an inverse problem for moore gibson thompson...

An inverse problem for Moore Gibson Thompsonequation

arising in high intensity ultrasound

Rodrigo Lecaros

Universidad Tecnica Federico Santa Marıa

Workshop on Applied & Interdisciplinary Mathematics19-20-21 March, 2019

supported by FONDECYT project 11180874

R. Lecaros (USM) Inverse problems in MGT U. Chile & U. Bath 1 / 21

Thank you for coming at this presentation

And thank you at the organizers

for the opportunity to present my work in this workshop

http://mat.utfsm.cl/

http://eventos.cmm.uchile.cl/cmmubath2019/

http://eventos.cmm.uchile.cl/cmmubath2019/

Joint work with

Alberto Mercado, Universidad Tecnica Federico Santa Marıa.

Sebastian Zamorano, Universidad de Santiago.


Thank the audience for being awake.

The model

A models for wave propagation in viscous thermally relaxing fluids.

It is well known that the use the classical Fourier’s law to describe theheat flux leads to an infinite signal speed paradox.

Moore Gibson Thompson (MGT) equationτuttt + αutt − c2∆u − b∆ut = f , Ω× (0,T )u = g , ∂Ω× (0,T )u(·, 0) = u0, ut(·, 0) = u1, utt(·, 0) = u2, Ω,

(1)

In this work, we consider the case α = α(x) and b > 0.



The model


(2)

In this work, we consider the case α = α(x) and b > 0.

α(x) > 0, is a coefficient depending on a viscosity of the fluid.

τ is the relaxation time.

c is the speed of sound

b = δ + τc2, where δ ≥ 0 is the diffusivity of sound.

Henceforth we will consider τ = 1.


The model


(3)

The equation models different phenomena depending on theparameters.

If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.A therapeutic method of non–invasive ablation of tumors.If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.If b = 0, there does not exist an infinitesimal generator of thesemigroup.If γ := α− c2

b > 0, the group associated to the equation isexponentially stable, and for γ = 0, the group is conservative.


The model


(3)

The equation models different phenomena depending on theparameters.If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.

A therapeutic method of non–invasive ablation of tumors.If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.If b = 0, there does not exist an infinitesimal generator of thesemigroup.If γ := α− c2



The model


(3)

The equation models different phenomena depending on theparameters.If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.A therapeutic method of non–invasive ablation of tumors.

If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.If b = 0, there does not exist an infinitesimal generator of thesemigroup.If γ := α− c2



The model


(3)

The equation models different phenomena depending on theparameters.If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.A therapeutic method of non–invasive ablation of tumors.If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.

If b = 0, there does not exist an infinitesimal generator of thesemigroup.If γ := α− c2



The model


(3)

The equation models different phenomena depending on theparameters.If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.A therapeutic method of non–invasive ablation of tumors.If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.If b = 0, there does not exist an infinitesimal generator of thesemigroup.

If γ := α− c2



The model


(3)

The equation models different phenomena depending on theparameters.If b = 0 and f = β(u2)t is the Westervelt equation, which is used asa model of finite-amplitude nonlinear wave propagation in soft tissues.A therapeutic method of non–invasive ablation of tumors.If b > 0, the well–posedness and exponential decay of the equationhas been proved by Kaltenbacher et al.If b = 0, there does not exist an infinitesimal generator of thesemigroup.If γ := α− c2

b > 0, the group associated to the equation isexponentially stable, and for γ = 0, the group is conservative.R. Lecaros (USM) Inverse problems in MGT U. Chile & U. Bath 5 / 21

Inverse Problem

The inverse problem is to recover the unknown coefficient α(x)

α(x)→

uttt + αutt − c2∆u − b∆ut = fu = gu(·, 0) = u0, ut(·, 0) = u1, utt(·, 0) = u2

→ u(α)

from partial knowledge of some trace of the solution u(α) at the boundary,where Γ0 ⊂ ∂Ω is a relatively open subset, called the observationregion,and n is the outward unit normal vector on Γ.

α→ ∂u(α)

∂non Γ0 × (0,T ),



The aim tasks

Under appropriate hypotheses:

Uniqueness:

∂u(α1)

∂n=∂u(α2)

∂non Γ0 × (0,T ) implies α1 = α2 in Ω.

Stability:

‖α1 − α2‖X (Ω) ≤ C

∥∥∥∥∂u(α1)

∂n− ∂u(α2)

∂n

∥∥∥∥Y (Γ0)

,

for some appropriate spaces X (Ω) and Y (Γ0).

Reconstruction: Design an algorithm to recover the coefficient α

from the knowledge of∂u(α)

∂non Γ0.


Difficulties

Third-order in time.

Energies is not preserved, α represent a dissipation coefficient.

M-G-T is not controllable with interior control.

Improve the energies estimates.


Under certain conditions for α, Γ0 and the time T .

The admissible coefficients:

AM =

α ∈ L∞(Ω),

c2

b≤ α(x) ≤ M ∀x ∈ Ω

, (4)

and we consider the assumptions:

∃x0 /∈ Ω such that Γ0 ⊃ x ∈ Γ : (x − x0) · n ≥ 0, (5)

andT > sup

x∈Ω|x − x0|. (6)

Also, we suppose that the data satisfies

(u0, u1, u2) ∈ (L2(Ω)× H−1(Ω)× H−2(Ω)), |u2| ≥ η > 0,f ∈ L1(0,T ; L2(Ω), g ∈ L2(0,T ; L2(∂Ω)).

(7)


Principal Results

Our main result, concerning the stability, is the following:

Theorem (R. L., A. Mercado, S. Zamorano)

Suppose that Γ0 ⊂ ∂Ω and T > 0 satisfy (5)-(6) and the data satisfy (7).Let M > 0, and α2 ∈ AM be such that the corresponding solution u(α2)of (3) (with α = α2) satisfies

u(α2) ∈ H3(0,T ; L∞(Ω)).

Then there exists a constant C > 0 such that

C−1‖α1−α2‖2L2(Ω) ≤

∥∥∥∥∂u(α1)

∂n− ∂u(α2)

∂n

∥∥∥∥2

H2(0,T ;L2(Γ0))

≤ C‖α1−α2‖2L2(Ω)

(8)for all α1 ∈ AM .


Remarks

The hypothesis u(α2) ∈ H3(0,T ; L∞(Ω)) in Theorem 1 is satisfied ifmore regularity is imposed on the data.

The inverse problem studied in this paper was previously consideredby Liu and Triggiani.The results obtained in this work requires less regularity.

The hypotheses (5) and (6) on Γ0 and T typically arises in the studyof stability or observability inequalities for the wave equation.

The assumption of the positiveness for u2 appearing in Theorem 1 isclassical when applying the Bukhgeim-Klibanov method and Carlemanestimates for inverse problems with only one boundary measurement.


Well–posedness

Theorem ([1], Theorem 2.2)

Let b > 0 and α ∈ L∞(Ω). Then the solution u(α) is generated by astrongly continuous group on the state space

H = (H2(Ω) ∩ H10 (Ω))× H1

0 (Ω)× L2(Ω).

That is, for each (u0, u1, u2) ∈ (H2(Ω) ∩ H10 (Ω))× H1

0 (Ω)× L2(Ω) andf ∈ L1(0,T ; L2(Ω)), there exists a unique solutionU = (u(α), ut(α), utt(α)) ∈ C ([0,T ];H).

Kaltenbacher, Barbara and Lasiecka, Irena

Exponential decay for low and higher energies in the third order linearMoore-Gibson-Thompson equation with variable viscosity

Palest. J. Math 1 (2012) 1–10.


Hidden regularity


The unique solution(u, ut , utt) ∈ C ([0,T ]; (H2(Ω) ∩ H1

0 (Ω))× H10 (Ω)× L2(Ω)) of (3) satisfies

∂u

∂n∈ H1(0,T ; L2(∂Ω)). (9)

Moreover, the normal derivative satisfies∥∥∥∥∂u∂n∥∥∥∥2

H1(0,T ;L2(∂Ω))

≤ C (‖u0‖2H2(Ω)∩H1

0 (Ω)+‖u1‖2H1

0 (Ω)+‖u2‖2L2(Ω)+‖f ‖2

L1(0,T ;L2(Ω))).

(10)Consequently, the mapping (f , u0, u1, u2) 7→ ∂u

∂n is linear continuous fromL1(0,T ; L2(Ω))× (H2(Ω) ∩ H1

0 (Ω))× H10 (Ω)× L2(Ω)) into

H1(0,T ; L2(∂Ω)).


Carleman estimates

For x0 ∈ RN \ Ω and λ > 0, we define the weight functions φ and ϕλ asfollows

ϕλ(x , t) = eλφ(x ,t), (11)

whereφ(x , t) = |x − x0|2 − βt2 + M0, 0 < β < 1, (12)

and M0 is chosen such that

∀(x , t) ∈ Ω× (−T ,T ), φ(x , t) ≥ 1. (13)



Carleman estimates


Suppose that Γ0 and T satisfies (5)–(6). Let M > 0 and α ∈ AM . Letβ ∈ (0, 1) such that

βT > supx∈Ω‖x − x0‖. (11)

Then, there exists s0 > 0, λ > 0 and a positive constant C such that forall s ≥ s0

√s

∫Ωe2sϕλ(0)|ytt (0)|2dx

+sλc4∫ T

0

∫Ωe2sϕλϕλ(|yt |2 + |∇y|2)dxdt + s3

λ3c4

∫ T

0

∫Ωe2sϕλϕ

3λ|y|

2dxdt

+sλ

∫ T

0

∫Ωe2sϕλϕλ(|ytt |2 + |∇yt |2)dxdt + s3

λ3∫ T

0

∫Ωe2sϕλϕ

3λ|yt |

2dxdt

≤ C

∫ T

0

∫Ωe2sϕλ |f |2dxdt + Csλ

∫ T

0

∫Γ0

e2sϕλ(|∇yt · n|2 + c4|∇y · n|2

)dσdt,

for all y ∈ L2(0,T ;H10 (Ω)) satisfying f ∈ L2(Ω× (0,T )),

y(·, 0) = yt(·, 0) = 0 in Ω, and ytt(·, 0) ∈ L2(Ω).



Sketch of the proof for Carleman estimates

For the wave equation we have

Ew (u) :=√

s

∫Ωe2sϕλ(0)|ut (0)|2dx + sλ

∫ T

0

∫Ωe2sϕλϕλ(|ut |2 + |∇u|2)dxdt + s3

λ3∫ T

0

∫Ωe2sϕλϕ

3λ|u|

2dxdt

≤ C

∫ T

0

∫Ωe2sϕλ |L0u|

2dxdt + Csλ

∫ T

0

∫Γ0

e2sϕλ(|∇u · n|2

)dσdt,

where L0u = utt − b∆u is the classical wave operator.

We consider theoperator Lαu = uttt + αutt − c2∆u − b∆ut , and we have

Lαu = L0ut +c2

bL0u + (α− c2

b)utt .

Now we consider the weight norm ‖f ‖2w =

∫ T0

∫Ω e2sϕλ |f |2dxdt. And we

compute

‖Lαu − (α− c2

b)utt‖2

w = ‖L0ut‖2w +

c4

b2‖L0u‖2

w + 2c2

b(e2sϕλ∂tL0u, L0u).

Using the Carleman estimates for the waver equation,

C‖Lαu‖2w+C‖(α−c2

b)utt‖2

w ≥ Ew (ut)+c4

b2Ew (u)+

c2

b

∫ T

0

∫Ωe2sϕλ∂t |L0u|2dxdt.


Sketch of the proof for Carleman estimates

For the wave equation we have

Ew (u) :=√

s

∫Ωe2sϕλ(0)|ut (0)|2dx + sλ

∫ T

0

∫Ωe2sϕλϕλ(|ut |2 + |∇u|2)dxdt + s3

λ3∫ T

0

∫Ωe2sϕλϕ

3λ|u|

2dxdt

≤ C

∫ T

0

∫Ωe2sϕλ |L0u|

2dxdt + Csλ

∫ T

0

∫Γ0

e2sϕλ(|∇u · n|2

)dσdt,

where L0u = utt − b∆u is the classical wave operator. We consider theoperator Lαu = uttt + αutt − c2∆u − b∆ut , and we have

Lαu = L0ut +c2

bL0u + (α− c2

b)utt .

Now we consider the weight norm ‖f ‖2w =

∫ T0

∫Ω e2sϕλ |f |2dxdt. And we

compute

‖Lαu − (α− c2

b)utt‖2

w = ‖L0ut‖2w +

c4

b2‖L0u‖2

w + 2c2

b(e2sϕλ∂tL0u, L0u).

Using the Carleman estimates for the waver equation,

C‖Lαu‖2w+C‖(α−c2

b)utt‖2

w ≥ Ew (ut)+c4

b2Ew (u)+

c2

b

∫ T

0

∫Ωe2sϕλ∂t |L0u|2dxdt.


Sketch of the proof for the stability Theorem

Bukhgeim-Klibanov method.


Algorithm to find the coefficient α

This algorithm will be based by the work of Baudouin, Buhan andErvedoza [?], in which they propose a reconstruction algorithm for thepotential of the wave equation.Let us consider the following functional

J[µ, f ](y) =1

2

∫ T

0

∫Ωe2sϕλ |Lαy − f |2dxdt

+1

2

∫ T

0

∫Γ0

e2sϕλ

(∣∣∣∣∂y∂n − µ∣∣∣∣2 +

∣∣∣∣∂yt∂n − µt∣∣∣∣2)dσdt, (12)

where α ∈ AM , g ∈ L2(Ω× (0,T )), µ ∈ H1(0,T ; L2(Γ0)).

L. Baudouin, M. De Buhan, and S. Ervedoza.

Global Carleman estimates for waves and applications.

Communications in Partial Differential Equations, 38(5):823–859, 2013.


Algorithm:

1 Initialization: α0 = c2

b .2 Iteration: From k to k + 1Step 1 - Given αk we consider µk = ∂t

(∂u(αk )∂n − ∂u(α)

∂n

)on Γ0 × (0,T )

where u(αk) and u(α) are the solution of the problems Lαku = f ,Ω× (0,T )u = g , ∂Ω× (0,T )u(·, 0) = u0, ut(·, 0) = u1, utt(·, 0) = u2,Ω

(13)

and Lαu = f ,Ω× (0,T )u = g , ∂Ω× (0,T )u(·, 0) = u0, ut(·, 0) = u1, utt(·, 0) = u2,Ω.

(14)

Step 2 - Minimize the functional J[µk , 0] on the admissible trajectories y .Step 3 - Let y∗,k the minimizer of J[µk , 0] and

αk+1 = αk +y∗,ktt (·, 0)

u2. (15)

Step 4 - Finally, consider αk+1 = T (αk+1), where

T (α) =

M if α > M

α ifc2

b≤ α ≤ M

c2

bif α <

c2

b.

(16)

This function T is to guarantee at each step that αk belongs to theadmissible set AM .


Algorithm:

1 Initialization: α0 = c2

b .2 Iteration: From k to k + 1Step 1 - Given αk we consider µk = ∂t

(∂u(αk )∂n − ∂u(α)

∂n

)on Γ0 × (0,T )

Step 2 - Minimize the functional J[µk , 0] on the admissible trajectories y .Step 3 - Let y∗,k the minimizer of J[µk , 0] and

αk+1 = αk +y∗,ktt (·, 0)

u2. (13)

Step 4 - Finally, consider αk+1 = T (αk+1), where

T (α) =

M if α > M

α ifc2

b≤ α ≤ M

c2

bif α <

c2

b.

(14)

This function T is to guarantee at each step that αk belongs to theadmissible set AM .


The convergence of this algorithm


Assume the same hypotheses of observability Theorem, and the followingassumption of u(α) :

u(α) ∈ H3(0,T ; L∞(Ω)) and |u2| ≥ η > 0. (15)

Then, there exists a constant C > 0 and s0 > 0 such that for all s ≥ s0

and k ∈ N∫Ωe2sϕλ(0)(αk+1 − α)2dx ≤ C√

s

∫Ωe2sϕλ(0)(αk − α)2dx . (16)


Working in progress

Consider a new numerical approach

Baudouin, L. and de Buhan, M. and Ervedoza, S.

Convergent Algorithm Based on Carleman Estimates for the Recovery of aPotential in the Wave Equation.

SIAM Journal on Numerical Analysis, 55(4):1578-1613, 2017.


Thank you for your attention!



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