Numerical control of waves

Transcription

1 Numerical control of waves Convergence 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, / 74

2 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

3 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

4 Formulation of exact controllability The wave equation with Dirichlet control: ( 2 t c 2 ) u = 0 in Ω (0, T), u t=0 = u 0, t u t=0 = u1 in Ω, (1) u = { g on Γc (0, T), 0 on Γ \ Γ c (0, T), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

5 Formulation of exact controllability The wave equation with Dirichlet control: ( 2 t c 2 ) u = 0 in Ω (0, T), u t=0 = u 0, t u t=0 = u1 in Ω, (1) u = { g on Γc (0, T), 0 on Γ \ Γ c (0, T), 1 Geometry and control time: T Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

6 Formulation of exact controllability The wave equation with Dirichlet control: ( 2 t c 2 ) u = 0 in Ω (0, T), u t=0 = u 0, t u t=0 = u1 in Ω, (1) u = { g on Γc (0, T), 0 on Γ \ Γ c (0, T), 1 Geometry and control time: T 2 Initial excitation: {u 0, u 1 } Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

7 Formulation of exact controllability The wave equation with Dirichlet control: ( 2 t c 2 ) u = 0 in Ω (0, T), u t=0 = u 0, t u t=0 = u1 in Ω, (1) u = { g on Γc (0, T), 0 on Γ \ Γ c (0, T), 1 Geometry and control time: T 2 Initial excitation: {u 0, u 1 } 3 Control function: g Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

8 Existence and uniqueness of g by the HUM The problem of exact boundary controllability: "Given T, u 0, u 1, find a control g such that the solution of (1) satisfies, at time T, u(x, T) = t u(x, T) = 0 in Ω" Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

9 Existence and uniqueness of g by the HUM The problem of exact boundary controllability: "Given T, u 0, u 1, find a control g such that the solution of (1) satisfies, at time T, Theorem [Lions 88] u(x, T) = t u(x, T) = 0 in Ω" For T large enough, there exists a unique control g that minimizes the L 2 (Γ c (0, T)) norm. This control can be constructed by the Hilbert Uniqueness Method. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

10 Existence and uniqueness of g by the HUM The problem of exact boundary controllability: "Given T, u 0, u 1, find a control g such that the solution of (1) satisfies, at time T, Theorem [Lions 88] u(x, T) = t u(x, T) = 0 in Ω" For T large enough, there exists a unique control g that minimizes the L 2 (Γ c (0, T)) norm. This control can be constructed by the Hilbert 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 of the forward equation and f is the final condition of the backward equation. Finally, compute g by a conjugate gradient algorithm. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

11 Geometric controllability GCC [Bardos, Lebeau, Rauch - SICON 92] Every ray of geometrical optics, starting at any point x Ω, at time t = 0, hits Γ c before time T at a nondiffractive point. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

12 Geometric controllability GCC [Bardos, Lebeau, Rauch - SICON 92] Every ray of geometrical optics, starting at any point x Ω, at time t = 0, hits Γ c before time T at a nondiffractive point. No glancing nor trapped rays. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

13 Geometric controllability GCC [Bardos, Lebeau, Rauch - SICON 92] Every ray of geometrical optics, starting at any point x Ω, at time t = 0, hits Γ c before time T at a nondiffractive point. No glancing nor trapped rays. Uncontrollable and controllable geometries Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

14 So where s the catch? Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

15 So where s the catch? Numerically, the HUM produces an ill-posed problem... Ill-posedness stability + consistency / convergence Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

16 So where s the catch? Numerically, the HUM produces an ill-posed problem... Ill-posedness stability + consistency / convergence Numerical PDE (h, t) (h, t)! 0 PDE Discrete exact Controllability Exact Controllability Discrete control Convergence? Control g h (h, t) (h, t)! 0 g Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

17 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 these methods? Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

18 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

19 Observability Controllability-Observability Controllability of the direct wave equation is equivalent to observability of the adjoint wave equation. So we prefer (for technical facility) to try and prove the observability condition which says that the energy of the system is bounded by the trace of the normal derivative on the boundary. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

20 High frequency modes In general, any discrete dynamics associated to the wave equation generates spurious high-frequency oscillations that do not exist at the continuous level. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

21 High frequency modes In general, any discrete dynamics associated to the wave equation generates spurious high-frequency oscillations that do not exist at the continuous level. Moreover, a numerical dispersion phenomenon appears and the velocity of propagation of some high frequency numerical waves may possibly converge to zero when the mesh size h does. This is a localization phenomenon... In this case, the controllability/observability property for the discrete system will not be uniform, as h 0, for a fixed time T and, consequently, there will be initial data (even very regular ones) for which the corresponding controls of the discrete model will diverge in the L 2 -norm as h tends to zero. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

22 Theoretical analysis Infante, Zuazua, (detailed analysis of the 1-D case) and Zuazua, (2-D case), considered a semi-discretization of the wave equation with the classical finite differences or finite element method: 1 J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the one-dimensional wave equation, M2AN 33 (2) (1999), E. Zuazua, Propagation, Observation, Control of Waves Approximated by Finite Difference Methods, SIAM Review, 47 (2) (2005), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

23 Theoretical analysis Infante, Zuazua, (detailed analysis of the 1-D case) and Zuazua, (2-D case), considered a semi-discretization of the wave equation with the classical finite differences or finite element method: after filtering the high frequency modes, a uniform observability inequality for the adjoint system holds - this is equivalent to the uniform controllability of the projection of the solutions over the space generated by the remaining eigenmodes; 1 J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the one-dimensional wave equation, M2AN 33 (2) (1999), E. Zuazua, Propagation, Observation, Control of Waves Approximated by Finite Difference Methods, SIAM Review, 47 (2) (2005), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

24 Theoretical analysis Infante, Zuazua, (detailed analysis of the 1-D case) and Zuazua, (2-D case), considered a semi-discretization of the wave equation with the classical finite differences or finite element method: after filtering the high frequency modes, a uniform observability inequality for the adjoint system holds - this is equivalent to the uniform controllability of the projection of the solutions over the space generated by the remaining eigenmodes; the dimension of this space tends to infinity as the step size h goes to zero and, in the limit, we would obtain the control of the continuous system - however, in practice these projection methods are not very efficient. 1 J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the one-dimensional wave equation, M2AN 33 (2) (1999), E. Zuazua, Propagation, Observation, Control of Waves Approximated by Finite Difference Methods, SIAM Review, 47 (2) (2005), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

25 Theoretical analysis (contd.) In Micu, the problem with finite difference approximation was considered again. It was proved that, if the high frequency modes of the discrete initial data are filtered out in an appropriate manner (or if the initial data are sufficiently regular), there are controls of the semi-discrete model which converge to a control of the continuous wave equation. 3 S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik 91 (4) (2002), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

26 Theoretical analysis (contd.) In Micu, the problem with finite difference approximation was considered again. It was proved that, if the high frequency modes of the discrete initial data are filtered out in an appropriate manner (or if the initial data are sufficiently regular), there are controls of the semi-discrete model which converge to a control of the continuous wave equation. This is one of the ways of taking care of the spurious high frequency oscillations that the numerical method introduces. 3 S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik 91 (4) (2002), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

27 Theoretical analysis (contd.) In Micu, the problem with finite difference approximation was considered again. It was proved that, if the high frequency modes of the discrete initial data are filtered out in an appropriate manner (or if the initial data are sufficiently regular), there are controls of the semi-discrete model which converge to a control of the continuous wave equation. This is one of the ways of taking care of the spurious high frequency oscillations that the numerical method introduces. Note that in this case the uniform controllability of the entire discrete solutions is ensured and not only that of the projections, as in Infante, Zuazua. 3 S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik 91 (4) (2002), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

28 Theoretical analysis (contd.) In Micu, the problem with finite difference approximation was considered again. It was proved that, if the high frequency modes of the discrete initial data are filtered out in an appropriate manner (or if the initial data are sufficiently regular), there are controls of the semi-discrete model which converge to a control of the continuous wave equation. This is one of the ways of taking care of the spurious high frequency oscillations that the numerical method introduces. Note that in this case the uniform controllability of the entire discrete solutions is ensured and not only that of the projections, as in Infante, Zuazua. Moreover, it was also shown that the norm of the discrete HUM controls may increase exponentially with the number of points in the mesh if no filtering is applied. 3 S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik 91 (4) (2002), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

29 Other numerical methods From a numerical point of view, several techniques have been proposed as possible cures of the high frequency spurious oscillations. In [Glowinski, Li, Lions, 1990] 4 a Tychonoff regularization procedure was successfully implemented. Roughly speaking, this method introduces an additional control, tending to zero with the mesh size, but acting on the interior of the domain. Another proposed numerical technique is the mixed finite element method (see [Glowinski, Kinton, Wheeler, 1989] 5 ). 4 R. Glowinski, C. H. Li, J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods, Japan J. Math., 68 (7), 1-76, Glowinski R., Kinton W. and Wheeler M. F., A mixed finite element formulation for the boundary controllability of the wave equation, Int. J. Numer. Methods Eng. 27(3), (1989), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

30 Other numerical methods (contd.) In [Castro, Micu, Münch, 2008] 6 a mixed finite element method is considered for 2-D wave equations. The main advantage of this method is that it is uniformly controllable as the discretization parameter h goes to zero and allows to construct a convergent sequence of approximate controls (as h 0) without filtering. It consists of a different space discretization scheme of the wave equation derived from a mixed finite element method, which is based on different discretizations for the position and velocity. More precisely, while classical first order splines are used for the former, discontinuous elements approximate the latter. This method is different to the one used in [Glowinski, Kinton, Wheeler] where u and u are approximated in different finite dimensional spaces. The major disadvantage of the approach is a stricter stability condition: t h 2 instead of t h for the other methods. 6 C. Castro, S. Micu and A. Münch, Numerical approximation of the boundary control of the wave equation in a square with a mixed finite element method, IMA J. Numerical Analysis 28 (3), (2008) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

31 Other numerical methods (contd.) Uniformly controllable, implicit finite difference schemes have been analyzed in: [Münch, 2005] 7 in the 1D case [Asch, Münch, 2009] 8 in the 2D case Major disadvantage: computational complexity (implicit, high-order) 7 Münch A., A uniformly controllable and implicit scheme for the 1-D wave equation, M2AN, 39(2), (2005). 8 Asch M., Münch A., Uniformly Controllable Schemes for the Wave Equation on the Unit Square. J. Optimization Theory and Applications. 143, 3, , Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

32 Summary: Convergence What is known? 1D 2D in a periodic region finite differences in space and time structured/uniform finite-element meshes What is (was) still to be done? General, unstructured finite-element meshes on arbitrary geometries. Two possible approaches: analytical approach based on observability estimations - this question was open until November ; numerical analysis approach based on finite-element convergence estimations - see below. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

33 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

34 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

35 The original idea (Glowinski, 1992) The bi-grid method, first suggested in Glowinski, uses two (space) grids, a coarse and a fine grid. The main idea is to consider a space discretization scheme of the wave equations on the fine grid but taking the initial (or final) data on the coarse one. In this way, the high frequencies associated to the fine grid are eliminated. The basic idea is a small change in the definition of the controllability operator: the input is posed on the coarse grid, then interpolated onto the fine grid, the usual Λ T -mapping (composition of direct- and adjoint-wave operatiors) is done on the fine grid, and the end-result is finally restricted onto the coarse grid again. 9 Roland Glowinski. Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys., 103(2): , Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

36 How does it work? The bi-grid method attenuates the short wavelength components of the initial data, which are responsible for the spurious high frequency oscillations. The justification for the method is, roughly, that the high-frequency components on the fine grid are minimal since the state on the fine grid comes from a state on the coarse grid. The method was used in Asch and Lebeau (1998), 10 where different numerical experiments were conducted for the wave equation in two dimensions, using a finite difference discretization in both time and space, the CG algorithm and the bi-grid method. 10 Asch M. and Lebeau G., Geometrical aspects of exact boundary controllability for the wave equation: a numerical study. ESAIM Control, Optimisation and Calculus of Variations, 3: , Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

37 Theoretical analysis After work on the 1-D case, it was proved, in the 2-D semi-discrete case, that the bi-grid method actually leads to uniform observability, but with a minimal control time which is twice the correct time (see Negreanu and Zuazua (2004) 11 ). This was improved by Loreti, Mehrenberger (2008), 12 who obtained the sharp control/observation time, T = 2 2 on the unit square. These proofs were restricted to uniform meshes (finite difference or finite element). 11 Mihaela Negreanu and Enrique Zuazua. Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Acad. Sci. Paris, 338(5), P. Loreti and M. Mehrenberger. An Ingham type proof for a two-grid observability theorem. ESAIM: COCV, 14(3), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

38 Theoretical analysis: eg. 1-D theorem Consider the following finite difference approximation of the 1D wave equation,u j 1 [ ] uj+1 2u h 2 j + u j 1 = 0, 0 < t < T, j = 1,..., N, u j (t) = 0, j = 0, N + 1, 0 < t < T, u j (0) = u 0 j, u j (0) = u 1 j, j = 1,..., N. Theorem For N N an odd integer and T > h, for any initial data pair (u 0 h, u1 h ) V h, the solution u h of the semi-discrete system satisfies, 2 T E h (0) T (2 u N 2 dt h) h 0 Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

39 Theoretical analysis: the way to go... Zuazua gave away the solution in the conclusion of Ignat and Zuazua (2009) 13 : 13 L. Ignat and E. Zuazua. Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

40 Theoretical analysis: the way to go... Zuazua gave away the solution in the conclusion of Ignat and Zuazua (2009) 13 : It would be interesting to see if these spectral methods can be adapted in order to guarantee uniform observability results for numerical methods based on the two-grid method. 13 L. Ignat and E. Zuazua. Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

41 Theoretical analysis: the way to go... Zuazua gave away the solution in the conclusion of Ignat and Zuazua (2009) 13 : It would be interesting to see if these spectral methods can be adapted in order to guarantee uniform observability results for numerical methods based on the two-grid method. He was referring to work by Miller, Tucsnak, Russel and Weiss - see below. 13 L. Ignat and E. Zuazua. Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

42 Theoretical analysis: spectral method Based on spectral characterizations of admissibility and observability for abstract systems. Original work done by Weiss (1989), Russel and Weiss (1994), Miller (2005), Tucsnak and Weiss (2010). 14 S. Ervedoza. Spectral conditions for admissibility and observability of wave systems, Num. Math., 113(3), L. Miller. Resolvent conditions for the control of unitary grooups and their approximations. J. Spectral Th. 2, 1, Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

43 Theoretical analysis: spectral method Based on spectral characterizations of admissibility and observability 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 there were some gaps. 14 S. Ervedoza. Spectral conditions for admissibility and observability of wave systems, Num. Math., 113(3), L. Miller. Resolvent conditions for the control of unitary grooups and their approximations. J. Spectral Th. 2, 1, Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

44 Theoretical analysis: spectral method Based on spectral characterizations of admissibility and observability 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 there were some gaps. Final improvements made by Miller (2012) S. Ervedoza. Spectral conditions for admissibility and observability of wave systems, Num. Math., 113(3), L. Miller. Resolvent conditions for the control of unitary grooups and their approximations. J. Spectral Th. 2, 1, Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

45 Theoretical analysis: spectral method Based on spectral characterizations of admissibility and observability 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 there were some gaps. Final improvements made by Miller (2012) 15. Conclusion Observability results are now valid for any conservative, fully discrete (space-time) finite element approximation of the linear wave equation. 14 S. Ervedoza. Spectral conditions for admissibility and observability of wave systems, Num. Math., 113(3), L. Miller. Resolvent conditions for the control of unitary grooups and their approximations. J. Spectral Th. 2, 1, Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

46 Spectral method Recall definition of filtering: restrict the semi-discretized equation to modes with eigenvalues lower than η/h σ for some positive η and σ. Ignat, Liviu Zuazua showed that σ = 2 is optimal for the boundary observation of 1-D wave equations on uniform meshes Ervedoza generalized to non-uniform meshes (but not for case of boundary control), with σ = 2/3 Miller improved this to σ = 1 for interior observability and to σ = 2/3 for boundary observability (4/3 and 2/3 for 2nd order systems). In addition, he deduced from the uniform exact observability of the filtered approximations that the minimal control provided by the Hilbert Uniqueness Method is the limit of the minimal controls for the filtered approximations. Reaches the optimal value, σ = 2, for the simplest 1-D case. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

47 Resolvent condition Let X, Y be Hilbert spaces, A : D(A) X be a self-adjoint operator ia generates a strongly continuous group ( e ita) of unitary t R operators on X X 1 denote D(A) with the norm x 1 = (A β)x for β / σ(a), the spectrum of A C L(X 1, Y) be the observation operator, B L(Y, X 1 ) be the control operator For simplicity, we consider the first-order, dual control observation system with output function y and input function u ẋ(t) iax(t) = 0, x(0) = x 0 X, y(t) = Cx(t) (2) ξ(t) ia ξ(t) = Bu(t), ξ(0) = ξ 0 X, u L 2 loc (R; Y ) (3) and these equations have unique solutions x C(R, X) and ξ C(R, X ) given by x(t) = e ita x 0, ξ(t) = e ita ξ Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74 t e i(t s)a Bu(s) ds

48 Resolvent condition (contd.) Definition The system (2) is admissible if for some time T > 0 there is an admissibility cost K T such that T 0 Ce ita x 0 2 dt KT x 0 2 x 0 D(A). The system (2) is exactly observable in time T at cost κ T if the following observability inequality holds: T x 0 2 κ T y(t) 2 dt, x 0 D(A). (4) 0 The system (3) is exactly controlable in time T at cost κ T if for all ξ 0 X there is a u L 2 (R; Y ) such that u(t) = 0 for t / [0, T], ξ(t) = 0 and T 0 u(t) 2 dt κ T ξ 0 2. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

49 Resolvent condition (contd.) 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 These conditions are reminiscent of relative boundedness of C with respect to A and resolvent estimates for A (6) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

50 Resolvent condition (contd.) The following theorems say that the resolvent conditions are necessary and sufficient for admissibility and exact controllability respectively: Theorem The system (2) is admissible if and only if the resolvent condition (5) holds. Theorem Let the system (2) be admissible. Then it is exactly observable if and only if the resolvent condition (6) holds. In fact, (4) implies (6) with M = T 2 κ T K T and m = 2Tκ T. Conversely, (6) implies (4) for all T > π M with κ T = 2mT/(T 2 Mπ 2 ). Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

51 Resolvent condition (2nd order system) The second-order, boundary, dual control-observation system with output function y and input function u is: ζ(t) = Bu(t), ζ(0) = ζ 0 H 0, = ζ1 H 1, u L 2 loc (R; Y z(t)+az(t) = 0, z(0) = z 0 H 1, ż(0) = z 1 H 0, y(t) = Cz(t) ζ(t)+a ζ(0) (7) which is the model wave equation. If we now define the variables x, ξ and the spaces X, X as follows, ( x(t) = (z(t), ż(t)) X = H 1 H 0, ξ(t) = ζ(t), ζ(t) ) X = H 0 H 1 Then we can relatively easily apply the first-order results to this system... Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

52 Semi-discretization Let A be a particular self-adjoint, positive, unbounded operator with bounded inverse. Let H s be the Hilbert space D(A s/2 ) with norm x s = A s/2 x 0. Consider the observation system ẋ(t) iax(t) = 0, x(0) = x 0 X, y(t) = Cx(t) (8) and resolvent condition x D(A), λ σ(a), Cx 2 L(λ) (A λ)x l(λ) x 2 0 Introduce a finite-dimensional approximation of this system which encompasses a wide range of numerical schemes where the state space H 0 is a space of functions on the continuum R d discretized on non-uniform meshes. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

53 Semi-discretization (contd.) Let (V h ) h>0 be a family of finite-dimensional vector spaces with injections J h : V h H 0. Let π h be the H 1 -orthogonal projection from H 1 onto H h. = J h V h. The only approximation assumption we make is: (I π h )A 1/2 1 = O(h), i.e. C 0 > 0, x π h x 1 c 0 h x 2, x D(A), h > 0, (9) or A 1/2 x A 1/2 π h x 0 c 0 h Ax 0 since x s = A s/2 x 0. This assumption is satisfied, for example, with a P 1 Lagrange finite element on a conformal mesh. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

54 Semi-discretization (contd.) We can then state a simplified version of the Aubin-Nitsche Lemma. Theorem The approximation assumption (9) (in H 1 ) implies the approximation in H 0 : (I π h )A 1/2 L(H0 = O(h), and ) (I π h )A 1 L(H0 ) = O(h2 ). More precisely, (9) implies, with the same constant c 0, x π h x 0 c 0 h x 1, x H 1, h > 0, x π h x 0 c 2 0h 2 x 2, x H 2, h > 0. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

55 Semi-discretization (contd.) By a classical Galerkin approximation, we obtain the semi-discrete. ( ) ( ) system for the generator G h = AJh AJh : V h V h, v h (t) ig h v h (t) = 0, v h (0) = v h 0 Hh, y h (t) = C h v h (t), (10) where C h = CJ h, C L(H 1 ; Y). We consider uniform resolvent conditions for the semi-discrete system (10) restricted to the filtered space 1 Gh <η/h σvh, where η and σ are positive filtering parameters: C h v 2 L s λ s (G h λ)v 2 0 +l s v 2 0, v 1 G h <η/h σvh, η/h σ λ inf A v 2 0 M s λ s (G h λ)v 2 0 +m s C h v 2 0, v 1 G h <η/h σvh, η/h σ λ inf A Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

56 Semi-discretization (contd.) Theorem Assume that C h = CJ h, C L(H 1 ; Y) and the approximation assumption (9) are valid. If the continuous first-order system (8) is admissible (exactly observable) then for all η > 0 (small enough), there exists T > 0 such that the semi-discrete system (10) restricted to the filtered space 1 Gh <η/hv h is admissible (exactly observable) in time T uniformly in h (0, η). If the second-order system (7) is admissible and exactly observable then for all η > 0 small enough, for all T > 0, the semi-discrete system (10) restricted to the filtered space 1 Gh <η/hv h is admissible and exactly observable in time T uniformly in h (0, η). The semi-discrete admissibility (observability) resolvent condition implies the continuous resolvent condition. NOTE: in the second-order, boundary control case, the filtering parameter σ = 2/3. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

57 Bi-Grid HUM Finite differences or finite elements (in space) Bi-grid filter: wave equations are solved on a fine mesh of size h residuals are computed on a coarse mesh of size 2h Conjugate gradient iteration for inversion of the HUM operator. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

58 HUM simulations full-view on square and disc, partial-view on square and disc. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

59 Correspondence with the filtered space The key idea of this two-grid filtering mechanism consists of using two grids: one, the computational one in which the discrete wave equations are solved, with step size h and a coarser one of size 2h. On the fine grid, the eigenvalues satisfy the sharp upper bound λ 4/h 2. (1-D) The coarse grid selects half of the eigenvalues, the ones corresponding toλ 2/h 2. (1-D) This indicates that on the fine grid the solutions obtained on the coarse grid will behave as filtered solutions. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

60 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

61 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

62 HUM Meshes Replace: Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

63 HUM Meshes by: or: Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

64 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

65 HUM Convergence Recall some classical FEM convergence results. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

66 HUM Convergence Recall some classical FEM convergence results. Define mesh quality and its connection with FEM error. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

67 HUM Convergence Recall some classical FEM convergence results. Define mesh quality and its connection with FEM error. How is all this related to BiGrid HUM? Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

68 HUM Convergence 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. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

69 Convergence vs. mesh quality Convergence of the FEM depends on mesh size, h mesh quality, Q = 4 3A l 2 1 +l2 2 +l2 3 l 1 A l 2 l 3 Bi-grid HUM is strongly dependent on mesh quality because of the brutal nature of the filtering, h h/2 Error norms used for convergence of the HUM: u 0 u 0 h L2 (Ω) u(t) u h (T) L2 (Ω) other possibilities: H 1 -norm of u t; L 2 -norm of g (see A., Lebeau. ESAIM COCV 98); L 1 -norms... (see A., Lebeau. ESAIM COCV 98). Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

70 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 = ) made of isoceles tiangles. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

71 Numerical results: quality (contd.) u 0 u(t) error quality (min) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

72 Numerical results: quality (contd.) u 0 u(t) error quality (min) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

73 O s and X s represent the classical finite-difference method: red for original mesh, black for refined mesh (2x) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74 Numerical results: quality (contd.) 0.06 u 0 u(t) error quality (min)

74 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

75 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

76 Convergence of 2 ideas Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

77 Convergence of 2 ideas HUM: Controllability of the wave equation Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

78 Convergence of 2 ideas HUM: Controllability of the wave equation Imaging algorithms for small-volume imperfections, based on Fourier techniques. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

79 Convergence of 2 ideas HUM: Controllability of the wave equation Imaging algorithms for small-volume imperfections, based on Fourier techniques. Result We can perform transient imaging with limited-view data, using waves as probes and employing practical imaging techniques Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

80 Convergence of 2 ideas HUM: Controllability of the wave equation Imaging algorithms for small-volume imperfections, based on Fourier techniques. Result We can perform transient imaging with limited-view data, using waves as probes and employing practical imaging techniques Key reference H. Ammari, An inverse IBVP for the wave equation in the presence of imperfections of small volume, SICON 41, 4 (2002). Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

81 Acoustic wave equation IBVP for the wave equation, in the presence of the imperfections, 2 u α t 2 (γ α u α ) = 0 in Ω (0, T), (W α ) u α = f on Ω (0, T), u α t=0 = u 0, u α t = u 1 in Ω. t=0 Define u to be the solution in the absence of any imperfections, 2 u t 2 (γ 0 u) = 0 in Ω (0, T), (W 0 ) u = f on Ω (0, T), u t=0 = u 0, u t = u 1 in Ω. t=0 Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

82 Control function Suppose that T and the part Γ c of the boundary Ω are such that they geometrically control Ω. Then, for any η R d, we can construct by the Hilbert uniqueness method (HUM), a unique g η H 1 0 (0, T; L2 (Γ)) in such a way that the unique weak solution w η of the wave equation (W c ) 2 w η t 2 (γ 0 w η ) = 0 in Ω (0, T), w η = g η in Γ c (0, T), w η = 0 in Ω\ Γ c (0, T), w η t=0 = β(x)e iη x H0 1(Ω), w η t = 0 in Ω, t=0 satisfies w η (T) = t w η (T) = 0. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

83 Key theorem (H. Ammari) Theorem Let η R d, d = 2, 3. Let u α be the unique solution to the wave equation (W α) with u 0 (x) = e iη x, u 1 (x)n = i γ 0 η e iη x, f (x, t) = e iη x i γ 0 η t. Suppose that Γ c and T geometrically control Ω; then we have T [ ( uα θη 0 Γ c n ) ( u n + tθ uα η t n )] u n = ( ) T 0 Γ c e i γ 0 η t t e i γ ( 0 η t g uα η n ) u n = α d ( ) [ ] m γ0 j=1 γ j 1 e 2iη z j M j (η) η η 2 B j + o(α d ). = Λ α(η), (11) where θ η is the unique solution to the ODE (12) { ( ) tθ η θ η = e i γ 0 η t t e i γ 0 η t g η for x Γ c, t (0, T), θ η(x, 0) = tθ η(x, T) = 0 for x Γ c, (12) with g η the boundary control in (W c), and M j the polarization tensor of B j. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

84 Idea of the proof asymptotic formula for uα ν on B j in terms of Φ ν, γ 0 γ j and u; introduce auxilliary solution, v α, of homogeneous wave equation with initial condition t v α = the asymptotic principal term; show that the asymptotic equals the boundary average of v α n g η; finally, use θ η and the Volterra equation (12) to replace v α by u α (using Taylor expansions and asymptotics in α) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

85 Asymptotic formula Λ α (η) = T 0 Γ c e i γ 0 η t ( t e i γ 0 η t ) ( g uα η n u ) n α d ( ) m γ0 j=1 γ j 1 e 2iη z j [M j (η) η η 2 ] B j. = α d m j=1 C j(η)e 2iη z j. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

86 Asymptotic formula Λ α (η) = T 0 Γ c e i γ 0 η t ( t e i γ 0 η t ) ( g uα η n u ) n α d ( ) m γ0 j=1 γ j 1 e 2iη z j [M j (η) η η 2 ] B j. = α d m j=1 C j(η)e 2iη z j. Fundamental observation: the inverse Fourier transform gives a linear combination of (derivatives of) delta functions, Λα 1 (x) α d m j=1 L ( ) j δ 2zj (x) Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

87 Fourier algorithm Reconstruction Algorithm Sample values of Λ α (η) at some discrete set of points and then calculate the corresponding discrete inverse Fourier transform. After a rescaling by 1 2, the support of this discrete inverse Fourier transform yields the location of the small imperfections B α. Once the locations are known, we may calculate the polarization tensors ( ) m M j by solving an appropriate linear j=1 system arising from (56). These polarization tensors give ideas on the orientation and relative size of the imperfections. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

88 Numerical results Objective: Recover Dirac masses situated at the points 2z j from the inverse transform of the asymptotic formula Λ α (η) = α d m j=1 ( ) γ0 1 e 2iη z j [M j (η) η η 2 ] B j, d = 2, 3 γ j Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

89 Numerical results Objective: Recover Dirac masses situated at the points 2z j from the inverse transform of the asymptotic formula Λ α (η) = α d m j=1 ( ) γ0 1 e 2iη z j [M j (η) η η 2 ] B j, d = 2, 3 γ j Numerous numerical issues arise: resolution/cost - aliasing/rippling - instabilities for large η - Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

90 Numerical results Objective: Recover Dirac masses situated at the points 2z j from the inverse transform of the asymptotic formula Λ α (η) = α d m j=1 ( ) γ0 1 e 2iη z j [M j (η) η η 2 ] B j, d = 2, 3 γ j Numerous numerical issues arise: resolution/cost - no free lunch... use parallel processing. aliasing/rippling - instabilities for large η - Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

91 Numerical results Objective: Recover Dirac masses situated at the points 2z j from the inverse transform of the asymptotic formula Λ α (η) = α d m j=1 ( ) γ0 1 e 2iη z j [M j (η) η η 2 ] B j, d = 2, 3 γ j Numerous numerical issues arise: resolution/cost - no free lunch... use parallel processing. aliasing/rippling - avoid truncation, respect periodicity of DFT. instabilities for large η - Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

92 Numerical results Objective: Recover Dirac masses situated at the points 2z j from the inverse transform of the asymptotic formula Λ α (η) = α d m j=1 ( ) γ0 1 e 2iη z j [M j (η) η η 2 ] B j, d = 2, 3 γ j 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. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

93 Example: Fourier resolution thresholding and windowing Three imperfections: z 1 = (0.63, 0.28), z 2 = (0.39, 0.66), z 3 = (0.73, 0.65), α = 0.03, γ j = 10. Sampling: N = 256, η m = 33. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

94 Conclusionetetperspectives Implémentation Implémentation paralléle paralléle mérique Example: 2DFourier : sans seuillage resolution ni fenêtrage thresholding and windowing mperfections re de Blackman : z 1 = (0.63, 0.28), z 2 = (0.39, 0.66), Seuillage Three imperfections: z 1 (0.63, 0.28), z 2 = (0.39, 0.66), 3 = (0.73, 0.65), α = 0.03 et γ j = 10 ; (η z 3 = (0.73, 0.65), α = 0.03, γ j = 10., η ) = (9, 9) chantillonnage : N e = 256 et η max = 33. Sampling: N = 256, η m = 33. tiste Détection numérique de petites imperfections en 2D et 3D 39 / 54 At left: without; at right: with thresholding (see [Asch, Mefire. IJNAM, 6 (1), Mark ste Asch (GT Contrôle, Détection LJLL, numérique Parris-VI) de petitesnumerical imperfections control enof2dwaves et 3D June 15th, / / ])

95 Example: Transient imaging by Fourier in 3D Four imperfections centered at z 1 = (0.66, 0.32, 0.47), z 2 = (0.55, 0.71, 0.39), z 3 = (0.39, 0.63, 0.31), z 4 = (0.71, 0.42, 0.74). N e = 64, η max = 40 and η = 9. Radius α = 0.01 and conductivity γ j = 10. [Asch, Darbas, Duval. ESAIM-COCV, 2010.] Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

96 Limited-view data in 2- and 3D Repeat above computations, but compute measurements on a part of the boundary that respects the geometric control condition (GCC) We obtain perfect reconstruction! Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

97 Limited-view data in 2- and 3D Repeat above computations, but compute measurements on a part of the boundary that respects the geometric control condition (GCC) We obtain perfect reconstruction! Conclusion: As long as we can accurately compute the boundary control function, g, we can reconstruct equally well with limited-view data. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

98 Contents 1 HUM Formulation Analysis of the non-convergence 2 Bi-Grid HUM Convergence of Bi-Grid HUM 3 Numerical analysis & results for HUM Meshes Convergence 4 Applications Dynamic detection of small imperfections Detection of point sources by imaging techniques 5 Conclusions and perspectives Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

99 Time domain imaging of point sources In our recent paper 16 we studied limited-view imaging of point-sources in a (quite) realistic setting (eg. photo-acoustics...). We seek to reconstruct the initial conditions, p 0 and p 1 from (partial) measurements of p/ n on the boundary, where p satisfies the acoustic wave equation 1 2 p c 2 t 2 p) = 0 in Ω (0, T), p = 0 on Ω (0, T), p t=0 = p 0, p t = p 1 in Ω. t=0 16 Ammari, Asch, Jugnon, Kang. SIAM J. Imaging Sci. 4 (4), Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

100 Time domain imaging (contd.) From the boundary measurements, we would like to obtain information of the type M(v 0, v 1 ) = p 0 (x)v 1 (x) p 1 (x)v 0 (x) dx Ω For example, if v 0 = 0, v 1 = e ik x, then M(0, e ik x ) = p 0 (x)e ik x dx = F[p 0 ](k) Ω If we can find a boundary control, g v0,v 1 (x, t), x Γ Ω from the wave equation for v with initial conditions v 0, v 1 such that v(x, T) = 0 and v/ t(x, T) = 0, then multiplying the p-equation by v and integrating, we have T 0 Γ g v0,v 1 (x, t) v n (x, t) ds dt = M(v 0, v 1 ). Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

101 Time domain imaging (contd.) We apply this to the case of point sources, with p 0 (x) = 0, p 1 (x) = δ z (x) and we seek the position(s) z. For each imaging algorithm: Choose v 0, compute the control g v0 (x, t), x Γ Ω and calculate the measurement functional N(v 0 ) = v 0 (x)δ z (x) dx = v 0 (z) for some well-chosen parametrizations of v 0. Ω Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

102 Time domain imaging (contd.) The imaging algorithms used were: Arrival time and delay: v 0 (x; x r ) = x x r Back propagation: v 0 (x; θ) = e iωθ x Kirchoff: v 0 (x; ω, x r ) = e iωθ x xr MUSIC: v 0 (x; θ, θ ) = e iω(θ+θ ) x Then we define an imaging functional for each algorithm, whose maximum (peaks) passes through the point source These algorithms are computationally inexpensive: AT costs n center BP costs n θ MUSIC costs n 2 θ Kirchoff costs n ω n receiver Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

103 Imagerie par onde transitoire avec vue limitée Résultats Time domain imaging: Conlusion results virtual receivers 1 real acoustic receivers true position of the source intersections of the circles Séminaire des doctorants Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

104 Time domain imaging: results Figure 10: Kirchhoff results for the geometry sqreg2 with several inclusions Figure 11: Back-propagation results for the geometry sqreg2 with several inclusions. References Mark Asch (GT Contrôle, LJLL, [1] H. Parris-VI) Ammari, An inversenumerical initial boundary control valueof problem wavesfor the wave equation in thejune 15th, / 74

105 Time domain imaging: results Figure 12: MUSIC results for the geometry sqreg2 with several inclusions. Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th, / 74

106 Time domain imaging: results 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 the Mark Asch (GT Contrôle, LJLL, source. Parris-VI) Numerical control of waves June 15th, / 74