficients and (λ̃, µ̃) the viscosity coefficients.
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Inverse problem
Hypothesis
µ̃(x, t) = p(x)h(t)
in Ω × (0, +∞)
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Inverse problem
Hypothesis
µ̃(x, t) = p(x)h(t)
in Ω × (0, +∞)
Inverse problem
Given (λ, µ, λ̃, h, u0 , u1 , f ), we would like to recover p in Ω from measurements of
∇u · n on Γ × (0, T )
where n is the unit normal on ∂Ω, Γ is a part of ∂Ω and T > 0.
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Stability result
Let u (resp. ū) be the solution of (1) associated to p (resp. p̄).
Theorem 1 (Logarithmic Stability) MdB, A. Osses (2010)
Under Hypothesis 1, 2 and 3, there exists κ ∈ (0, 1) such that

−κ

 
 
kp − p̄kH 2 (Ω) ≤ C log 2 +
 
C
X
k∂xα (u − ū)k2L2 (Γ×(0,T ))




1≤|α|≤2
where C > 0 depends on the C 2 (Ω)-norm of p and p̄ and on the norm
W 8,∞ (Ω × (0, T ))3 of u and ū.
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Stability result
Let u (resp. ū) be the solution of (1) associated to p (resp. p̄).
Theorem 1 (Logarithmic Stability) MdB, A. Osses (2010)
Under Hypothesis 1, 2 and 3, there exists κ ∈ (0, 1) such that

−κ

 
 
kp − p̄kH 2 (Ω) ≤ C log 2 +
 
C
X
k∂xα (u − ū)k2L2 (Γ×(0,T ))




1≤|α|≤2
where C > 0 depends on the C 2 (Ω)-norm of p and p̄ and on the norm
W 8,∞ (Ω × (0, T ))3 of u and ū.
Corollary 1 (Uniqueness)
∂xα u = ∂xα ū
in Γ × (0, T ) |α| = 1, 2
=⇒
p = p̄
in Ω
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Assumptions
Hypothesis 1 on the coefficients
(λ, µ) ∈ C 2 (Ω)2
(λ̃, µ̃) ∈ C 2 (Ω × (0, +∞))2
u ∈ W 8,∞ (Ω × (0, +∞))3
p = p̄ known in a neighborhood ω of ∂Ω
h(0) 6= 0
h0 (0) = 0
µ and λ + 2µ satisfy the pseudo-convexity condition
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Assumptions
Hypothesis 1 on the coefficients
(λ, µ) ∈ C 2 (Ω)2
(λ̃, µ̃) ∈ C 2 (Ω × (0, +∞))2
u ∈ W 8,∞ (Ω × (0, +∞))3
p = p̄ known in a neighborhood ω of ∂Ω
h(0) 6= 0
h0 (0) = 0
µ and λ + 2µ satisfy the pseudo-convexity condition
Hypothesis 2 on the observation part
Γ arbitrarly small
and
T sufficiently large
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Assumptions
Hypothesis 1 on the coefficients
(λ, µ) ∈ C 2 (Ω)2
(λ̃, µ̃) ∈ C 2 (Ω × (0, +∞))2
u ∈ W 8,∞ (Ω × (0, +∞))3
p = p̄ known in a neighborhood ω of ∂Ω
h(0) 6= 0
h0 (0) = 0
µ and λ + 2µ satisfy the pseudo-convexity condition
Hypothesis 2 on the observation part
Γ arbitrarly small
and
T sufficiently large
Hypothesis 3 on the initial data
There exist x0 ∈ R3 \ Ω such that
|(∇ū0 + ∇ū0T )(x) · (x − x0 )| ≥ M > 0
in Ω \ ω
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Idea of the proof
Write a Carleman estimate for system (1) by
decoupling the system in equations for u, ∇ ∧ u and ∇ · u as in
Imanuvilov and Yamamoto (2005)
using the change of variable of Cavaterra, Lorenzi and Yamamoto
(2006) to treat the integral term
applying a pointwise Carleman estimate of Klibanov and Timonov
(2004) for a scalar hyperbolic equation
+ Apply the method of Bukhgeim and Klibanov (1981)
=⇒ Theorem 2 (Hölder Stability) MdB, A. Osses (2009)
kp − p̄kH 2 (Ω) ≤ C ku − ūk2κ
H 2 (ω×(0,T /6))
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Idea of the proof
+ Prove a sharp unique continuation result for system (1) by
applying the method of Bellassoued (2001) which uses some results
of Robbiano (1995),
introducing a new transformation inspired from the
Fourier-Bros-Iagolnitzer transform but which is able to treat the
integral term of (1)
 
−1
 
 
kuk2H 2 (ω×(0,T /6)) ≤ C log 2 +
 
C
X
k∂xα uk2L2 (Γ×(0,T ))




1≤|α|≤2
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Example
µ(x) = λ(x) = 1200,
λ̃(x, t) = 400h(t) = 400e −t/τ with τ = 1
400
in the healthy tissue
p̄(x) =
> 400 in the tumor
T = 50, δt = 1, δ = 2%
computational mesh
p̄
observation area
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Numerical resolution
Direct problem
Discretization of system (1)
in space by P1 Lagrange Finite Elements
in time by a θ-scheme with θ = 0.5 (implicit centered scheme)
with the trapezium formula for the integral term
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Numerical resolution
Direct problem
Discretization of system (1)
in space by P1 Lagrange Finite Elements
in time by a θ-scheme with θ = 0.5 (implicit centered scheme)
with the trapezium formula for the integral term
Inverse problem
We consider the non-quadratic functional
1
J(p) =
2
Z
0
T
Z
|u(p) − uobs |2 + |∇(u(p) − uobs )|2 dxdt
ω
where uobs is the observed displacement in ω × (0, T ).
We minimize J by a BFGS algorithm. As regularization method, we look
for the numerical solution p ∗ in a finite dimensional space PK .
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Mesh and basis adaptation
Iterative process
pi∗ ∈ PK = {p ∈ P, p = p̄|∂Ω +
K
X
pi ϕ i }
i=1
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Mesh and basis adaptation
Iterative process
pi∗ ∈ PK = {p ∈ P, p = p̄|∂Ω +
K
X
pi ϕ i }
i=1
where
−∇ · (a∇ϕ ) = σ ϕ
i
i i
ϕi = 0
in Ω
on ∂Ω
(
with
a(x) =
1
1
∗ (x)
∇pi−1
i =0
i >0
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Mesh and basis adaptation
Iterative process
pi∗ ∈ PK = {p ∈ P, p = p̄|∂Ω +
K
X
pi ϕ i }
i=1
where
−∇ · (a∇ϕ ) = σ ϕ
i
i i
ϕi = 0
in Ω
on ∂Ω
(
with
a(x) =
1
1
∗ (x)
∇pi−1
i =0
i >0
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Our current result in recovering the coefficient
p̄
p0∗ with K = 100
p2∗ with K = 50
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