inverse problems in lithospheric exure and viscoelasticity€¦ · viscoelasticity axel osses in...

Inverse problems in lithospheric flexure andviscoelasticity

Axel Ossesin coll with M. de Buhan (CNRS, France),

E. Contreras (Geophysics Department, Chile),

B. Palacios (DIM, Chile)

DIM - Departamento de Ingenierıa Matematica

CMM - Centro de Modelamiento Matematico

Facultad de Ciencias Fısicas y Matematicas, Universidad de Chile, Santiago.

[matematika mugaz bestalde] - [bcam]Bilbao, July 4th 2011

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Outline

1 IntroductionThe Maxwell viscoelastic modelRelationship to other viscoelastic models

2 Inverse problems

3 Motivation

4 Stability Result (interior measurement)

5 Stability Result (boundary measurement)

6 Numerical resolution

7 Examples

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IntroductionThe Maxwell viscoelastic model

σ → stress, ε→ strain

springs: σ0 = E0ε, σe = Eεe , dashpot: σv = ηε′v

(E ,E0 = Young’s modulus, η = viscosity)

in parallel: σe = σv , ε = εe + εv ⇒ ηε′v = Eεe = E (ε− εv )

⇒ εv =

∫ t

0

1

τe−

t−sτ ε(s) ds, τ =

η

E(relaxation time)

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σ = σ0 + σe = E0ε+ Eεe

= E0ε+ E (ε− εv )

= (E0 + E )ε− Eεv

= E 0ε︸︷︷︸elasticity

−E

∫ t

0

1

τe−

t−sτ ε(s) ds︸︷︷︸

viscoelasticity

PDE: ∂2t u −∇ · σ = f

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Let x ∈ Ω bounded domain in R3, t ≥ 0, we consider the 3D-viscoelasticmodel (density ρ = 1), stress tensor: ε(u) = ∇u +∇uT

(1)

Pu := ∂2t u −∇ ·

(µ0ε(u) + λ0(∇ · u)I

)︸︷︷︸elasticity

+ ∇ ·∫ t

0

(µ(t − s)ε(u(s)) + λ(t − s)(∇ · u)(s)I )ds︸︷︷︸viscoelasticity (one branch)

= 0,

u(0) = u0, ∂tu(0) = u1 in Ω,

u = 0, on ∂Ω× (0,+∞).

µ(x , t) = µ(x)h(t), λ(x , t) = λ(x)h(t) (e.g. h(t) = τ−1e−t/τ ).

u(x , t) : displacement vector,(λ0(x), µ0(x), λ(x), µ(x)) : coefficients.

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IntroductionRelationship to other viscoelastic models

Changing u by us in the viscoelastic part...

(1′)

Pu := ∂2t u −∇ · (µ0ε(u) + λ0(∇ · u)I )︸︷︷︸

elasticity

− ∇ ·∫ t

0

(µ(t − s)ε(us(s)) + λ(t − s)(∇ · us)(s)I )ds︸︷︷︸viscoelasticity (one branch)

= 0,

u(0) = u0, ∂tu(0) = u1 in Ω,

u = 0, on ∂Ω× (0,+∞).

µ(x , t) = µ(x)h(t), λ(x , t) = λ(x)h(t) (e.g. h(t) = e−t/τ ).

µ0(x) = µ0(x) + µ(x), λ0(x) = λ0(x) + λ(x),

(λ0(x), µ0(x), λ(x), µ(x)) : coefficients.

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generalized Maxwell model...

(1′′)

Pu := ∂2t u −∇ ·

(µ0ε(u) + λ0(∇ · u)I


+ ∇ ·∫ t

0

N∑j=1

µj(t − s)ε(u(s))ds︸︷︷︸viscoelasticity (N branchs)

= 0,

u(0) = u0, ∂tu(0) = u1 in Ω,

u = 0, on ∂Ω× (0,+∞).

µ(x , t) =N∑j=1

µj(x)hj(t), (e.g. hj(t) = τ−1j e−t/τj ).

(λ0(x), µ0(x), µj(x)Nj=1) : coefficients.

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fractional Maxwell models (α ∈ (0, 1))...

(1′′)

Pu := ∂2t u −∇ ·

(µ0ε(u) + λ0(∇ · u)I


+ ∇ ·∫ t

0

∫ ∞0

µ(ξ, t − s)dξ ε(u(s))ds︸︷︷︸viscoelasticity (∞ branchs)

= 0,

u(0) = u0, ∂tu(0) = u1 in Ω,

u = 0, on ∂Ω× (0,+∞).

µα(x , t) =

∫ ∞0

µα(ξ, x)h(ξ, t) dξ, (e.g. h(ξ, t) = ξ e−ξ t , ξ = 1/τ).

(λ0(x), µ0(x), µα(ξ, x)) : coefficients.

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viscoelastic Kirchhoff plates... (D, u) = Dyyuxx − 2Dxyuxy + Dxxuyy

(2)

Qu :=∂2ttu − ε2∂2

tt∆u + D0∆2u(s) + (D0, u)︸︷︷︸elasticity

+

∫ t

0

D(t − s)∆2u + (D(t − s), u(s))ds︸︷︷︸viscoelasticity

= 0,

+i.c and b.c., on ∂Ω× (0,+∞).

D(x , t) = D(x)h(t), (e.g. hj(t) = τ−1e−t/τ ).

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Outline

1 Introduction

2 Inverse problemsRecover viscoelastic parameter from local displacementRecover viscoelastic parameter for plates (open problem)

3 Motivation




7 Examples

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Inverse problemsRecover viscoelastic parameter from local displacement

Recover p := µ(x) or p := λ(x) from measurements of the solution :

u(x , t) in ω × (0,T )︸︷︷︸(single time dependent interior measurements)

where ω is a (small) subset of Ω and T > 0 or

u(x , t) on Γ× (0,T )︸︷︷︸(single time dependent boundary measurements)

where Γ is a (small) part of ∂Ω and T > 0.

boundary measure

ω

Γ

T

0Ω

internal measure

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Inverse problemsRecover viscoelastic parameter for plates (open problem)

• Recover (uniqueness, stability) p := D0(x) (or p := D(x) for t →∞) inthe stationary flexure plate model from the Cauchy data:(

u,∂u

∂n,∆u,

∂∆u

∂n

)|∂Ω

• Recover (uniqueness, stability) p := D(x) in the stationary flexure platemodel from the Cauchy data: from internal (ω × (0,T )) or boundary(Γ× (0,T )) local measurements.

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Outline

1 Introduction

2 Inverse problems

3 MotivationMotivation: elastographyLithospheric flexure




7 Examples

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MotivationMotivation: elastography

McLaughlin and Yoon, Inverse Problems 2004 [1]elastography: diagram for identification of stiffness profile

M. Fink, presentation in Valparaiso, 2010quantitative shear wave elastography, Airexplorer 2008

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MotivationLithospheric flexure

Watts and Zhong, Geophys. J. Int. 2000flexure of a two-layer viscoelastic plate model, relaxation time=1Myr

Contreras and Osses, Geophys. J. Int. 2010 [2]Nazca plate variable thickness flexure (1D)

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Contreras, Manriquez, Osses, 2011, work in progressBathymetry

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Contreras, Manriquez, Osses, 2011, work in progressNazca plate variable thickness flexure (2D)

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Outline

1 Introduction

2 Inverse problems

3 Motivation

4 Stability Result (interior measurement)SettingTheoremAssumptions



7 Examples

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Stability Result (interior measurement)Setting

p in Ω ?

u in ω

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Stability Result (interior measurement)Theorem

Theorem (2) (Stability) [3] Let u (resp. u) be the solution of (1)associated to the coefficient p (resp. p). Under Hypothesis 1, 2 and 3,there exists κ ∈ (0, 1) such that:

‖p − p‖H2(Ω) ≤ C‖u − u‖κH2(ω×(0,T )),

where C depends on the W 2,∞(Ω)-norm of p and p and on the W 8,∞(Ω×(0,T ))-norm of u and u.

Corollary (Uniqueness from interior measurement)

u = u in ω × (0,T ) =⇒ p = p in Ω

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Stability Result (interior measurement)Assumptions

Hypothesis 1 on the coefficients and trajectories

(λ0, µ0) ∈ (W 2,∞(Ω))2 and (λ, µ) ∈ (W 2,∞(Ω× (0,T )))2 are suchthat u ∈W 8,∞(Ω× (0,T )),

µ0, λ0 + 2µ0, µ0 − h(0)µ, λ0 + 2µ0 − h(0)(λ+ 2µ) satisfyCondition 1,

p is known in a neighborhood V of ∂Ω,

h(0) 6= 0, u1 = − h′(0)h(0) u0, (u1 = τ−1u0).

Condition 1 The scalar function q satisfies Condition 1 if :

it exists K > 0 such that ∀x ∈ Ω, q(x) ≥ K ,

it exists x0 ∈ R3 \ Ω such that ∀x ∈ Ω :

1

2q(x)− |∇q(x) · (x − x0)| ≥ 0.

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Stability Result (interior measurement)Assumptions

Hypothesis 2 on the observation part

T > T0 =d√β︸︷︷︸

large enough

and ω ⊃ V︸︷︷︸arbitrarily narrow, non trapping

,

d = supx∈Ω|x − x0|, β > 0 small.

Ω

V

∂Ω

Hypothesis 3 on the initial dataWe suppose that u0 is such that there exists M > 0 such that :

|ε(u0(x))(x − x0)| ≥ M, ∀x ∈ Ω.

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Stability Result (interior measurement)Proof: Carleman estimate for the viscoleastic system (1) u(0)∂t u(0) ≤ 0

1) Carleman weight: ϕ(x , t) = |x − x0|2 − βt2 → level sets → cones.

weighted energy ≤ source terms + internal measurements + ...

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Write (1) as a system of 7 scalar equations for u, ∇ · u and ∇∧ u(see McLaughlin-Yoon 2004 [1], Imanuvilov-Yamamoto 2005 [4]),

∂2t ui (x , t)−pi (x)∆ui (x , t)+

∫ t

0

h(t−s) pi (x)∆ui (x , s)ds = Li (u,∇·u,∇∧u)

χ = 0

εε

χ = 1Ω

Change of variables to drop the integral term (similar to Cavaterraet al. 2005 [5]):

ui (x , t) = (1+αt)

(pi (x)ui (x , t)−

∫ t

0

h(t − s) pi (x)ui (x , s)ds

), α large.

=⇒ ∂2t ui (x , t)− qi (x)∆ui (x , t) = Li (u,∇ · u,∇∧ u)

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Apply a modified pointwise Carleman inequality for a scalarhyperbolic equation (Klibanov and Timonov, 2004 [6]) in a cone.

weigth = eσϕ(x,t), ϕ(x , t) = |x−x0|2−βt2, x0 ∈ R3\Ω, β, σ > 0

0x

t

ϕ(x , t) = 0

Q

δ2δ

χ = 1

χ=

0

εε

Return to the initial variable and bound/absorb the integral terms:∫Q

(∫ t

0

|u(x , s)|ds

)2

e2σϕ(x,t)dx dt ≤ C

σ

∫Q

|u(x , t)|2e2σϕ(x,t)dx dt

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Stability Result (interior measurement)Proof: Bukhgeim and Klibanov method

2) We apply the method of Bukhgeim and Klibanov 1981 [7] :

w = ∂2t (u − u), w = ∂2

t u,

Pw = −∫ t

0

h(t − s)∇ · (2(p − p) ε(w(s))) ds

−h′(t)∇ · (2(p − p) ε(u0)) , in Ω× (0,+∞),

w(0) = 0, in Ω,

∂tw(0) = −h(0)∇ · (2(p − p) ε(u0)), in Ω,

w = 0, on ∂Ω× (0,+∞).

+ Apply Carleman inequality :

weighted initial energy ≤ interior measurement + sources

=⇒ the stability result : ‖p − p‖H2(Ω) ≤ C‖u − u‖κH2(ω×(0,T ))

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Outline

1 Introduction

2 Inverse problems

3 Motivation


5 Stability Result (boundary measurement)SettingTheoremAssumptionsProof: unique continuationMain scheme of the proof - elastic caseMain scheme of the proof - viscoelastic case


7 Examples

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Stability Result (boundary measurement)Setting

p in Ω ?

u on Γ

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Stability Result (boundary measurement)Theorem

Theorem (3) (Logarithmic stability) [8] Let u (resp. u) be the solutionof (1) associated to the coefficient p (resp. p). Under Hypothesis 1’, 2’and 3’, there exists κ ∈ (0, 1) such that:

‖p − p‖H2(Ω) ≤ C

[log

(2 +

C∑1≤|α|≤2 ‖δαx (u − u)‖2

L2(Γ×(0,T ))

)]−κ

where C depends on the W 2,∞(Ω)-norm of p and p and on the W 8,∞(Ω×(0,T ))-norm of u and u.

Corollary (Uniqueness from boundary measurement)

∂αx u = ∂αx u on Γ× (0,T ), |α| = 1, 2 =⇒ p = p in Ω.

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Stability Result (boundary measurement)Assumptions

Hypothesis 1’ on the coefficientsthe same.

Hypothesis 2’ on the observation part

T > T0︸︷︷︸large enough

and Γ ⊂ ∂Ω︸︷︷︸arbitrarily small

.

Hypothesis 3’ on the initial datathe same.

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Stability Result (boundary measurement)Proof: unique continuation

Theorem (1) (Unique continuation) [8] Let u be a regular solution of(1) starting from rest and with right hand side vanishing in a neighborhoodV of ∂Ω. Then

‖u‖H2(V×(0,T/3)) ≤ C

[log

(2 +

C∑1≤|α|≤2 ‖δαx u‖2

L2(Γ×(0,3T ))

)]−1

where C depends on the W 2,∞(Ω)-norm of coefficients and on the W 4,∞(Ω×(0,T ))-norm of u.

Corollary (Unique continuation)

∂αx u = 0 on Γ× (0, 3T ), |α| = 1, 2 =⇒ u = 0 in V × (0,T/3).

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Stability Result (boundary measurement)Proof: unique continuation

This is done by

using a variant of Fourier-Bros-Iagolnitzer transform [9] for changingequations from hyperbolic to elliptic character as in Bellassoued2008 [4].

applying a Carleman inequality for the resulting elliptic sytem.

using interpolation results as in Robbiano 1995 [10].

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Stability Result (boundary measurement)Main scheme of the proof - elastic case

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Stability Result (boundary measurement)Main scheme of the proof - viscoelastic case

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Outline

1 Introduction

2 Inverse problems

3 Motivation




7 Examples

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Numerical resolutionDirect problem: finite elements

We discretize the system (1):

in space by P1 Lagrange Finite Elements,

in time by a θ-scheme with θ = 0.5 (implicit centered scheme),

by using the Trapezium Formula for the integral term.

We consider :

µ0(x) = λ0(x) = 3,

λ(x) = µ(x) = 1,

h(t) = e−t/τ with τ = 0.3.

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Numerical resolutionInverse problem: variational approach

We are looking for the minimizer of the non-quadratic functional :

J(p) =1

2

∫ T

0

∫ω

(|u(p)− uobs |2 + |∇(u(p)− uobs)|2

)dxdt

with u(p) =M(p), M being the nonlinear operator associated to system(1). We solve the minimization problem by a BFGS algorithm, so we need:

∇J(p, δp) = limε→0

1

ε(J(p + εδp)− J(p))

=

∫ T

0

∫ω

Mpδp · ((u − uobs)−∆(u − uobs))

with Mp the linearized operator ofM around p, i.e. Mpδp = δu satisfies :

(2)

Pδu = −∇ ·

∫ t

0

2δp h(t − s)ε(u(s))ds

δu(0) = 0, in Ω,

∂tδu(0) = 0, in Ω,

δu = 0, on ∂Ω× (0,+∞).

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Numerical resolutionInverse problem: adjoint problem and sensitivity

We introduce the adjoint problem of (2) :

(2)∗

P∗δu∗ := ∂2t δu∗ −∇ ·

(2µ0ε(δu∗) + λ0(∇ · δu∗)I

)+

∫ T

t

∇ ·(

2p h(s − t)ε(δu∗(s)) + λ(s)(∇ · δu∗)(s)I)

ds

P∗δu∗ =

(u(p)− uobs)−∆(u(p)− uobs) in ω × (0,T )0 in (Ω \ ω)× (0,T )

δu∗(T ) = 0, in Ω,

∂tδu∗(T ) = 0, in Ω,

δu∗ = 0, on ∂Ω× (0,+∞).

So we can compute :

∇J(p, δp) =

∫ T

0

∫Ω

δu · P∗δu∗ =

∫ T

0

∫Ω

Pδu · δu∗

=

∫Ω

δp

(∫ T

0

∫ t

0

2h(t − s)ε(u(s)) : ε(δu∗(t))ds dt

)dx

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Outline

1 Introduction

2 Inverse problems

3 Motivation




7 ExamplesExample 1: Reference dataA first result in recovering the coefficientExample II: More realistic domain

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ExamplesExample 1: Reference data

Resolution with FreeFem 2D, Visualization with Medit,

µ(x) = λ(x) = 3, λ(x , t) = h(t) = e−t/τ with τ = 0.3,

p(x) =

1, in the healthy tissu,3, in the tumor,

T = 4, N =T

δt= 20, K = 25, δ = 0.

tumor measurement region mesh

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ExamplesA first result in recovering the coefficient

p ΠK p p∗

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ExamplesChanging the tumor size

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

Error in the parameter in L2!norm

number of eigenvectors in the basis

% o

f erro

r

big tumormedium tumorsmall tumor

=⇒ K optimal depends on the tumor size

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ExamplesChanging the noise level

δ

2%

5%

10%

‖uobs − u

u‖L∞(ω×(0,T )) ≤ δ

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50


% error in the data

% e

rror i

n th

e pa

ram

eter

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ExamplesChanging the time of observation

T

2

4

6

Error in the parameter with respect toT = Nδt with N fixed

1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

50


observation time

% e

rror i

n th

e pa

ram

eter

=⇒ T0

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ExamplesChanging the measurement region

Error in the parameter withrespect to ω

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

50


observation zone

% e

rror i

n th

e pa

ram

eter

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ExamplesExample II: More realistic domain

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ExamplesInitial condition and trajectory

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ExamplesRegularization: finite number K of eigenfunctions

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ExamplesRecovering with fixed eigenfunctions

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ExamplesEigen and mesh adaptation

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ExamplesRecovering with adaptative eigenfunctions

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ExamplesComparison

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References:

J.-R. Yoon J. R. McLaughlin.

Unique identifiability of elastic parameters from time-dependent interior displacement measurement.

Inverse Problems, 20:25–45, 2004.

A. Osses J. Contreras-Reyes.

Lithospheric flexure modeling seaward of the chile trench: Implications for oceanic plate weakening in thetrench outer rise region.

Geophys. J. Int., 182:97–112., 2010.

M. de Buhan and A. Osses.

A stability result in parameter estimation of the 3d viscoelasticity system.

C. R. Acad. Sci. Paris, Ser. 347:1373–1378, 2009.

M. Bellassoued, O. Imanuvilov, and M. Yamamoto.

Inverse problem of determining the density and the Lame coefficients by boundary data.

SIAM J. Math. Anal., 40, 2008.

C. Cavaterra, A. Lorenzi, and M. Yamamoto.

A stability result via Carleman estimates for an inverse source problem related to a hyperbolicintegro-differential equation.

Computational and Applied Mathematics, 25:229–250, 2006.

M.V. Klibanov and A. Timonov.

Carleman estimates for coefficient inverse problems and numerical applications.

VSP, Utrecht, 2004.

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A.L. Bukhgeim and M.V. Klibanov.

Global uniqueness of class of multidimensional inverse problems.

Soviet Math. Dokl., 24, 1981.

M. de Buhan and A. Osses.

Logarithmic stability in determination of a 3d viscoelastic coefficient and numerical examples.

Inverse Problems, 26:38pp., 2010.

D. Iagolnitzer.

Microlocal essential support of a distribution and local decompositions - an introduction. Hyperfunctionsand theoretical physics, volume 449, pp. 121-132.

Lecture Notes in Mathematics, Springer-Verlag, 1975.

L. Robbiano.

Fonction de cout et controle des solutions des equations hyperboliques.

Asymptot. Anal., 10, 1995.

inverse problems in lithospheric exure and viscoelasticity€¦ · viscoelasticity axel osses in...

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