Back and forth observers: application to TAT

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(14) . Haine, Ghislain Back and forth observers: application to TAT. (2017) In: Seminar of applied mathematics, 10 February. 2017 (Londres, United Kingdom). (Unpublished) .

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(24) Institut Supérieur de l’Aéronautique et de l’Espace. Back and forth observers: application to TAT. Ghislain Haine ISAE-SUPAERO. 10 February 2017. G. Haine. Seminar. Back and forth observers. 1/ 38.

(25) 1. Introduction Motivation An abstract setting, a simple idea A (very) brief history. 2. The finite dimensional setting Finite dimensional (autonomous) linear systems Come back to the intuitive definitions Backward in time. 3. Infinite dimensional systems Disasters Solutions A good case. 4. Application to TAT Modelling the problem Writing the wave system as z˙ = Az, y = Cz 3D Simulations. 5. Conclusion. G. Haine. Back and forth observers. 2/ 38.

(26) Outline. 1. Introduction Motivation An abstract setting, a simple idea A (very) brief history. 2. The finite dimensional setting. 3. Infinite dimensional systems. 4. Application to TAT. 5. Conclusion. G. Haine. Back and forth observers. 3/ 38.


(28) Motivation. Image : RECENDT. G. Haine. Back and forth observers. 5/ 38.

(29) Motivation. Image : RECENDT. The generated outward wave w satisfies:  2 d w   (x, t) = c2 (x)∆w(x, t),   dt2 w(x, 0) = w0 (x),   dw   (x, 0) = 0, dt G. Haine. ∀t ≥ 0, x ∈ R3 , ∀x ∈ R3 , ∀x ∈ R3 ,. Back and forth observers. 5/ 38.

(30) Motivation The generated outward wave w satisfies:  2   d w (x, t) = c2 (x)∆w(x, t),   dt2 w(x, 0) = w0 (x),  dw    (x, 0) = 0, dt. ∀t ≥ 0, x ∈ R3 , ∀x ∈ R3 , ∀x ∈ R3 ,. c is the known velocity of the wave,. G. Haine. Back and forth observers. 5/ 38.

(31) Motivation The generated outward wave w satisfies:  2   d w (x, t) = c2 (x)∆w(x, t),   dt2 w(x, 0) = w0 (x),  dw    (x, 0) = 0, dt. ∀t ≥ 0, x ∈ R3 , ∀x ∈ R3 , ∀x ∈ R3 ,. c is the known velocity of the wave, (w0 , 0) is the unknown containing information on the distribution of energy absorption (which is related to cell’s health).. G. Haine. Back and forth observers. 5/ 38.

(32) Motivation The generated outward wave w satisfies:  2   d w (x, t) = c2 (x)∆w(x, t),   dt2 w(x, 0) = w0 (x),  dw    (x, 0) = 0, dt. ∀t ≥ 0, x ∈ R3 , ∀x ∈ R3 , ∀x ∈ R3 ,. c is the known velocity of the wave, (w0 , 0) is the unknown containing information on the distribution of energy absorption (which is related to cell’s health). y(t, x) will be the observation of this outward wave outside the body, somewhere, during some finite time interval.. G. Haine. Back and forth observers. 5/ 38.

(33) Motivation The generated outward wave w satisfies:  2   d w (x, t) = c2 (x)∆w(x, t),   dt2 w(x, 0) = w0 (x),  dw    (x, 0) = 0, dt. ∀t ≥ 0, x ∈ R3 , ∀x ∈ R3 , ∀x ∈ R3 ,. c is the known velocity of the wave, (w0 , 0) is the unknown containing information on the distribution of energy absorption (which is related to cell’s health). y(t, x) will be the observation of this outward wave outside the body, somewhere, during some finite time interval.. Question What can be recover of w0 from the knowledge of y?. G. Haine. Back and forth observers. 5/ 38.


(35) An abstract setting, a simple idea Formally, we consider systems that can be written under the form  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ], where A and C are linear operators (think about matrices).. G. Haine. Back and forth observers. 7/ 38.

(36) An abstract setting, a simple idea Formally, we consider systems that can be written under the form  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ], where A and C are linear operators (think about matrices).. Two “intuitive definitions”: Detectability : we can estimate the state.. G. Haine. Back and forth observers. 7/ 38.

(37) An abstract setting, a simple idea Formally, we consider systems that can be written under the form  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ], where A and C are linear operators (think about matrices).. Two “intuitive definitions”: Detectability : we can estimate the state. Observability : we can distinguish two different initial states.. G. Haine. Back and forth observers. 7/ 38.

(38) An abstract setting, a simple idea The iterative algorithm is based on forward and backward observers.. •. •. z0. 0. z(τ ). τ y(t) is available only on [0, τ ].. G. Haine. Back and forth observers. 7/ 38.

(39) An abstract setting, a simple idea The iterative algorithm is based on forward and backward observers.. •. •. z0. z(τ ). z1+ •. •. z0+ G. Haine. •. τ y(t) is available only on [0, τ ]. Back and forth observers. 7/ 38.

(40) An abstract setting, a simple idea The iterative algorithm is based on forward and backward observers.. • z2+ •. •. z0. z(τ ). • z1+ •. •. z0+ G. Haine. •. τ y(t) is available only on [0, τ ]. Back and forth observers. 7/ 38.


(42) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters.. G. Haine. Back and forth observers. 9/ 38.

(43) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation.. G. Haine. Back and forth observers. 9/ 38.

(44) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation. 2009-2011: Uhlmann et al. (SIAM J. Imaging Sciences, Inverse Problems, ...) use time reversal methods for solving TAT, leading to a Neumann series expansion.. G. Haine. Back and forth observers. 9/ 38.

(45) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation. 2009-2011: Uhlmann et al. (SIAM J. Imaging Sciences, Inverse Problems, ...) use time reversal methods for solving TAT, leading to a Neumann series expansion. 2010: Ramdani, Tucsnak and Weiss (Automatica) generalized the TRF, based on the generalization of Luenberger’s observers.. G. Haine. Back and forth observers. 9/ 38.

(46) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation. 2009-2011: Uhlmann et al. (SIAM J. Imaging Sciences, Inverse Problems, ...) use time reversal methods for solving TAT, leading to a Neumann series expansion. 2010: Ramdani, Tucsnak and Weiss (Automatica) generalized the TRF, based on the generalization of Luenberger’s observers. 2014: Oksanen and Uhlmann (Math. Res. Lett.) generalized the previous results of Uhlmann et al. with uncertain wave speed.. G. Haine. Back and forth observers. 9/ 38.

(47) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation. 2009-2011: Uhlmann et al. (SIAM J. Imaging Sciences, Inverse Problems, ...) use time reversal methods for solving TAT, leading to a Neumann series expansion. 2010: Ramdani, Tucsnak and Weiss (Automatica) generalized the TRF, based on the generalization of Luenberger’s observers. 2014: Oksanen and Uhlmann (Math. Res. Lett.) generalized the previous results of Uhlmann et al. with uncertain wave speed. 2016: Chervova and Oksanen (Inverse Problems) proposed a new method that can be viewed as a modification of the back and forth nudging.. G. Haine. Back and forth observers. 9/ 38.

(48) A (very) brief history 2005: Auroux and Blum (C. R. Math. Acad. Sci. Paris) introduced the Back and Forth Nuding (BFN), based on the generalization of Kalman’s filters. 2008: Phung and Zhang (SIAM J. Appl. Math.) introduced the Time Reversal Focusing (TRF), for the Kirchhoff plate equation. 2009-2011: Uhlmann et al. (SIAM J. Imaging Sciences, Inverse Problems, ...) use time reversal methods for solving TAT, leading to a Neumann series expansion. 2010: Ramdani, Tucsnak and Weiss (Automatica) generalized the TRF, based on the generalization of Luenberger’s observers. 2014: Oksanen and Uhlmann (Math. Res. Lett.) generalized the previous results of Uhlmann et al. with uncertain wave speed. 2016: Chervova and Oksanen (Inverse Problems) proposed a new method that can be viewed as a modification of the back and forth nudging. The algorithm can lead to a Neumann series expansion, even in ill-posed cases, and only need direct wave solver in practice. G. Haine. Back and forth observers. 9/ 38.

(49) Outline. 1. Introduction. 2. The finite dimensional setting Finite dimensional (autonomous) linear systems Come back to the intuitive definitions Backward in time. 3. Infinite dimensional systems. 4. Application to TAT. 5. Conclusion. G. Haine. Back and forth observers. 10/ 38.


(51) Finite dimensional (autonomous) linear systems Let us consider the mass–spring problem:. M. G. Haine. Back and forth observers. x. 12/ 38.

(52) Finite dimensional (autonomous) linear systems Let us consider the mass–spring problem:. M. If we observe the velocity of the mass, then  x ¨(t) + x(t) = 0,     − − − − − − − − −−    x(0) = x0 , x(0) ˙ = x1 ,     − − − − − − − − −−    y(t) = x(t), ˙. G. Haine. x. ∀t ≥ 0, − − −−,. − − −−, ∀t ∈ [0, τ ].. Back and forth observers. 12/ 38.

(53) Finite dimensional (autonomous) linear systems Let us consider the mass–spring problem:. M. Let:. so that:. x x0 0 z= , z0 = ,A = x˙ x1 −1 . 1 ,C = 0 0. x. 1 ,. ∀t ≥ 0, ∀t ∈ [0, τ ],. z(t) ˙ = Az(t), y(t) = Cz(t),. with the unkown initial state z(0) = z0 .. G. Haine. Back and forth observers. 12/ 38.

(54) Finite dimensional (autonomous) linear systems Let us consider the mass–spring problem:. M. Let:. so that:. x x0 0 z= , z0 = ,A = x˙ x1 −1 . z(t) ˙ = Az(t), y(t) = Cz(t),. 1 ,C = 0 0. x. 1 ,. ∀t ≥ 0, ∀t ∈ [0, τ ],. with the unkown initial state z(0) = z0 . It is a realisation of the linear system described by the ordinary differential equation. Every n-th order linear autonomous ODE can be realized (not uniquely!).. G. Haine. Back and forth observers. 12/ 38.


(56) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. G. Haine. ∀t ≥ 0, ∀t ≥ 0,. Back and forth observers. 14/ 38.

(57) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Detectability: there exists H + such that A − H + C is Hurwitz.. G. Haine. Back and forth observers. 14/ 38.

(58) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Detectability: there exists H + such that A − H + C is Hurwitz. Indeed, if this is the case, let the Luenberger’s observer  + ∀t ≥ 0,  z˙ (t) = Az + (t) + H + (y(t) − y + (t)), z + (0) = z0+ ,  + y (t) = Cz + (t), ∀t ≥ 0.. G. Haine. Back and forth observers. 14/ 38.

(59) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Detectability: there exists H + such that A − H + C is Hurwitz. Indeed, if this is the case, let the Luenberger’s observer  + ∀t ≥ 0,  z˙ (t) = Az + (t) + H + (y(t) − y + (t)), z + (0) = z0+ ,  + y (t) = Cz + (t), ∀t ≥ 0. then e+ = z + − z satisfises (since y = Cz) e˙ + (t) = (A − H + C)e+ (t),. ∀t ≥ 0.. Detectability implies that e+ is exponentially stable. So we can estimate the state in long time. G. Haine. Back and forth observers. 14/ 38.

(60) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Detectability: there exists H + such that A − H + C is Hurwitz. Indeed, if this is the case, let the Luenberger’s observer  + ∀t ≥ 0,  z˙ (t) = Az + (t) + H + (y(t) − y + (t)), z + (0) = z0+ ,  + y (t) = Cz + (t), ∀t ≥ 0. then e+ = z + − z satisfises (since y = Cz) e˙ + (t) = (A − H + C)e+ (t),. ∀t ≥ 0.. Detectability implies that e+ is exponentially stable. So we can estimate the state in long time (but remember: in applications, we have τ < ∞ !). G. Haine. Back and forth observers. 14/ 38.

(61) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Observability: the linear map Ψ : z0 7→ y is injective.. G. Haine. Back and forth observers. 14/ 38.

(62) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Observability: the linear map Ψ : z0 7→ y is injective. Z τ kCetA z0 k2 dt = 0 ⇐⇒ z0 = 0. Equivalently, this means 0. From Cayley-Hamilton’s theorem, this is  C  CA  Rank  .  ... equivalent to . CAn−1.    = n, . This is the Kalman’s criteria for observability. It says in particular that it does not depend on the time τ > 0. G. Haine. Back and forth observers. 14/ 38.

(63) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Pole–assignement theorem ¯ ∈ Λ. Let Λ = {λi ∈ C | i = 1, · · · , n} such that λ ∈ Λ ⇔ λ If (A, C) is observable, then there exists H + such that the spectrum of A − H + C is exactly Λ.. G. Haine. Back and forth observers. 14/ 38.

(64) Come back to the intuitive definitions Let a system on Rn :  ˙ = Az(t),  z(t) z(0) = z0 ,  y(t) = Cz(t),. ∀t ≥ 0, ∀t ≥ 0,. Pole–assignement theorem ¯ ∈ Λ. Let Λ = {λi ∈ C | i = 1, · · · , n} such that λ ∈ Λ ⇔ λ If (A, C) is observable, then there exists H + such that the spectrum of A − H + C is exactly Λ. So observability (τ -independent) implies detectability with arbitrary exponential decay of the error: we can estimate the state as soon as t > 0 even if y is only known on an arbitrary small finite time interval [0, τ ]: ke+ (t)k ≤ ket(A−H. G. Haine. +. C). kke+ (0)k ≤ et max <eΛ kz0+ − z0 k,. Back and forth observers. ∀t ∈ [0, τ ].. 14/ 38.


(66) Backward in time Let 0 < τ < ∞ be fixed. We have shown that if (A, C) is detectable: kz + (τ ) − z(τ )k ≤ e−σ. +. τ. kz0+ − z0 k,. for some σ + > 0, which can be made arbitrary large if (A, C) is observable.. G. Haine. Back and forth observers. 16/ 38.

(67) Backward in time Let 0 < τ < ∞ be fixed. We have shown that if (A, C) is detectable: kz + (τ ) − z(τ )k ≤ e−σ. +. τ. kz0+ − z0 k,. for some σ + > 0, which can be made arbitrary large if (A, C) is observable. The next step is to come back in time, from the final state z + (τ ).. G. Haine. Back and forth observers. 16/ 38.

(68) Backward in time Let 0 < τ < ∞ be fixed. We have shown that if (A, C) is detectable: kz + (τ ) − z(τ )k ≤ e−σ. +. τ. kz0+ − z0 k,. for some σ + > 0, which can be made arbitrary large if (A, C) is observable. The next step is to come back in time, from the final state z + (τ ).. Backward detectability We says that (A, C) is backward detectable if there exists H − such that −A − H − C is Hurwitz.. G. Haine. Back and forth observers. 16/ 38.

(69) Backward in time Let 0 < τ < ∞ be fixed. We have shown that if (A, C) is detectable: kz + (τ ) − z(τ )k ≤ e−σ. +. τ. kz0+ − z0 k,. for some σ + > 0, which can be made arbitrary large if (A, C) is observable. The next step is to come back in time, from the final state z + (τ ).. Backward detectability We says that (A, C) is backward detectable if there exists H − such that −A − H − C is Hurwitz. Under this assumption, one construct a backward Luenberger’s observer  − ∀t ∈ [0, τ ],  z˙ (t) = Az − (t) − H − (y(t) − y − (t)), z − (τ ) = z + (τ ),  − y (t) = Cz − (t), ∀t ≥ 0. By putting Z(t) = z − (τ − t), we come back to a direct system fully determined by y(τ − t) and gets the same result as above. G. Haine. Back and forth observers. 16/ 38.

(70) Backward in time Under this assumption, one construct a backward Luenberger’s observer  − ∀t ∈ [0, τ ],  z˙ (t) = Az − (t) − H − (y(t) − y − (t)), z − (τ ) = z + (τ ),  − ∀t ≥ 0. y (t) = Cz − (t), By putting Z(t) = z − (τ − t), we come back to a direct system fully determined by y(τ − t) and gets the same result as above. k z − (0) −z0 k ≤ e−σ | {z } z1+. −. τ. −. +. kz + (τ ) − z(τ )k ≤ e|−(σ {z+σ )τ} kz0+ − z0 k, 0<α<1. for some σ − > 0, which can be made arbitrary large under backward observability assumption (i.e. observability for (−A, C)). This says: z1+ is a better guess of z0 than z0+ .. G. Haine. Back and forth observers. 16/ 38.

(71) Backward in time Under this assumption, one construct a backward Luenberger’s observer  − ∀t ∈ [0, τ ],  z˙ (t) = Az − (t) − H − (y(t) − y − (t)), z − (τ ) = z + (τ ),  − ∀t ≥ 0. y (t) = Cz − (t), By putting Z(t) = z − (τ − t), we come back to a direct system fully determined by y(τ − t) and gets the same result as above. k z − (0) −z0 k ≤ e−σ | {z }. −. τ. −. +. kz + (τ ) − z(τ )k ≤ e|−(σ {z+σ )τ} kz0+ − z0 k, 0<α<1. z1+. for some σ − > 0, which can be made arbitrary large under backward observability assumption (i.e. observability for (−A, C)). This says: z1+ is a better guess of z0 than z0+ . Iterating the process gives the algorithm in finite dimension kzk+ − z0 k ≤ αk kz0+ − z0 k, G. Haine. ∀k ∈ N.. Back and forth observers. 16/ 38.

(72) Outline. 1. Introduction. 2. The finite dimensional setting. 3. Infinite dimensional systems Disasters Solutions A good case. 4. Application to TAT. 5. Conclusion. G. Haine. Back and forth observers. 17/ 38.


(74) Disasters 1. G. Haine. Linear 6⇒ continuous: differential operators (as ∆).. Back and forth observers. 19/ 38.

(75) Disasters 1. 2. G. Haine. Linear 6⇒ continuous: differential operators (as ∆). P 1 etA := k≥0 (tA)k does not make sense if A is not continuous. k!. Back and forth observers. 19/ 38.

(76) Disasters 1. 2. 3. G. Haine. Linear 6⇒ continuous: differential operators (as ∆). P 1 etA := k≥0 (tA)k does not make sense if A is not continuous. k! Solution of: z˙ = Az + f when z(0) = z0 ?. Back and forth observers. 19/ 38.

(77) Disasters 1. 2. 3 4. G. Haine. Linear 6⇒ continuous: differential operators (as ∆). P 1 etA := k≥0 (tA)k does not make sense if A is not continuous. k! Solution of: z˙ = Az + f when z(0) = z0 ? Ker F = 0 6⇒ injectivity of F : two concepts of observability.. Back and forth observers. 19/ 38.

(78) Disasters 1. 2. 3 4 5. G. Haine. Linear 6⇒ continuous: differential operators (as ∆). P 1 etA := k≥0 (tA)k does not make sense if A is not continuous. k! Solution of: z˙ = Az + f when z(0) = z0 ? Ker F = 0 6⇒ injectivity of F : two concepts of observability. If C is not continuous, two concepts for “detectability” (a priori). Back and forth observers. 19/ 38.

(79) Disasters 1. 2. 3 4 5. Linear 6⇒ continuous: differential operators (as ∆). P 1 etA := k≥0 (tA)k does not make sense if A is not continuous. k! Solution of: z˙ = Az + f when z(0) = z0 ? Ker F = 0 6⇒ injectivity of F : two concepts of observability. If C is not continuous, two concepts for “detectability” (a priori). A priori, these concepts are not τ -independent! Geometric Optic Condition (Bardos, Lebeau et Rauch 1992). G. Haine. Back and forth observers. 19/ 38.


(81) Solutions 1. G. Haine. Under some assumptions (on a Hilbert space, operator densely defined with non-empty resolvant set), we can “recover some continuity” from the domain of an operator to the Hilbert space (thanks to “the graph norm”).. Back and forth observers. 21/ 38.

(82) Solutions 1 2. Problem with continuity OK. We can construct families (Tt )t≥0 on a Hilbert space X that behave like the exponential group (etA )t∈R (family of matrices) on Rn , except that elements are not invertible in general: C0 -semigroups. We construct generators for these families, which fit the first point. However, we only have. ∃ω0 ∈ R, ∀ω > ω0 , ∃Mω ≥ 1,. kTt z0 kX ≤ Mω eωt kz0 kX , | {z }. ∀z0 ∈ X, t ≥ 0.. <1?. G. Haine. Back and forth observers. 21/ 38.

(83) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. We have an equivalent to the variation of the constant formula (if T is the semigroup generated by A on the Hilbert space X) Z z(t) = Tt z0 +. t. Tt−s f (s)ds,. ∀t ≥ 0.. 0. But the regularity of this solution is more complicated as in the finite dimensional setting, think it is only continuous in time with value in the Hilbert space X.. G. Haine. Back and forth observers. 21/ 38.

(84) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. Let τ > 0 be fixed and Ψτ : z0 7→ y, then (A, C) is exactly (resp. approximatly) observable in time τ if Ψτ is injective (resp. Ker Ψτ = 0). The injectivity being equivalent, since Ψτ is linear, to the boundedness from below: in the case C ∈ L(X, Y ), this reads Z τ 2 ∃kτ > 0, ∀z0 ∈ X, kΨτ z0 kL2 (0,τ ;Y ) = kCTt z0 k2Y dt ≥ kτ kz0 k2X . 4. 0. G. Haine. Back and forth observers. 21/ 38.

(85) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. 4. Problem with observability OK.. 5. G. Haine. Regarding the detectability problem, it involves duality to define the concept of estimatability which is a priori weaker, but known to be equivalent when C is continuous: we consider C ∈ L(X, Y ) here!. Back and forth observers. 21/ 38.

(86) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. 4. Problem with observability OK.. 5. Problem with detectability OK.. If we suppose that (A, C) is detectable: there exists H + ∈ L(Y, X) such that the C0 -semigroup T+ generated by A − H + C satisfies ω0+ < 0.. G. Haine. Back and forth observers. 21/ 38.

(87) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. 4. Problem with observability OK.. 5. Problem with detectability OK.. If we suppose that (A, C) is detectable: there exists H + ∈ L(Y, X) such that the C0 -semigroup T+ generated by A − H + C satisfies ω0+ < 0. (A, C) is backward detectable: there exists H − ∈ L(Y, X) such that the C0 -semigroup T− generated by −A − H − C satisfies ω0− < 0.. G. Haine. Back and forth observers. 21/ 38.

(88) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. 4. Problem with observability OK.. 5. Problem with detectability OK.. If we suppose that (A, C) is detectable: there exists H + ∈ L(Y, X) such that the C0 -semigroup T+ generated by A − H + C satisfies ω0+ < 0. (A, C) is backward detectable: there exists H − ∈ L(Y, X) such that the C0 -semigroup T− generated by −A − H − C satisfies ω0− < 0. τ > 0 is large enough to get: for some max(ω0+ , ω0− ) < ω < 0 + − + 2ωτ kzkX , kT− τ Tτ zkX ≤ Mω Mω e {z } |. ∀z ∈ X,. α<1!. then the algorithm exponentially converges! G. Haine. Back and forth observers. 21/ 38.

(89) Solutions 1. Problem with continuity OK.. 2. Problem with exponential OK.. 3. Problem with differential equation OK.. 4. Problem with observability OK.. 5. Problem with detectability OK.. If we suppose that (A, C) is detectable: there exists H + ∈ L(Y, X) such that the C0 -semigroup T+ generated by A − H + C satisfies ω0+ < 0. (A, C) is backward detectable: there exists H − ∈ L(Y, X) such that the C0 -semigroup T− generated by −A − H − C satisfies ω0− < 0. τ > 0 is large enough to get: for some max(ω0+ , ω0− ) < ω < 0 + − + 2ωτ kzkX , kT− τ Tτ zkX ≤ Mω Mω e {z } |. ∀z ∈ X,. α<1!. then the algorithm exponentially converges! But: H + and H − ? G. Haine. Back and forth observers. 21/ 38.


(91) A good case Assume that A∗ = −A (think about skew-hermitian matrices), C ∈ L(X, Y ) and that (A, C) is exactly observable in time τ > 0.. G. Haine. Back and forth observers. 23/ 38.

(92) A good case Assume that A∗ = −A (think about skew-hermitian matrices), C ∈ L(X, Y ) and that (A, C) is exactly observable in time τ > 0. Then, for all γ > 0, (A, C) is (backward) detectable with H + = H − = γC ∗ (Liu, 1997).. G. Haine. Back and forth observers. 23/ 38.

(93) A good case Assume that A∗ = −A (think about skew-hermitian matrices), C ∈ L(X, Y ) and that (A, C) is exactly observable in time τ > 0. Then, for all γ > 0, (A, C) is (backward) detectable with H + = H − = γC ∗ (Liu, 1997). Let us consider  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ].. G. Haine. Back and forth observers. 23/ 38.

(94) A good case Assume that A∗ = −A (think about skew-hermitian matrices), C ∈ L(X, Y ) and that (A, C) is exactly observable in time τ > 0. Then, for all γ > 0, (A, C) is (backward) detectable with H + = H − = γC ∗ (Liu, 1997). Let us consider  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ]. The algorithm then reads  + + +  z˙k (t) = Azk (t) − γC ∗ Czk (t) + γC ∗ y(t), + − z (0) = zk−1 (0),  k+ z0 (0) = z0+ ,. ∀t ≥ 0, ∀k ≥ 1,. z˙k− (t) = Azk+ (t) + γC ∗ Czk+ (t) − γC ∗ y(t), zk− (τ ) = zk+ (τ ),. ∀t ≥ 0, ∀k ∈ N,. . G. Haine. Back and forth observers. 23/ 38.

(95) A good case Assume that A∗ = −A (think about skew-hermitian matrices), C ∈ L(X, Y ) and that (A, C) is exactly observable in time τ > 0. Then, for all γ > 0, (A, C) is (backward) detectable with H + = H − = γC ∗ (Liu, 1997). Let us consider  ˙ = Az(t), ∀t ≥ 0,  z(t) z(0) = z0 ,  y(t) = Cz(t), ∀t ∈ [0, τ ]. The algorithm then reads  + + +  z˙k (t) = Azk (t) − γC ∗ Czk (t) + γC ∗ y(t), + − z (0) = zk−1 (0),  k+ z0 (0) = z0+ ,. ∀t ≥ 0, ∀k ≥ 1,. z˙k− (t) = Azk+ (t) + γC ∗ Czk+ (t) − γC ∗ y(t), zk− (τ ) = zk+ (τ ),. ∀t ≥ 0, ∀k ∈ N,. . From Ito, Ramdani and Tucsnak (2011), α < 1: convergence!. G. Haine. Back and forth observers. 23/ 38.

(96) Outline. 1. Introduction. 2. The finite dimensional setting. 3. Infinite dimensional systems. 4. Application to TAT Modelling the problem Writing the wave system as z˙ = Az, y = Cz 3D Simulations. 5. Conclusion. G. Haine. Back and forth observers. 24/ 38.


(98) Modelling the problem We perform external observation =⇒ w(x, t) on a “boundary” S.. G. Haine. Back and forth observers. 26/ 38.

(99) Modelling the problem We perform external observation =⇒ w(x, t) on a “boundary” S. Observation during a finite time interval =⇒ measurement until time 0 < τ < ∞.. G. Haine. Back and forth observers. 26/ 38.

(100) Modelling the problem We perform external observation =⇒ w(x, t) on a “boundary” S. Observation during a finite time interval =⇒ measurement until time 0 < τ < ∞.. We choose τ such that all information “comes out” (Huygens’ principle).. G. Haine. Back and forth observers. 26/ 38.

(101) Modelling the problem We perform external observation =⇒ w(x, t) on a “boundary” S. Observation during a finite time interval =⇒ measurement until time 0 < τ < ∞.. We choose τ such that all information “comes out” (Huygens’ principle). Hence y(x, t) = w(x, t),. G. Haine. ∀ x ∈ S, t ∈ [0, τ ].. Back and forth observers. 26/ 38.


(103) Writing the wave system as z˙ = Az, y = Cz 1. G. Haine. Main issue =⇒ Unbounded domain R3 : the exact observability property can not hold !. Back and forth observers. 28/ 38.

(104) Writing the wave system as z˙ = Az, y = Cz 1. 2. G. Haine. Main issue =⇒ Unbounded domain R3 : the exact observability property can not hold ! w0 ∈ C ∞ (R3 ) compactly supported in Ω (support = body).. Back and forth observers. 28/ 38.

(105) Writing the wave system as z˙ = Az, y = Cz 1. 2. Main issue =⇒ Unbounded domain R3 : the exact observability property can not hold ! w0 ∈ C ∞ (R3 ) compactly supported in Ω (support = body). Poisson-Kirchhoff formula w(x, t) = with Sf (x)(t) =. ∂ (tSw0 (x)) , ∂t. ∀ x ∈ R3 , t ≥ 0,. Z f (x + tv)dσ(v). |v|=1. G. Haine. Back and forth observers. 28/ 38.

(106) Writing the wave system as z˙ = Az, y = Cz 1. 2. Main issue =⇒ Unbounded domain R3 : the exact observability property can not hold ! w0 ∈ C ∞ (R3 ) compactly supported in Ω (support = body). Poisson-Kirchhoff formula w(x, t) = with Sf (x)(t) =. ∂ (tSw0 (x)) , ∂t. ∀ x ∈ R3 , t ≥ 0,. Z f (x + tv)dσ(v). |v|=1. 3. Huygens’ principle =⇒ for all t ≥ 0, the support of w(x, t) is in Ωt = {y ∈ R3 | |x − y| ≤ t, x ∈ Ω}.. G. Haine. Back and forth observers. 28/ 38.

(107) Writing the wave system as z˙ = Az, y = Cz 1. 2. Main issue =⇒ Unbounded domain R3 : the exact observability property can not hold ! w0 ∈ C ∞ (R3 ) compactly supported in Ω (support = body). Poisson-Kirchhoff formula w(x, t) = with Sf (x)(t) =. ∂ (tSw0 (x)) , ∂t. ∀ x ∈ R3 , t ≥ 0,. Z f (x + tv)dσ(v). |v|=1. 3. Huygens’ principle =⇒ for all t ≥ 0, the support of w(x, t) is in Ωt = {y ∈ R3 | |x − y| ≤ t, x ∈ Ω}.. 4. Since we measure during τ > 0 seconds =⇒ we bound “the computation domain” by (for some fixed ε > 0) Ωτ + = {y ∈ R3 | |x − y| ≤ τ + ε, x ∈ Ω}.. G. Haine. Back and forth observers. 28/ 38.

(108) Writing the wave system as z˙ = Az, y = Cz On Ωτ + ,                 . G. Haine. w(x, t) is also the solution of ∂2 w(x, t) = ∆w(x, t), ∂t2 w(x, t) = 0, w(x, 0) = w0 (x), w(x, 0) = 0, ∂ w(x, 0) = 0, ∂t. ∀ x ∈ Ωτ + , t ∈ [0, τ ], ∀ x ∈ ∂Ωτ + , t ∈ [0, τ ], ∀ x ∈ Ω, ∀ x ∈ Ωτ + \ Ω, ∀ x ∈ Ωτ + .. Back and forth observers. 28/ 38.

(109) Writing the wave system as z˙ = Az, y = Cz On Ωτ + ,         . w(x, t) is also the solution of. ∂2 w(x, t) = ∆w(x, t), ∀ x ∈ Ωτ + , t ∈ [0, τ ], ∂t2 w(x, t) = 0, ∀ x ∈ ∂Ωτ + , t ∈ [0, τ ], w(x, 0) = w0 (x), ∀ x ∈ Ω,   w(x, 0) = 0, ∀ x ∈ Ωτ + \ Ω,     ∂   w(x, 0) = 0, ∀ x ∈ Ωτ + . ∂t 1 Let γ0 ∈ L H01 (Ωτ + ), H 2 (∂Ω) be the Dirichlet operator on ∂Ω. We define D (A0 ) = H 2 (Ωτ + ) ∩ H01 (Ωτ + ),. H = L2 (Ωτ + ),. A0 = −∆ : D (A0 ) −→ H,. G. Haine. Back and forth observers. 28/ 38.

(110) Writing the wave system as z˙ = Az, y = Cz On Ωτ + ,         . w(x, t) is also the solution of. ∂2 w(x, t) = ∆w(x, t), ∀ x ∈ Ωτ + , t ∈ [0, τ ], ∂t2 w(x, t) = 0, ∀ x ∈ ∂Ωτ + , t ∈ [0, τ ], w(x, 0) = w0 (x), ∀ x ∈ Ω,   w(x, 0) = 0, ∀ x ∈ Ωτ + \ Ω,     ∂   w(x, 0) = 0, ∀ x ∈ Ωτ + . ∂t 1 Let γ0 ∈ L H01 (Ωτ + ), H 2 (∂Ω) be the Dirichlet operator on ∂Ω. and 1 D A02 = H01 (Ωτ + ), Y = L2 (∂Ω), 1 1 C0 = γ0 : D A02 → H 2 (∂Ω) ,→ Y.. G. Haine. Back and forth observers. 28/ 38.

(111) Writing the wave system as z˙ = Az, y = Cz Then. G. Haine.  ¨ + A0 w(t) =   w(t) 0,1 w(0) = w0 ∈ D A02 ,   w(0) ˙ = 0 ∈ H.. ∀ t ∈ [0, τ ],. Back and forth observers. 28/ 38.

(112) Writing the wave system as z˙ = Az, y = Cz Then.  ¨ + A0 w(t) =   w(t) 0,1 w(0) = w0 ∈ D A02 ,   w(0) ˙ = 0 ∈ H.. ∀ t ∈ [0, τ ],. Finally, rewriting the model as a first-order system w(t) w0 z(t) = , z0 = , w(t) ˙ 0 1 0 I A= , D (A) = D (A0 ) × D A02 , −A0 0. G. Haine. Back and forth observers. 1 X = D A02 × H, C = C0. 0 ,. 28/ 38.

(113) Writing the wave system as z˙ = Az, y = Cz Then.  ¨ + A0 w(t) =   w(t) 0,1 w(0) = w0 ∈ D A02 ,   w(0) ˙ = 0 ∈ H.. ∀ t ∈ [0, τ ],. Finally, rewriting the model as a first-order system w(t) w0 z(t) = , z0 = , w(t) ˙ 0 1 0 I A= , D (A) = D (A0 ) × D A02 , −A0 0 and then. . z(t) ˙ = Az(t), z(0) = z0 ∈ X,. 1 X = D A02 × H, C = C0. 0 ,. ∀ t ∈ [0, τ ],. with y(t) = Cz(t),. G. Haine. ∀ t ∈ [0, τ ].. Back and forth observers. 28/ 38.

(114) Reconstruction algorithm We show easily that. G. Haine. 1. A is skew-adjoint. 2. C ∈ L(X, Y ). 3. (A, C) is not exactly observable (∀τ > 0). Back and forth observers. 29/ 38.

(115) Reconstruction algorithm We show easily that 1. A is skew-adjoint. 2. C ∈ L(X, Y ). 3. (A, C) is not exactly observable (∀τ > 0). Indeed. Some rays are trapped (Bardos, Lebeau, Rauch 1992). G. Haine. Back and forth observers. 29/ 38.

(116) Reconstruction algorithm Decomposition of X:. G. Haine. Back and forth observers. 29/ 38.

(117) Reconstruction algorithm Decomposition of X: Let us denote Ψτ the following continuous linear operator Ψτ. G. Haine. :. X z0. −→ 7→. L2 ([0, τ ], Y ) , y(t).. Back and forth observers. 29/ 38.

(118) Reconstruction algorithm Decomposition of X: Let us denote Ψτ the following continuous linear operator Ψτ. :. X z0. −→ 7→. L2 ([0, τ ], Y ) , y(t).. Intuitively, if z0 is in Ker Ψτ , then y(t) ≡ 0, and we have no information on z0 !. G. Haine. Back and forth observers. 29/ 38.

(119) Reconstruction algorithm Decomposition of X: Let us denote Ψτ the following continuous linear operator Ψτ. :. X z0. −→ 7→. L2 ([0, τ ], Y ) , y(t).. Intuitively, if z0 is in Ker Ψτ , then y(t) ≡ 0, and we have no information on z0 ! We decompose X = Ker Ψτ ⊕ (Ker Ψτ ) VUnobs = Ker Ψτ ,. G. Haine. ⊥. and define ⊥. VObs = (Ker Ψτ ) = Ran Ψ∗τ .. Back and forth observers. 29/ 38.

(120) Reconstruction algorithm Decomposition of X: Let us denote Ψτ the following continuous linear operator Ψτ. :. X z0. −→ 7→. L2 ([0, τ ], Y ) , y(t).. Intuitively, if z0 is in Ker Ψτ , then y(t) ≡ 0, and we have no information on z0 ! We decompose X = Ker Ψτ ⊕ (Ker Ψτ ) VUnobs = Ker Ψτ ,. ⊥. and define ⊥. VObs = (Ker Ψτ ) = Ran Ψ∗τ .. Note that the exact observability assumption is equivalent to Ψτ is bounded from below and then to ⇒ X = Ran Ψ∗τ = VObs .. G. Haine. Back and forth observers. 29/ 38.

(121) Reconstruction algorithm Theorem Denote by Π the orthogonal projection from X onto VObs . Then the following statements hold true for all z0 ∈ X and z0+ ∈ VObs : 1. 2. For all k ≥ 1, (I − Π) zk− (0) − z0 = (I − Π) z0 .. The sequence Π zk− (0) − z0 k≥1 is strictly decreasing and. Π z − (0) − z0 = z − (0) − Πz0 −→ 0. k k k→∞. 3. There exists a constant α ∈ (0, 1), independent of z0 and z0+ , such that for all k ≥ 1,. Π z − (0) − z0 ≤ αk z + − Πz0 , 0 k if and only if Ran Ψ∗τ is closed in X.. G. Haine. Back and forth observers. 29/ 38.

(122) Reconstruction algorithm The forward observer reads  + w˙ (t) = −γC0∗ C0 wk+ (t) + w ek+ (t) + γC0∗ y(t),   k+    e˙ k (t) = −A0 wk+ (t),  w  w1+ (0) = 0,  e1+ (0) = 0,  w  + −    wk (0) = wk−1 (0),  + − w ek (0) = w ek−1 (0),. G. Haine. Back and forth observers. ∀ t ∈ [0, τ ], ∀ t ∈ [0, τ ],. ∀ k ≥ 2, ∀ k ≥ 2,. 29/ 38.

(123) Reconstruction algorithm The forward observer reads  + w˙ (t) = −γC0∗ C0 wk+ (t) + w ek+ (t) + γC0∗ y(t),   k+    e˙ k (t) = −A0 wk+ (t),  w  w1+ (0) = 0,  e1+ (0) = 0,  w  + −    wk (0) = wk−1 (0),  + − w ek (0) = w ek−1 (0), and the backward observer is  − w˙ k (t) = γC0∗ C0 wk− (t) + w ek− (t) − γC0∗ y(t),    ˙− w ek (t) = −A0 wk− (t)(t), − +    wk− (τ ) = wk+ (τ ), w ek (τ ) = w ek (τ ),. G. Haine. Back and forth observers. ∀ t ∈ [0, τ ], ∀ t ∈ [0, τ ],. ∀ k ≥ 2, ∀ k ≥ 2,. ∀ t ∈ [0, τ ], ∀ t ∈ [0, τ ], ∀ k ≥ 1, ∀ k ≥ 1.. 29/ 38.


(125) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time. G. Haine. Back and forth observers. 31/ 38.

(126) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time We know that there exists an optimal number of iterations (Haine and Ramdani, 2012). Furthermore, under some hypothesis, we have kw0 − w0,h,∆t k 12 ≤ Mτ (h + ∆t) ln2 (h + ∆t) kw0 k 23 + |ln(h + ∆t)| ∆t. K X. y(t` ) − yh` . `=0. G. Haine. Back and forth observers. 31/ 38.

(127) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time We know that there exists an optimal number of iterations (Haine and Ramdani, 2012). Furthermore, under some hypothesis, we have kw0 − w0,h,∆t k 12 ≤ Mτ (h + ∆t) ln2 (h + ∆t) kw0 k 23 + |ln(h + ∆t)| ∆t. K X. y(t` ) − yh` . `=0. We simulate the outward wave and measure it on a sphere. G. Haine. Back and forth observers. 31/ 38.

(128) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time We know that there exists an optimal number of iterations (Haine and Ramdani, 2012). Furthermore, under some hypothesis, we have kw0 − w0,h,∆t k 12 ≤ Mτ (h + ∆t) ln2 (h + ∆t) kw0 k 23 + |ln(h + ∆t)| ∆t. K X. y(t` ) − yh` . `=0. We simulate the outward wave and measure it on a sphere We add gaussian noise with 0.25 of standard deviation on the observation. G. Haine. Back and forth observers. 31/ 38.

(129) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time We know that there exists an optimal number of iterations (Haine and Ramdani, 2012). Furthermore, under some hypothesis, we have kw0 − w0,h,∆t k 12 ≤ Mτ (h + ∆t) ln2 (h + ∆t) kw0 k 23 + |ln(h + ∆t)| ∆t. K X. y(t` ) − yh` . `=0. We simulate the outward wave and measure it on a sphere We add gaussian noise with 0.25 of standard deviation on the observation We use this noisy observation on several configurations:. G. Haine. Back and forth observers. 31/ 38.

(130) 3D Simulations We use Gmsh and GetDP to simulate our problem: P1 FEM scheme in space and BFD1 scheme in time We know that there exists an optimal number of iterations (Haine and Ramdani, 2012). Furthermore, under some hypothesis, we have kw0 − w0,h,∆t k 12 ≤ Mτ (h + ∆t) ln2 (h + ∆t) kw0 k 23 + |ln(h + ∆t)| ∆t. K X. y(t` ) − yh` . `=0. We simulate the outward wave and measure it on a sphere We add gaussian noise with 0.25 of standard deviation on the observation We use this noisy observation on several configurations: 1 2. G. Haine. We test the influence of the gain parameter γ We test ill-posed cases: lack of observation Back and forth observers. 31/ 38.

(131) Simulations with observation on a sphere. Simulations with observation on a sphere: well-posed inverse problem ! G. Haine. Back and forth observers. 32/ 38.

(132) Simulations with observation on a half-sphere. Simulations with observation on a half-sphere G. Haine. Back and forth observers. 33/ 38.

(133) What about real life applications?. Small Animal Scanner – 2D Array TAT – Wikipedia (EN). G. Haine. Back and forth observers. 34/ 38.

(134) What about real life applications?. Small Animal Scanner – 2D Array TAT – Wikipedia (EN). G. Haine. Back and forth observers. 34/ 38.

(135) Simulations with observation on a 2D array. Simulations with observation on a 2D array on the half-sphere G. Haine. Back and forth observers. 35/ 38.

(136) Influence of parameter γ. Relative errors in L2 with gain parameter γ = 1 (left) and γ = 5 (right). G. Haine. Back and forth observers. 36/ 38.

(137) Outline. 1. Introduction. 2. The finite dimensional setting. 3. Infinite dimensional systems. 4. Application to TAT. 5. Conclusion. G. Haine. Back and forth observers. 37/ 38.

(138) Conclusion Read more on the subject? G. Haine Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint operator (Mathematics of Control, Signals, and Systems (MCSS), January 2014 ) G. Haine and K. Ramdani Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations (Numerische Mathematik (Numer. Math.), 2012 ). G. Haine. Back and forth observers. 38/ 38.

(139) Conclusion Read more on the subject? G. Haine Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint operator (Mathematics of Control, Signals, and Systems (MCSS), January 2014 ) G. Haine and K. Ramdani Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations (Numerische Mathematik (Numer. Math.), 2012 ) But there is still a lot to be done: Stability of VObs and VUnobs with noisy observation y Generalization (A∗ 6= −A) Uncertainty in the model (sound speed for instance) Optimization of γ G. Haine. Back and forth observers. 38/ 38.

(140)