convergence issues and some applications mark asch · mark asch (gt contrôle, ljll, parris-vi)...

Numerical control of wavesConvergence issues and some applications

Mark Asch

U. Amiens, LAMFA UMR-CNRS 7352

June 15th, 2012

Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 1 / 74

Contents

1 HUMFormulationAnalysis of the non-convergence

2 Bi-Grid HUMConvergence of Bi-Grid HUM

3 Numerical analysis & results for HUMMeshesConvergence

4 ApplicationsDynamic detection of small imperfectionsDetection of point sources by imaging techniques

5 Conclusions and perspectives


Contents







Formulation of exact controllability

The wave equation with Dirichlet control:(∂2t − 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),

1 Geometry and control time: T2 Initial excitation: {u0,u1}3 Control function: g




)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),

1 Geometry and control time: T

2 Initial excitation: {u0,u1}3 Control function: g




)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),

1 Geometry and control time: T2 Initial excitation: {u0,u1}

3 Control function: g




)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),

1 Geometry and control time: T2 Initial excitation: {u0,u1}3 Control 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 conjugate gradient algorithm.


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

Numerical PDE PDE(h, �t) (h, �t) ! 0

Discrete control Control

(h, �t) ! 0gh(h, �t) g

ExactControllability

Discrete exactControllability

Convergence?


Don’t worry...

There are numerous solutions that have been proposed:Tykhonov regularization.Mixed finite elements.Bi-grid filtering.Uniformly controllable schemes.Spectral approach.

But how do we analyze and prove the convergence of thesemethods?


Contents







Observability

Controllability-ObservabilityControllability of the direct wave equation is equivalent toobservability of the adjoint wave equation.

So we prefer (for technical facility) to try and prove theobservability condition which says that the energy of the system isbounded by the trace of the normal derivative on the boundary.


High frequency modes

In general, any discrete dynamics associated to the waveequation generates spurious high-frequency oscillations that donot exist at the continuous level.

Moreover, a numerical dispersion phenomenon appears and thevelocity of propagation of some high frequency numericalwaves may possibly converge to zero when the mesh size hdoes. This is a localization phenomenon...

In this case, the controllability/observability property for thediscrete system will not be uniform, as h→ 0, for a fixed time Tand, consequently,there will be initial data (even very regular ones) for which thecorresponding controls of the discrete model will diverge in theL2-norm as h tends to zero.


Theoretical analysis

Infante, Zuazua, 19991 (detailed analysis of the 1-D case) andZuazua, 20052 (2-D case), considered a semi-discretization of thewave equation with the classical finite differences or finite elementmethod:

after filtering the high frequency modes, a uniformobservability inequality for the adjoint system holds - this isequivalent to the uniform controllability of the projection of thesolutions over the space generated by the remainingeigenmodes;

the dimension of this space tends to infinity as the step size hgoes to zero and, in the limit, we would obtain the control of thecontinuous system - however, in practice these projectionmethods are not very efficient.

1J.A. Infante and E. Zuazua, Boundary observability for the spacediscretization of the one-dimensional wave equation, M2AN 33 (2) (1999), 407-438.

2E. Zuazua, Propagation, Observation, Control of Waves Approximated byFinite Difference Methods, SIAM Review, 47 (2) (2005), 197–243


Theoretical analysis (contd.)

In Micu, 2002 3 the problem with finite difference approximationwas considered again.

It was proved that, if the high frequency modes of the discrete initialdata are filtered out in an appropriate manner (or if the initial dataare sufficiently regular), there are controls of the semi-discrete modelwhich converge to a control of the continuous wave equation.

This is one of the ways of taking care of the spurious high frequencyoscillations that the numerical method introduces.

Note that in this case the uniform controllability of the entirediscrete solutions is ensured and not only that of the projections, asin Infante, Zuazua.

Moreover, it was also shown that the norm of the discrete HUMcontrols may increase exponentially with the number of points in themesh if no filtering is applied.

3S. Micu, Uniform boundary controllability of a semi-discrete 1-D waveequation, Numerische Mathematik 91 (4) (2002), 723-768.


Other numerical methods

From a numerical point of view, several techniques have beenproposed as possible cures of the high frequency spuriousoscillations.

In [Glowinski, Li, Lions, 1990]4 a Tychonoff regularizationprocedure was successfully implemented. Roughly speaking,this method introduces an additional control, tending to zerowith the mesh size, but acting on the interior of the domain.Another proposed numerical technique is the mixed finiteelement method (see [Glowinski, Kinton, Wheeler, 1989]5).

4R. Glowinski, C. H. Li, J.-L. Lions, A numerical approach to the exactboundary controllability of the wave equation (I). Dirichlet controls: Description ofthe numerical methods, Japan J. Math., 68 (7), 1-76, 1990.

5Glowinski R., Kinton W. and Wheeler M. F., A mixed finite elementformulation for the boundary controllability of the wave equation, Int. J. Numer.Methods Eng. 27(3), (1989), 623-636.


Other numerical methods (contd.)

In [Castro, Micu, Münch, 2008]6 a mixed finite element methodis considered for 2-D wave equations.

The main advantage of this method is that it is uniformlycontrollable as the discretization parameter h goes to zero andallows to construct a convergent sequence of approximatecontrols (as h→ 0) without filtering.It consists of a different space discretization scheme of the waveequation derived from a mixed finite element method, which isbased on different discretizations for the position and velocity.More precisely, while classical first order splines are used for theformer, discontinuous elements approximate the latter. Thismethod is different to the one used in [Glowinski, Kinton,Wheeler] where u and ∇u are approximated in different finitedimensional spaces.The major disadvantage of the approach is a stricter stabilitycondition: ∆t ≤ h2 instead of ∆t ≤ h for the other methods.

6C. Castro, S. Micu and A. Münch, Numerical approximation of the boundarycontrol of the wave equation in a square with a mixed finite element method, IMAJ. Numerical Analysis 28 (3), 186–214. (2008)


Other numerical methods (contd.)

Uniformly controllable, implicit finite difference schemes havebeen analyzed in:

[Münch, 2005]7in the 1D case[Asch, Münch, 2009]8 in the 2D case

Major disadvantage: computational complexity (implicit,high-order)

7Münch A., A uniformly controllable and implicit scheme for the 1-D waveequation, M2AN, 39(2), 377-418 (2005).

8Asch M., Münch A., Uniformly Controllable Schemes for the Wave Equation onthe Unit Square. J. Optimization Theory and Applications. 143, 3, 417-438, 2009.


Summary: Convergence

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

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

Two possible approaches:“analytical” approach based on observability estimations - thisquestion was open until November 2011...;“numerical analysis” approach based on finite-elementconvergence estimations - see below.


Contents







The original idea (Glowinski, 1992)

The bi-grid method, first suggested in Glowinski, 19929 uses two(space) grids, a coarse and a fine grid.

The main idea is to consider a space discretization scheme ofthe wave equations on the fine grid but taking the initial (orfinal) data on the coarse one.In this way, the high frequencies associated to the fine grid areeliminated.The basic idea is a small change in the definition of thecontrollability operator:

the input is posed on the coarse grid, then interpolated onto thefine grid,the usual ΛT-mapping (composition of direct- and adjoint-waveoperatiors) is done on the fine grid,and the end-result is finally restricted onto the coarse grid again.

9Roland Glowinski. Ensuring well-posedness by analogy; Stokes problem andboundary control for the wave equation. J. Comput. Phys., 103(2):189–221, 1992.


How does it work?

The bi-grid method attenuates the short wavelengthcomponents of the initial data, which are responsible for thespurious high frequency oscillations. The justification for themethod is, roughly, that the high-frequency components on thefine grid are minimal since the state on the fine grid comesfrom a state on the coarse grid.The method was used in Asch and Lebeau (1998),10wheredifferent numerical experiments were conducted for the waveequation in two dimensions, using a finite differencediscretization in both time and space, the CG algorithm andthe bi-grid method.

10Asch M. and Lebeau G., Geometrical aspects of exact boundary controllabilityfor the wave equation: a numerical study. ESAIM Control, Optimisation andCalculus of Variations, 3:163–212, 1998.


Theoretical analysis

After work on the 1-D case, it was proved, in the 2-Dsemi-discrete case, that the bi-grid method actually leads touniform observability, but with a minimal control time whichis twice the correct time (see Negreanu and Zuazua (2004)11).This was improved by Loreti, Mehrenberger (2008),12 whoobtained the sharp control/observation time, T = 2

√2 on the

unit square.These proofs were restricted to uniform meshes (finitedifference or finite element).

11Mihaela Negreanu and Enrique Zuazua. Convergence of a multigrid methodfor the controllability of a 1-d wave equation. C. R. Acad. Sci. Paris, 338(5), 2004.

12P. Loreti and M. Mehrenberger. An Ingham type proof for a two-gridobservability theorem. ESAIM: COCV, 14(3), 2008.


Theoretical analysis: eg. 1-D theorem

Consider the following finite difference approximation of the 1Dwave equation,u′′j − 1h2

[uj+1 − 2uj + uj−1

]= 0, 0 < t < T, j =

1, . . . ,N,uj(t) = 0, j = 0,N + 1, 0 < t < T,uj(0) = u0j , u′j(0) =

u1j , j = 1, . . . ,N.

TheoremFor N ∈ N an odd integer and T > 2

√2 + 2h, for any initial data

pair (u0h,u1h) ∈ Vh, the solution uh of the semi-discrete system

satisfies,

Eh(0) ≤2

T − (2√

2 + 2h)

∫ T0

∣∣∣uNh ∣∣∣2 dt.


Theoretical analysis: the way to go...

Zuazua gave away the solution in the conclusion of Ignat andZuazua (2009)13 :

“It would be interesting to see if these spectral methods can beadapted in order to guarantee uniform observability results fornumerical methods based on the two-grid method.”

He was referring to work by Miller, Tucsnak, Russel and Weiss- see below.

13L. Ignat and E. Zuazua. Convergence of a two-grid algorithm for the control ofthe wave equation. J. Eur. Math. Soc. 11 (2), 2009.


Theoretical analysis: spectral method

Based on spectral characterizations of admissibility andobservability for abstract systems.Original work done by Weiss (1989), Russel and Weiss (1994),Miller (2005), Tucsnak and Weiss (2010).

A big step forward was made by Ervedoza (2009)14, but therewere some gaps.

Final improvements made by Miller (2012)15.

ConclusionObservability results are now valid for any conservative, fullydiscrete (space-time) finite element approximation of the linearwave equation.

14S. Ervedoza. Spectral conditions for admissibility and observability of wavesystems, Num. Math., 113(3), 2009.

15L. Miller. Resolvent conditions for the control of unitary grooups and theirapproximations. J. Spectral Th. 2, 1, 2012.


Spectral method

Recall definition of filtering: restrict the semi-discretizedequation to modes with eigenvalues lower than η/hσ for somepositive η and σ.

Ignat, Liviu Zuazua showed that σ = 2 is optimal for theboundary observation of 1-D wave equations on uniform meshesErvedoza generalized to non-uniform meshes (but not for case ofboundary control), with σ = 2/3Miller improved this to σ = 1 for interior observability and toσ = 2/3 for boundary observability (4/3 and 2/3 for 2nd ordersystems).In addition, he deduced from the uniform exact observability ofthe filtered approximations that the minimal control provided bythe Hilbert Uniqueness Method is the limit of the minimalcontrols for the filtered approximations.Reaches the optimal value, σ = 2, for the simplest 1-D case.


Resolvent condition

LetX, Y be Hilbert spaces, A : D(A)→ X be a self-adjoint operatoriA generates a strongly continuous group

(eitA

)t∈R of unitary

operators on XX1 denote D(A) with the norm ‖x‖1 = ‖(A− β)x‖ for β /∈ σ(A),the spectrum of AC ∈ L(X1,Y) be the observation operator, B ∈ L(Y ′,X ′−1) be thecontrol operator

For simplicity, we consider the first-order, dual controlobservation system with output function y and input function u

ẋ(t)− iAx(t) = 0, x(0) = x0 ∈ X, y(t) = Cx(t) (2)ξ̇(t)− iA′ξ(t) = Bu(t), ξ(0) = ξ0 ∈ X ′, u ∈ L2loc(R; Y ′) (3)

and these equations have unique solutions x ∈ C(R,X) andξ ∈ C(R,X ′) given by

x(t) = eitAx0, ξ(t) = eitA′ξ0 +

∫ t0

ei(t−s)ABu(s) dsMark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 27 / 74

Resolvent condition (contd.)

DefinitionThe system (2) is admissible if for some time T > 0 there is anadmissibility cost KT such that∫ T

0

∥∥CeitAx0∥∥2 dt ≤ KT ‖x0‖2 x0 ∈ D(A).The system (2) is exactly observable in time T at cost κT if the followingobservability inequality holds:

‖x0‖2 ≤ κT∫ T

0‖y(t)‖2 dt, x0 ∈ D(A). (4)

The system (3) is exactly controlable in time T at cost κT if for all ξ0 ∈ X ′there is a u ∈ L2(R; Y ′) such that u(t) = 0 for t /∈ [0,T] , ξ(T) = 0 and∫ T

0‖u(t)‖2 dt ≤ κT ‖ξ0‖2 .



Resolvent conditions

∃ L, l > 0, ∀x ∈ D(A), λ ∈ R, ‖Cx‖2 ≤ L ‖(A− λ)x‖2 + l ‖x‖2 (5)

∃M,m > 0, ∀x ∈ D(A), λ ∈ R, ‖x‖2 ≤M ‖(A− λ)x‖2 + m ‖Cx‖2(6)

These conditions are reminiscent of relative boundedness of Cwith respect to A and resolvent estimates for A



The following theorems say that the resolvent conditions arenecessary and sufficient for admissibility and exact controllabilityrespectively:

TheoremThe system (2) is admissible if and only if the resolvent condition (5)holds.

TheoremLet the system (2) be admissible. Then it is exactly observable if andonly if the resolvent condition (6) holds. In fact, (4) implies (6) withM = T2κTKT and m = 2TκT. Conversely, (6) implies (4) for allT > π

√M with κT = 2mT/(T2 −Mπ2).


Resolvent condition (2nd order system)

The second-order, boundary, dual control-observation systemwith output function y and input function u is:

z̈(t)+Az(t) = 0, z(0) = z0 ∈ H1, ż(0) = z1 ∈ H0, y(t) = Cz(t)(7)

ζ̈(t)+A′ζ(t) = Bu(t), ζ(0) = ζ0 ∈ H0, ζ̇(0) = ζ1 ∈ H−1, u ∈ L2loc(R; Y)which is the model wave equation.If we now define the variables x, ξ and the spaces X, X ′ asfollows,

x(t) = (z(t), ż(t)) ∈ X = H1×H0, ξ(t) =(ζ(t), ζ̇(t)

)∈ X ′ = H0×H−1

Then we can relatively easily apply the first-order results tothis system...


Semi-discretization

Let A be a particular self-adjoint, positive, unbounded operatorwith bounded inverse. Let Hs be the Hilbert space D(As/2) withnorm ‖x‖s =

∥∥As/2x∥∥0 . Consider the observation systemẋ(t)− iAx(t) = 0, x(0) = x0 ∈ X, y(t) = Cx(t) (8)

and resolvent condition

∀x ∈ D(A), λ ∈ σ(A), ‖Cx‖2 ≤ L(λ) ‖(A− λ)x‖20 + l(λ) ‖x‖20

Introduce a finite-dimensional approximation of this systemwhich encompasses a wide range of numerical schemes wherethe state space H0 is a space of functions on the continuum Rddiscretized on non-uniform meshes.


Semi-discretization (contd.)

Let (Vh)h>0 be a family of finite-dimensional vector spaces withinjections Jh : Vh → H0.Let πh be the H1-orthogonal projection from H1 ontoHh .= JhVh. The only approximation assumption we make is:∥∥(I − πh)A−1/2∥∥1 = O(h), i.e. ∃C0 > 0,

‖x− πhx‖1 ≤ c0h ‖x‖2 , x ∈ D(A), h > 0, (9)

or∥∥A1/2x− A1/2πhx∥∥0 ≤ c0h ‖Ax‖0 since ‖x‖s = ∥∥As/2x∥∥0 .

This assumption is satisfied, for example, with a P1 Lagrangefinite element on a conformal mesh.



We can then state a simplified version of the Aubin-Nitsche Lemma.

TheoremThe approximation assumption (9) (in H1 ) implies theapproximation in H0 :

∥∥(I − πh)A−1/2∥∥L(H0) = O(h), and∥∥(I − πh)A−1∥∥L(H0) = O(h2). More precisely, (9) implies, with thesame constant c0,

‖x− πhx‖0 ≤ c0h ‖x‖1 , x ∈ H1, h > 0,

‖x− πhx‖0 ≤ c20h2 ‖x‖2 , x ∈ H2, h > 0.



By a classical Galerkin approximation, we obtain the semi-discretesystem for the generator Gh

.=(√

AJh)∗(√

AJh)

: Vh → Vh,

v̇h(t)− iGhvh(t) = 0, vh(0) = vh0 ∈ Hh, yh(t) = Chvh(t), (10)

where Ch = CJh, C ∈ L(H1; Y). We consider uniform resolventconditions for the semi-discrete system (10) restricted to the filteredspace 1Gh


TheoremAssume that Ch = CJh, C ∈ L(H1; Y) and the approximationassumption (9) are valid. If the continuous first-order system (8) isadmissible (exactly observable) then for all η > 0 (small enough),there exists T > 0 such that the semi-discrete system (10) restrictedto the filtered space 1Gh 0 small enough, for all T > 0, the semi-discretesystem (10) restricted to the filtered space 1Gh

Bi-Grid HUM

Finite differences or finite elements (in space)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.


HUM simulations

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


Correspondence with the filtered space

The key idea of this two-grid filtering mechanism consists of usingtwo grids: one, the computational one in which the discrete waveequations are solved, with step size h and a coarser one of size 2h.

On the fine grid, the eigenvalues satisfy the sharp upper boundλ ≤ 4/h2. (1-D)The coarse grid “selects” half of the eigenvalues, the onescorresponding toλ ≤ 2/h2. (1-D)This indicates that on the fine grid the solutions obtained onthe coarse grid will behave as filtered solutions.


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

or:0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1


Contents







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

Convergence of the FEM depends on

mesh size, hmesh quality,Q = 4

√3A

l21+l22+l

23

Al1

l2

l3

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 − u0h∥∥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);L1-norms... (see A., Lebeau. ESAIM COCV’98).


Numerical results: quality

We compare the controllability error for:finite element, unstructured meshes with increasing quality(0.65 < Q < 0.95),a finite difference mesh (Q = 0.8667) made of isoceles tiangles.


Numerical results: quality (contd.)

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)


Numerical results: quality (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)

O’s and X’s represent the “classical” finite-difference method:red for original mesh, black for refined mesh (2x)


Contents







Convergence of 2 ideas

HUM: Controllability of the wave equation

Imaging algorithms for small-volume imperfections, based onFourier techniques.

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).


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 ∈ H10(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\Γ̄c × (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

∑mj=1

(γ0γj− 1)

e2iη·zj[Mj(η) · η − |η|2 |Bj|

]+ o(αd) .= Λα(η),

(11)

where θη is the unique solution to the ODE (12){∂tθη − θη = ei

√γ0|η|t∂t


)for x ∈ Γc, t ∈ (0,T),

θη(x, 0) = ∂tθη(x,T) = 0 for x ∈ Γc,(12)

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 (12) 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

∑mj=1 Cj(η)e

2iη·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

Reconstruction AlgorithmSample values of Λα(η) at some discrete set of points and thencalculate the corresponding discrete inverse Fourier transform.After a rescaling by −12 , 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 (56). These polarization tensors give ideason the orientation and relative size of the imperfections.


Numerical results

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

Λα(η) = αd

m∑j=1

(γ0γj− 1)

e2iη·zj[Mj(η) · η − |η|2

∣∣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.


Numerical results


Λα(η) = αd

m∑j=1

(γ0γj− 1)


∣∣Bj∣∣] , d = 2,3Numerous numerical issues arise:

resolution/cost -

no free lunch... use parallel processing.

aliasing/rippling -



use thresholding.


Numerical results


Λα(η) = αd

m∑j=1

(γ0γj− 1)



resolution/cost - no free lunch... use parallel processing.aliasing/rippling -



use thresholding.


Numerical results


Λα(η) = αd

m∑j=1

(γ0γj− 1)



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

use thresholding.


Numerical results


Λα(η) = αd

m∑j=1

(γ0γj− 1)



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.

IntroductionProcédure d’identification

Implémentation de la localisation dynamiqueDétection numérique en 2D et 3D

Conclusion et perspectives

Tests de calibrationParamètres d’entréeRésultats numériques 2DRésultats numériques 3DImplémentation paralléle

Test numérique 2D : sans seuillage ni fenêtrage

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 Détection numérique de petites imperfections en 2D et 3D 39 / 54





Test numérique 2D

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


At left: without; at right: with thresholding (see [Asch, 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 numérique 2D : sans seuillage ni fenêtrage

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 numérique 2D

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


At left: without; at right: with thresholding (see [Asch, Mefire. IJNAM, 6 (1),2009.])Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 59 / 74

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.

[Asch, Darbas, Duval. ESAIM-COCV, 2010.]Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 60 / 74

Limited-view data in 2- and 3D

Repeat above computations, but compute measurements on apart of the boundary 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.


Contents







Time domain imaging of point sources

In our recent paper16 we studied limited-view imaging ofpoint-sources in a (quite) realistic setting (eg.photo-acoustics...).We seek to reconstruct the initial conditions, p0 and p1 from(partial) measurements of ∂p/∂n on the boundary, where psatisfies the acoustic wave equation

1c2∂2p∂t2−∆p) = 0 in Ω× (0,T),

p = 0 on ∂Ω× (0,T),p|t=0 = p0 ,

∂p∂t

∣∣∣∣t=0

= p1 in Ω.

16Ammari, Asch, Jugnon, Kang. SIAM J. Imaging Sci. 4 (4), 2011.Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 63 / 74

Time domain imaging (contd.)

From the boundary measurements, we would like to obtaininformation of the type

M(v0, v1) =∫

Ωp0(x)v1(x)− p1(x)v0(x) dx

For example, if v0 = 0, v1 = eik·x, then

M(0, eik·x) =∫

Ωp0(x)eik·x dx = F [p0](k)

If we can find a boundary control, gv0,v1(x, t), x ∈ Γ ⊂ ∂Ω fromthe wave equation for v with initial conditions v0, v1 such thatv(x,T) = 0 and ∂v/∂t(x,T) = 0, then multiplying thep-equation by v and integrating, we have∫ T

0

∫Γ

gv0,v1(x, t)∂v∂n

(x, t) ds dt = M(v0, v1).



We apply this to the case of point sources, with

p0(x) = 0, p1(x) = δz(x)

and we seek the position(s) z.For each imaging algorithm:

Choose v0,compute the control gv0(x, t), x ∈ Γ ⊂ ∂Ωand calculate the measurement functional

N(v0) =∫

Ω

v0(x)δz(x) dx = v0(z)

for some well-chosen parametrizations of v0.



The imaging algorithms used were:Arrival time and delay: v0(x; xr) = |x− xr|Back propagation: v0(x; θ) = eiωθ·xKirchoff: v0(x;ω, xr) = eiωθ|x−xr|MUSIC: v0(x; θ, θ′) = eiω(θ+θ

′)·x

Then we define an imaging functional for each algorithm,whose maximum (peaks) passes through the point sourceThese algorithms are computationally inexpensive:

AT costs ncenterBP costs nθMUSIC costs n2θKirchoff costs nω × nreceiver


Time domain imaging: resultsIntroduction physique

Imagerie par onde transitoire avec vue limitéeRésultats

Conlusion

−1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

virtual receiversreal acoustic receiverstrue position of the sourceintersections of the circles

0.099 0.1 0.101 0.102 0.103 0.104 0.1050.238

0.2385

0.239

0.2395

0.24

0.2405

0.241

0.2415

0.242

0.2425

0.243

Séminaire des doctorantsMark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, 2012 67 / 74

Time domain imaging: results

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

6

7

8

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

6

7

8

Figure 10: Kirchhoff results for the geometry sqReg2 with several inclusions.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 11: Back-propagation results for the geometry sqReg2 with several inclusions.

References

[1] H. Ammari, An inverse initial boundary value problem for the wave equation in thepresence of imperfections of small volume, SIAM J. Control Optim., 41 (2002), 1194–1211.

[2] H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Math-ematics and Applications, Volume 62, Springer-Verlag, Berlin, 2008.

[3] H. Ammari, E. Bossy, V. Jugnon, and H. Kang, Mathematical modelling in photo-acoustic imaging of small absorbers, SIAM Rev., 52 (2010), 677–695.

22



−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 12: MUSIC results for the geometry sqReg2 with several inclusions.

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.5

1

1.5

2

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.5

1

1.5

2

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 13: Kirchhoff results for the geometry sqReg2 with perturbed boundary (from left-to-right,� = 0.01, 0.025 and 0.05). The (black/white) x denotes the (numerical/theoretical)center of the source.

[4] H. Ammari, E. Bretin, V. Jugnon, and A. Wahab, Photoacoustic imaging for attenuat-ing acoustic media, in Mathematical Modeling in Biomedical Imaging II, Lecture Notesin Mathematics, Springer-Verlag, Berlin, to appear.

[5] H. Ammari, Y. Capdeboscq, H. Kang, and A. Kozhemyak, Mathematical models andreconstruction methods in magneto-acoustic imaging, Euro. J. Appl. Math., 20 (2009),303–317.

[6] H. Ammari, P. Garapon, L. Guadarrama Bustos, and H. Kang, Transient anomalyimaging by the acoustic radiation force, J. Diff. Equat., 249 (2010), 1579–1595.

23



−0.5 0 0.5−0.5

0

0.5

0

0.5

1

1.5

2

2.5

−0.5 0 0.5−0.5

0

0.5

0

0.5

1

1.5

2

2.5

−0.5 0 0.5−0.5

0

0.5

0

0.5

1

1.5

2

−0.5 0 0.5−0.5

0

0.5

0

0.5

1

1.5

2

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

0

0.5

1

1.5

2

2.5

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

0

0.5

1

1.5

2

2.5

Results in the full view setting Results in the partial view setting

Figure 4: Kirchhoff results for the geometries of Table 1 - from top to bottom: squareReg0,squareReg2, disc. The (black/white) x denotes the (numerical/theoretical) center of thesource.

We test our algorithms by perturbing the boundary nodes outwards

xi,perturbed = xi + �Unxi ,

where � is the magnitude of perturbation, U is a uniform random variable in [0 1] and nxiis the outward normal at the point xi. We simulate measurements on the perturbed mesh,which is then supposed unknown since we compute the geometric control on the unperturbedmesh.

To illustrate the results, we use squareReg2 with three levels of perturbation, � = 0.01,

17


Contents







Conclusions

Convergence of bi-grid algorithm is now completely proven.

Bi-grid performs well on unstructured meshes.Optimizing the mesh quality improves the convergence.Bi-grid is robust for applications in inverse imaging problemswith limited-view measurements.


Conclusions

Convergence of bi-grid algorithm is now completely proven.Bi-grid performs well on unstructured meshes.

Optimizing the mesh quality improves the convergence.Bi-grid is robust for applications in inverse imaging problemswith limited-view measurements.


Conclusions

Convergence of bi-grid algorithm is now completely proven.Bi-grid performs well on unstructured meshes.Optimizing the mesh quality improves the convergence.

Bi-grid is robust for applications in inverse imaging problemswith limited-view measurements.


Conclusions

Convergence of bi-grid algorithm is now completely proven.Bi-grid performs well on unstructured meshes.Optimizing the mesh quality improves the convergence.Bi-grid is robust for applications in inverse imaging problemswith limited-view measurements.


Perspectives

Robust numerical codes are “available” for wave control...


Further reading I

Mesh quality:Jonathan Shewchuck - consulthttp://www.cs.cmu.edu/~jrs/

FEM convergence:Many books - Ciarlet, Ern, Brenner, Zienckiwiecz, Babuska,Monk (Maxwell...), etc.

E. Zuazua.HUM - Consult http://www.bcamath.org/zuazua.

S. Ervedoza.Fine observability estimates for FEM discretizations - Consulthttp://www.math.univ-toulouse.fr/~ervedoza/.


HUMFormulationAnalysis of the non-convergence

Bi-Grid HUMConvergence of Bi-Grid HUM

Numerical analysis & results for HUMMeshesConvergence

ApplicationsDynamic detection of small imperfectionsDetection of point sources by imaging techniques

Conclusions and perspectivesAppendix

convergence issues and some applications mark asch · mark asch (gt contrôle, ljll, parris-vi)...

Documents