HUMming to the MUSIC - sorbonne-universite · HUMming to the MUSIC On the use of control methods for imaging small imperfections Mark Asch Université d’Amiens (LAMFA UMR-CNRS 6140)

HUMming to the MUSICOn the use of control methods for imaging small imperfections

Mark Asch

Université d’Amiens (LAMFA UMR-CNRS 6140)Université libre de Bruxelles (Environmental Hydroacoustics Lab)

December 2nd, 2011

Mark Asch (EHL & CNRS 6140 ) GCP-DHC December 2nd, 2011 1 / 51

Joint work with...

Habib Ammari, Ecole Normale Supérieure, Paris.Vincent Jugnon, Ecole Polytechnique, Palaiseau.Marion Darbas, Jean-Baptiste Duval, Amiens.


Convergence of 2 ideas

HUM: Controllability of the wave equation

MUSIC: Imaging algorithms for small-volume imperfections.

ResultWe can perform transient imaging with limited-view data, usingwaves as probes and employing practical imaging techniques

Keyreference

H. Ammari, An inverse IBVP for the waveequation in the presence of imperfections ofsmall volume, SICON 41, 4 (2002).






Keyreference







Keyreference







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Outline

Control forImaging

Control-lability

HUM

Geometry

Conver-gence

Imaging

Static

Transient

Multi-modeNumerics

Methods

Conver-gence

Results


Contents

1 HUMFormulationConvergence of Bi-Grid algorithm for HUM

2 Imaging of small imperfections and point sourcesStatic imagingTransient imaging

3 Numerical formulation

4 Numerical results for Imaging

5 Numerical analysis & results for HUMMeshesConvergence

6 Conclusions and perspectives


Contents








Formulation of exact controllability

The wave equation with Dirichlet control:(∂2

t − c2∆)

u = 0 in Ω× (0,T) ,

u|t=0 = u0, ∂tu|t=0 = u1 in Ω, (1)

u =

g on Γc × (0,T),

0 on Γ \ Γc × (0,T),

Geometry and control time: TInitial excitation: u0,u1Control function: g




t − c2∆)

u = 0 in Ω× (0,T) ,

u|t=0 = u0, ∂tu|t=0 = u1 in Ω, (1)

u =

g on Γc × (0,T),

0 on Γ \ Γc × (0,T),

Geometry and control time: T

Initial excitation: u0,u1Control function: g




t − c2∆)

u = 0 in Ω× (0,T) ,

u|t=0 = u0, ∂tu|t=0 = u1 in Ω, (1)

u =

g on Γc × (0,T),

0 on Γ \ Γc × (0,T),

Geometry and control time: TInitial excitation: u0,u1

Control function: g




t − c2∆)

u = 0 in Ω× (0,T) ,

u|t=0 = u0, ∂tu|t=0 = u1 in Ω, (1)

u =

g on Γc × (0,T),

0 on Γ \ Γc × (0,T),

Geometry and control time: TInitial excitation: u0,u1Control function: g


Existence and uniqueness of g by the HUM

The problem of exact boundary controllability:"Given T, u0, u1, find a control g such that the solution of (1)satisfies, at time T,

u(x,T) = ∂tu(x,T) = 0 in Ω"

Theorem [Lions’88]For T large enough, there exists a unique control g that minimizesthe L2(Γc × (0,T)) norm. This control can be constructed by theHilbert Uniqueness Method.

Proof.(constructive...) Set up the optimality system: equation (forward) plus adjoint (backward).Then show the invertibility of the HUM operator Λe = f , where e is the initial condition ofthe forward equation and f is the final condition of the backward equation. Finally, computeg by a preconditioned, conjugate gradient algorithm.




u(x,T) = ∂tu(x,T) = 0 in Ω"






u(x,T) = ∂tu(x,T) = 0 in Ω"




Geometric controllability

GCC [Bardos, Lebeau, Rauch - SICON’92]Every ray of geometrical optics, starting at any point x ∈ Ω, at timet = 0, hits Γc before time T at a nondiffractive point.

No “glancing” nor “trapped” rays.

Uncontrollable and controllable geometries












So where’s the catch?

Numerically, the HUM produces an ill-posed problem...

Ill-posedness

stability + consistency V/ convergence

Various solutions:Tykhonov regularization.Mixed finite elements.Bi-grid filtering.Uniformly controllable schemes.Spectral approach.




Ill-posedness






Ill-posedness




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Convergence

What is known?1D2D in a periodic regionfinite differences in space and timestructured finite-element meshes

What is still to be done?General, unstructured finite-element meshes on arbitrarygeometries.

Two possible approaches:“analytical” approach based on observability estimations;“numerical analysis” approach based on finite-elementconvergence estimations.


Contents








Contents








Electrical Impedance Imaging

Simplest example that contains ALL the ingredients...

Idea:Impose (a finite number of) boundary voltages and measure theinduced boundary currents to estimate the electrical conductivity.

Asymptotic approach:Use outer expansions in terms of the characteristic size of theanomaly.


Electrical Impedance Imaging

Simplest example that contains ALL the ingredients...

Idea:Impose (a finite number of) boundary voltages and measure theinduced boundary currents to estimate the electrical conductivity.

Asymptotic approach:Use outer expansions in terms of the characteristic size of theanomaly.


Identification of small-volume imperfectionsFormulation

Ω ⊂ Rd, d = 2,3 : bounded, Lipschitz domain, u background solutionof the homogeneous equation

∆u = 0 in Ω,

k0∂u∂ν

∣∣∂Ω

= g(2)

where g ∈ L20(∂Ω) = g ∈ L2(∂Ω),

∫∂Ω

g dσ = 0.m small inclusions, Ds = zs + εBs pairwise-separated andboundary-separated, centres zs, size ε, shapes Bs (Lipschitz),conductivities ks 6= k0 for 1 ≤ s ≤ m.

uε voltage potential, solution of the perturbed equation∇ ·(

k0χ(

Ω \⋃ms=1 Ds

)+∑m

s=1 ksχ(Ds))∇uε = 0 in Ω,

k0∂uε

∂ν

∣∣∂Ω

= g.(3)


Asymptotic expansion for uε − u

(uε − u)(y) = −εdm∑

s=1

(k0

ks− 1)∇xN(zs, y) ·M(s)∇u(zs) + o(εd) (4)

[Friedman, Vogelius, ARMA, 1989]N(x, y) is the Neumann function

∆xN(x, y) = −δy dans Ω,∂N∂νx

∣∣∣∂Ω

= − 1|∂Ω|

M(s) is a symmetric d× d, polarization tensor,

M(s)lm = |Bs| δlm −

∫∂Bs

k0

kl

∂φ+m

∂νdσy

φ+m is a “corrector”: harmonic inside and outside B, zero at∞,

with zero jumps across ∂B.Mark Asch (EHL & CNRS 6140 ) GCP-DHC December 2nd, 2011 17 / 51

Inversion Methodology

Apply the asymptotic formula in order to identify the location andcertains properties of the shape of the imperfections.Using methods based on taking suitable averages of the boundarymeasurements using special background solutions as weights.The term of order εd corresponds physically to a voltage potential dueto m polarized dipoles located at the points zs, 1 ≤ s ≤ m.


Numerous Algorithms

Current projection Apply (d =)2 currents, gi = k0νi, corresponding tobackground potentials ui = xi, i = 1, 2; then use 2 (harmonic) test functions,wi = xi, i = 1, 2Fourier Inversion : for η ∈ R2 arbitrary, apply currents

g = i(η + iη⊥) · νei(η+iη⊥)·x

and test functionsw = ei(η+iη⊥)·x

then Γ(η) is the Fourier transformation of a linear combination of derivativesof delta functions centered at the points 2zs.

Quadratic method.Least squares method.Linear sampling method.MUSIC method.Dynamical methods...


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Acoustic wave equation

IBVP for the wave equation, in the presence of the imperfections,

(Wα)

∂2uα∂t2 −∇ · (γα∇uα) = 0 in Ω× (0,T),

uα = f on ∂Ω× (0,T),

uα|t=0 = u0 ,∂uα∂t

∣∣∣∣t=0

= u1 in Ω.

Define u to be the solution in the absence of any imperfections,

(W0)

∂2u∂t2 −∇ · (γ0∇u) = 0 in Ω× (0,T),

u = f on ∂Ω× (0,T),

u|t=0 = u0 ,∂u∂t

∣∣∣∣t=0

= u1 in Ω.


Control function

Suppose that T and the part Γc of the boundary ∂Ω are suchthat they geometrically control Ω.

Then, for any η ∈ Rd, we can construct by the Hilbertuniqueness method (HUM), a unique gη ∈ H1

0(0,T; L2(Γ)) insuch a way that the unique weak solution wη of the waveequation

(Wc)

∂2wη

∂t2 −∇ · (γ0∇wη) = 0 in Ω× (0,T),

wη = gη in Γc × (0,T), wη = 0 in ∂Ω\Γc × (0,T),

wη|t=0 = β(x)eiη·x∈H10(Ω),

∂wη

∂t

∣∣∣∣t=0

= 0 in Ω,

satisfies wη(T) = ∂twη(T) = 0.


Key theorem (H. Ammari)

TheoremLet η ∈ Rd, d = 2, 3. Let uα be the unique solution to the wave equation (Wα) with

u0(x) = eiη·x, u1(x)n = −i√γ0 |η| eiη·x, f (x, t) = eiη·x−i√γ0|η|t.

Suppose that Γc and T geometrically control Ω; then we have∫ T0

∫Γc

[θη(∂uα∂n −

∂u∂n

)+ ∂tθη∂t

(∂uα∂n −

∂u∂n

)]= −

∫ T0

∫Γc

ei√γ0|η|t∂t

(e−i√γ0|η|tgη

) (∂uα∂n −

∂u∂n

)= αd∑m

j=1

(γ0γj− 1)

e2iη·zj[Mj(η) · η − |η|2 |Bj|

]+ o(αd)

.= Λα(η),

(5)

where θη is the unique solution to the ODE (6)∂tθη − θη = ei√γ0|η|t∂t


)for x ∈ Γc, t ∈ (0,T),

θη(x, 0) = ∂tθη(x,T) = 0 for x ∈ Γc,(6)

with gη the boundary control in (Wc), and Mj the polarization tensor of Bj.


Idea of the proof

asymptotic formula for ∂uα∂ν on ∂Bj in terms of ∂Φ

∂ν ,γ0γj

and u;

introduce auxilliary solution, vα, of homogeneous waveequation with initial condition ∂tvα = the asymptotic principalterm;show that the asymptotic equals the boundary average of∂vα∂n gη;

finally, use θη and the Volterra equation (6) to replace vα by uα(using Taylor expansions and asymptotics in α)


Asymptotic formula

Λα(η) = −∫ T

0∫

Γcei√γ0|η|t∂t


) (∂uα∂n − ∂u

∂n)

≈ αd∑mj=1

(γ0γj− 1)

e2iη·zj[Mj(η) · η − |η|2

∣∣Bj∣∣]

.= αd∑m

j=1 Cj(η)e2iη·zj .

Fundamental observation: the inverse Fourier transform givesa linear combination of (derivatives of) delta functions,

Λ−1α (x) ≈ αd∑m

j=1 Lj(δ−2zj

)(x)


Asymptotic formula

Λα(η) = −∫ T

0∫

Γcei√γ0|η|t∂t


) (∂uα∂n − ∂u

∂n)

≈ αd∑mj=1

(γ0γj− 1)


∣∣Bj∣∣]

.= αd∑m

j=1 Cj(η)e2iη·zj .

Fundamental observation: the inverse Fourier transform givesa linear combination of (derivatives of) delta functions,

Λ−1α (x) ≈ αd∑m

j=1 Lj(δ−2zj

)(x)


Fourier algorithm

Reconsctruction AlgorithmSample values of Λα(η) at some discrete set of points and thencalculate the corresponding discrete inverse Fourier transform.After a rescaling by −1

2 , the support of this discrete inverseFourier transform yields the location of the small imperfectionsBα.

Once the locations are known, we may calculate thepolarization tensors

(Mj)m

j=1 by solving an appropriate linearsystem arising from (6). These polarization tensors give ideason the orientation and relative size of the imperfections.


Other algorithms

By choosing other “currents”, we can develop a variety of differentmethods:

Kirchoff migrationBack-propagationArrival timeMUSIC

Keyreferences

H. Ammari, E. Bossy, V. Jugnon and H. Kang,Mathematical Modelling in Photo-AcousticImaging. SIAM Review 52. (2010),H. Ammari, M. Asch, V. Jugnon and H. Kang,Transient wave imaging with limited-viewdata, SIAM J. Imaging Sci., 4, pp. 1097-1121(2011).


Contents








Wave equation

Standard P1 −Q1 finite element discretization in space.

MU′′h(t) + KUh(t) = Bh 0 < t ≤ T,Uh(0) = U0

h and U′h(0) = V0h given,

(7)

where,mass matrix, M, has coefficients Mij =

∫Ωωi(x)ωj(x)dx,

stiffness matrix, Kij =∫

Ωγ(x)∇ωi(x)∇ωj(x)dx,

and Bh is the right hand side vector.

Newmark scheme in time (permits to pass from fully explicit tofully implicit schemes).


Bi-Grid HUM

Newmark wave-equation solver.Bi-grid filter:

wave equations are solved on a fine mesh of size hresiduals are computed on a coarse mesh of size 2h

Conjugate gradient iteration for inversion of the HUMoperator.

Keyreference

A., G. Lebeau, Geometrical aspects of exactboundary controllability for the wave equationa numerical study. ESAIM : Control,Optimisation and Calculus of Variations ; Vol.3 ; pp. 163-212 (1998).


HUM simulations

full-view on square and disc,partial-view on square and disc.


Fourier Imaging Algorithm

Take a finite number of imperfections, zj + αBj for j = 1, ...,m,with conductivities γj.

Then, for each η ∈ [−ηm, ηm]d in a discrete set D of values,1 Compute the solution uα of the wave equation (Wα) by a finite

element method to simulate the boundary data∂uα∂n

on Γ× (0,T)

for the inverse problem.2 Compute the quantity

∂u∂n

on Γc × (0,T) which is explicitlyknown from the initial conditions.

3 Calculate the control gη of (Wc ) via the BiGrid HUM method.4 Form the quantity Λα(η) from the left hand side of (6) with a

suitable quadrature formula.

Finally, apply the inverse Fourier transform to Λα(η) (η in theFourier space D) to compute Λα(x) (x in the physical space Ω).


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Fourier calibration

Objective:Recover Dirac masses situated at the points −2zj from the inversetransform of the asymptotic formula

Λα(η) = αdm∑

j=1

(γ0

γj− 1)


∣∣Bj∣∣] , d = 2,3

Numerous numerical issues arise:resolution/cost -

no free lunch... use parallel processing.

aliasing/rippling -

avoid truncation, respect periodicity of DFT.

instabilities for large |η| -

use thresholding.


Fourier calibration


Λα(η) = αdm∑

j=1

(γ0

γj− 1)


∣∣Bj∣∣] , d = 2,3

Numerous numerical issues arise:resolution/cost -

no free lunch... use parallel processing.

aliasing/rippling -



use thresholding.


Fourier calibration


Λα(η) = αdm∑

j=1

(γ0

γj− 1)


∣∣Bj∣∣] , d = 2,3

Numerous numerical issues arise:resolution/cost - no free lunch... use parallel processing.aliasing/rippling -



use thresholding.


Fourier calibration


Λα(η) = αdm∑

j=1

(γ0

γj− 1)


∣∣Bj∣∣] , d = 2,3

Numerous numerical issues arise:resolution/cost - no free lunch... use parallel processing.aliasing/rippling - avoid truncation, respect periodicity of DFT.instabilities for large |η| -

use thresholding.


Fourier calibration


Λα(η) = αdm∑

j=1

(γ0

γj− 1)


∣∣Bj∣∣] , d = 2,3

Numerous numerical issues arise:resolution/cost - no free lunch... use parallel processing.aliasing/rippling - avoid truncation, respect periodicity of DFT.instabilities for large |η| - use thresholding.


Example: Fourier resolutionthresholding and windowing

Three imperfections: z1 = (0.63,0.28), z2 = (0.39,0.66),z3 = (0.73,0.65), α = 0.03, γj = 10.Sampling: N = 256, ηm = 33.

IntroductionProcedure d’identification

Implementation de la localisation dynamiqueDetection numerique en 2D et 3D

Conclusion et perspectives

Tests de calibrationParametres d’entreeResultats numeriques 2DResultats numeriques 3DImplementation parallele

Test numerique 2D : sans seuillage ni fenetrage

Imperfections : z1 = (0.63, 0.28), z2 = (0.39, 0.66),z3 = (0.73, 0.65), α = 0.03 et γj = 10 ;Echantillonnage : Ne = 256 et ηmax = 33.

Duval Jean-Baptiste Detection numerique de petites imperfections en 2D et 3D 39 / 54





Test numerique 2D

Fenetre de Blackman Seuillage‖ (η∗, η∗) ‖ = ‖ (9, 9) ‖


At left: without; at right: with thresholding (see [A., Mefire. IJNAM, 6 (1), 2009.])


Example: Fourier resolutionthresholding and windowing

Three imperfections: z1 = (0.63,0.28), z2 = (0.39,0.66),z3 = (0.73,0.65), α = 0.03, γj = 10.Sampling: N = 256, ηm = 33.





Test numerique 2D : sans seuillage ni fenetrage

Imperfections : z1 = (0.63, 0.28), z2 = (0.39, 0.66),z3 = (0.73, 0.65), α = 0.03 et γj = 10 ;Echantillonnage : Ne = 256 et ηmax = 33.






Test numerique 2D

Fenetre de Blackman Seuillage‖ (η∗, η∗) ‖ = ‖ (9, 9) ‖


At left: without; at right: with thresholding (see [A., Mefire. IJNAM, 6 (1), 2009.])Mark Asch (EHL & CNRS 6140 ) GCP-DHC December 2nd, 2011 35 / 51

Example: Transient imaging by Fourier in 3D

Four imperfections centered at z1 = (0.66, 0.32, 0.47), z2 = (0.55, 0.71, 0.39),z3 = (0.39, 0.63, 0.31), z4 = (0.71, 0.42, 0.74) .Ne = 64, ηmax = 40 and η∗ = 9. Radius α = 0.01 and conductivity γj = 10.

[A., Darbas, Duval. ESAIM-COCV, 2010.]Mark Asch (EHL & CNRS 6140 ) GCP-DHC December 2nd, 2011 36 / 51

Limited-view data in 2- and 3D

Repeat above computations, but compute measurements on apart of the bopundary that respects the geometric controlcondition (GCC)We obtain “perfect” reconstruction!

Conclusion:As long as we can accurately compute the boundary controlfunction, g, we can reconstruct equally well with limited-view data.


Limited-view data in 2- and 3D

Repeat above computations, but compute measurements on apart of the bopundary that respects the geometric controlcondition (GCC)We obtain “perfect” reconstruction!

Conclusion:As long as we can accurately compute the boundary controlfunction, g, we can reconstruct equally well with limited-view data.


Some point source exampleshere comes the MUSIC...

Computations performed by V. JugnonMark Asch (EHL & CNRS 6140 ) GCP-DHC December 2nd, 2011 38 / 51

Contents








Contents








HUMMeshes

Replace:

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


HUMMeshes

by:

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1


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HUMConvergence

Recall some classical FEM convergence results.

Define mesh quality and its connection with FEM error.How is all this related to BiGrid HUM?Preliminary numerical convergence results.


HUMConvergence

Recall some classical FEM convergence results.Define mesh quality and its connection with FEM error.

How is all this related to BiGrid HUM?Preliminary numerical convergence results.


HUMConvergence

Recall some classical FEM convergence results.Define mesh quality and its connection with FEM error.How is all this related to BiGrid HUM?

Preliminary numerical convergence results.


HUMConvergence

Recall some classical FEM convergence results.Define mesh quality and its connection with FEM error.How is all this related to BiGrid HUM?Preliminary numerical convergence results.


Convergence vs. mesh quality

As seen previously, the convergence of the FEM depends onmesh sizemesh quality

Bi-grid HUM is strongly dependent on mesh quality because ofthe “brutal” nature of the filtering, h −→ h/2Error norms used for convergence of the HUM:∥∥u0 − uh

0

∥∥L2(Ω)∥∥u(T)− uh(T)

∥∥L2(Ω)

other possibilities:H−1-norm of ut;L2-norm of g (see A., Lebeau. ESAIM COCV’98)


(Preliminary) Numerical results

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

quality (min)

erro

r

u

0

u(T)


(Preliminary) Numerical results

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

quality (min)

err

or

u0

u(T)


(Preliminary) Numerical results (contd.)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.01

0.02

0.03

0.04

0.05

0.06

quality (min)

err

or

u0

u(T)


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Conclusions

Imaging:Precise, robust and fast (non-iterative) imaging algorithms.Can deal with limited-view data in a transient context.

HUM:Bi-grid performs well on unstructured meshes.Optimizing the mesh quality improves the convergence.


Perspectives

Imaging:multi-wave imaging techniques (PAT, MAT, RFAT, etc.)dissipative media which are far more realistic (especially formedical imaging)random media...

HUM:general proof of Bi-grid convergence using FEM error estimatesand mesh quality...


Thank you! Questions?

Further reading I

H. Ammari.Asymptotic, multi-wave imaging... Consulthttp:///www.cmap.polytechnique.fr/~ammari/

H. Ammari.An Introduction to Mathematics of Emerging BiomedicalImaging. Mathématiques et Applications, Volume 62,Springer-Verlag, Berlin, 2008.

E. Zuazua.HUM... Consulthttp://www.bcamath.org/zuazua.


Documents

HUMming to the MUSIC - sorbonne-universite · HUMming to the MUSIC On the use of control methods for imaging small imperfections Mark Asch Université d’Amiens (LAMFA UMR-CNRS 6140)