Reconstruction Scheme for Active Thermography

Transcription

1 Reconstruction Scheme for Active Thermography Gen Nakamura Department of Mathematics, Hokkaido University, Japan Newton Institute, Cambridge, Sept. 20, 2011

2 Contents.1.. Important Preliminary Estimates Gradient estimate of solutions for parabolic equations Gradient estimate of fundamental solution Remarks About the proof.2.. Active thermography Forward problem Dynamical probe method Seperated inclusions case result Outline of the proof Remark for non-separated inclusions case Identifying isotropic conductivity

3 Important Preliminary Estimates Important preliminary estimates Joint work with J.Fan, K.Kim and S.Nagayasu

4 Important Preliminary Estimates Gradient estimate of solutions for parabolic equations Domain and operators Ω R n : b dd domain (heat conductor), Ω : C 2. γ(x) = (γ jk (x)) : defined a.e. in Ω, symm, pos. def. (conductivity) λ ξ 2 γ jk (x)ξ j ξ k Λ ξ 2. We further assume the followings for Ω and γ.

5 Important Preliminary Estimates Gradient estimate of solutions for parabolic equations Domain and operators (continued) ( L ) Let Ω = D m \ Ω. m=1 γ (m) C µ (D m ) (0 < µ < 1), γ(x) = γ (m) (x) (x D m ). Each separated D m is of C 1,α smooth with 0 < α 1 and non-separated one is the limit of the separated one. D 1 D2 D 3 D 4

6 Important Preliminary Estimates Gradient estimate of solutions for parabolic equations Gradient estimate. Theorem 1 (Fan, Kim, Nagayasu and N).. Let Ω Ω, 0 < t 0 < T. Any sol u to (P): t u (γ u) = 0 in Ω (0, T ) has the following interior regularity est: sup t0 <t<t u(, t) C 1,α (Ω D m ) C u L 2 (Ω (0,T )), where 0 < α min(µ, inclusions.. α 2(α+1) ) and C is indep of the dist between Ω

7 Important Preliminary Estimates Gradient estimate of fundamental solution Gradient estimate of fundamental solution By applying our main theorem and a scaling argument, we obtain pointwise grad. est. for 0 < t s < T, x E(x, t; y, s) C T (t s) n+1 exp 2 of fund sol E(x, t; y, s) for t (γ ). ( ) c x y 2 t s

8 Important Preliminary Estimates Remarks Remarks (i) We can obtain a similar estimate for non-homog parabolic eq : t u (γ u) = g + f. (ii) H. Li-Y. Li extended the result to time dep. parabolic sys. (preprint). But the inclusions are indep. of time. As a simplification of the proof, it can be derived based on the time indep. case by considering F t (x, t; y, s) with F τ = τ E τ, E τ : fund. sol. for the op. with coeff. γ t=τ (iii) The time dependent inclusions case is an open problem. (iv) The elliptic case was proved by Li-Vogelius for scalar equations and Li-Nirenberg ([LN]) for systems, which answered to the Babuška s conjecture. Babuška et al (1999) numerically observed that the gradient est of sol is indep of the distances between inclusions.

9 Important Preliminary Estimates About the proof Idea of Proof Idea of proof: Some interior estimates (Lemma). (ref. Ladyzenskaja-Rivkind-Uralceva) Apply [LN] to (P).

10 Important Preliminary Estimates About the proof Proof. Lemma 2.. Let Ω Ω, 0 < t 0 < T. Any sol u to has the following estimates:. t u (γ u) = 0 in Ω (0, T ) =: Q sup u(, t) L 2 ( Ω) C u L 2 (Q) (standard), t 0<t<T u L ( Ω (t0,t )) C u L 2 (Q) (Di Giorgi s arg.), u t L 2 ( Ω (t0,t )) C u L 2 (Q) ([LRU]).. Remark 3.. (i)this lemma holds for any γ L (Ω). (ii). LRU=Ladyzenskaja-Rivkind-Uralceva.

11 Important Preliminary Estimates About the proof Proof Let Ω 3 Ω 2 Ω 1 Ω 0 := Ω, 0 < δ 1 < δ 2 < T. Then ( ) sup δ2<t<t u(, t) L2 ( Ω C u 2) L 2 (Q), ( ) u t L2 ( Ω 1 (δ 1,T )) C u L 2 (Q). Since t u t (γ u t ) = 0, we have ( ) u t L ( Ω2 (δ 2,T )) C u t L 2 ( Ω1 (δ 1,T )) C u L2 (Q). Now we fix t (δ 2, T ): (γ u) = u t L ( Ω 2 ).

12 Important Preliminary Estimates About the proof Proof Then by [LN], we have u(, t) C 1,α (D m Ω 3 ) ( ) C u(, t) L2 ( Ω + u 2) t(, t) L ( Ω 2). Taking sup δ2 <t<t, we have by (*), (***), sup δ2 <t<t u(, t) C 1,α (D m Ω 3 ) ( ) C sup δ2<t<t u(, t) L2 ( Ω + u 2) t L ( Ω 2 (δ 2,T )) C u L2 (Q).

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14 Active thermography Active thermography D Ω u(f) Ω A u Ω = f

15 Active thermography Principle of active thermography infrared camera heater / flash lamp inclusion

16 Active thermography Dynamical probe method for anisotropic heat conductors Joint work with K.Kim

17 Forward problem Mixed problem (set up) Ω R n (1 n 3) : bounded domain, Ω : C 2 (n = 2, 3), Ω = Γ D Γ N, where Γ D, Γ N are open subsets of Ω such that Γ D Γ N = and Γ D, Γ N are C 2 if they are nonempty. D Ω : open set (separated inclusion(s)), D Ω, D : C 1,α (0 < α 1), Ω \ D : connected. Heat conductivity: γ(x) = A(x) + (Ã(x) A(x))χ D : positive definite for each x Ω, where A, à C 1 (Ω) are positive definite and à A is always positive definite or negative in a neigh. of D, χ D is the char func of D.

18 Forward problem H p ( Ω), H p,q (Ω (0, T )): usual Sobolev spaces (p, q Z + := N {0} or p = 1 2 ) ex. For p, q Z +, g H p,q (Ω (0, T )) iff g H p,q (Ω (0,T )) := α +2k p k q Ω (0,T ) x α t k g 2 dtdx 1/2 < L 2 ((0, T ); H p ( Ω)) := {f ; T 0 f(, t) 2 H p ( Ω) dt < }

19 Forward problem Mixed problem (forward problem) Given f L 2 ((0, T ); H 1 2 (Γ D )), g L 2 ((0, T ); Ḣ 1 2 (Γ N )), (?)! weak solution u = u(f, g) W (Ω T ) := {u H 1,0 (Ω T ), t u L 2 ((0, T ); H 1 (Ω) )} : P D u(x, t) := t u(x, t) div x (γ(x) x u(x, t)) = 0 in Ω T u(x, t) = f(x, t) on Γ D T, Au(x, t) := ν A u(x, t) = g(x, t) on Γ N T u(x, 0) = 0 for x Ω, where ν is the outer unit normal of Ω, H 1 2 (Γ D ), Ḣ 1 2 (Γ N ) are Hörmander s notations of Sobolev sp, Ω T = Ω (0,T ) := Ω (0, T ), Ω T = Ω (0,T ) := Ω (0, T ). (cylindrical sets) This is a well-posed problem.

20 Forward problem Measured data Neumann-to-Dirichlet map Λ D : For fixed f L 2 ((0, T ); H 1 2 (Γ D )), define Λ D : L 2 ((0, T ); Ḣ 1 2 (ΓN )) L 2 ((0, T ); H 1 2 (Γ N )) g u(f, g) Γ N T. Inverse boundary value problem Reconstruct the unknown inclusion D from Λ D.

21 Forward problem Known results I H. Bellout (1992): Local uniqueness and stability. A. Elayyan and V. Isakov (1997): Global uniqueness using the localized Neumann-to-Dirichlet map. M. Di Cristo and S. Vessella (2010): Optimal stability estimate (i.e. log type stability estimate) even for time dependent inclusions. Y. Daido, H. Kang and G. Nakamura (2007) (Inverse Problems) : Introduced the dynamical probing method for 1-D case. Y. Daido, Y. Lei, J. Liu and G. Nakamura (2009) (Applied Mathematics and Computation) Numerical implementations of 1-D dynamical probe method for non-stationary heat equation.

22 Forward problem Known results II Y. Lei, K. Kim and G. Nakamura (2009) (Journal of Computational Mathematics) Theoretical and numerical studies for 2-D dynamical probe method. M. Ikehata and M. Kawashita (2009) (Inverse Problems) Extracted some geometric information of an unknown cavity using CGO solution and asymptotic analysis. V. Isakov, K. Kim and G. Nakamura (2010) (Ann. Scuola Superior di Pisa) Gave the theoretical basis of dynamical probe method. K. Kim and G. Nakamura (2011) (J. of Physics: Conf. Series, Vol. 290) Gave the argument of dynamical probe method for anisotropic conductivities.

23 Dynamical probe method Dynamical probe method (fundamental solutions) For (y, s), (y, s ) R n R, (x, t) Ω T, Γ(x, t; y, s) : fundamental solution of P := t (A(x) ) Γ (x, t; y, s ) : fundamental solution of P := t (A(x) ) G(x, t; y, s), G (x, t; y, s ): P G(x, t; y, s) = δ(x y)δ(t s) in Ω T, G(, ; y, s) = 0 on Γ D T, G(x, t; y, s) = 0 for x Ω, t s P G (x, t; y, s ) = δ(x y)δ(t s ) in Ω T, G (, ; y, s ) = 0 on Γ D T, G (x, t; y, s ) = 0 for x Ω, t s G(x, t; y, s) Γ(x, t; y, s), G (x, t; y, s ) Γ (x, t; y, s ) : Ct 1, Cx 2 in Ω T.

24 Dynamical probe method Dynamical probe method (Runge s approximation) {v 0j (y,s)}, {ψ0j (y,s ) } H2,1 (Ω ( ε,t +ε) ) for ε > 0 s.t. P v 0j (y,s) = 0 in Ω ( ε,t +ε), v 0j (y,s) = 0 on ΓD ( ε, T + ε), v 0j (y,s)(x, t) = 0 if ε < t 0, v 0j (y,s) G(, ; y, s) in H2,1 (U ( ε, T + ε )) as j, P ψ0j (y,s ) = 0 in Ω ( ε,t +ε), ψ 0j (y,s ) = 0 on ΓD ( ε, T + ε), ψ 0j (y,s )(x, t) = 0 if T t < T + ε, ψ 0j (y,s ) G (, ; y, s ) in H 2,1 (U ( ε, T + ε )) as j for 0 < ε < ε, U Ω : open s.t. U Ω, Ω \ U : connected, U : Lipschitz, U y, y, and ε < s, s < T + ε.

25 Dynamical probe method Dynamical probe method (Runge approx funcs) Let v, ψ satisfy P v = 0 in Ω T, v = f on Γ D T, A v = 0 on Γ N T, v(x, 0) = 0 for x Ω, P ψ = 0 in Ω T, ψ = 0 on Γ D T, A ψ = g on Γ N T, ψ(x, T ) = 0 for x Ω. For j = 1, 2,, we define { v j (y,s) := v + v0j (y,s) V (y,s) := v + G(, ; y, s) ψ j (y,s ) := ψ + ψ0j (y,s ) Ψ (y,s ) := ψ + G (, ; y, s ). in H 2,1 (U T ) as j. {v j (y,s) }, {ψj (y,s )} : Runge s approximation functions

26 Dynamical probe method Pre-indicator function. Definition 4.. (y, s), (y, s ) Ω T {v j (y,s) }, {ψj (y,s ) } W (Ω T ) : Runge s approximation functions Pre-indicator function : I(y, s ; y, s) = lim j Γ N T [ ] A v j (y,s) Γ N ψj T (y,s ) Γ N Λ D( T A v j (y,s) ) Γ N Aψ j T (y s ) Γ N T whenever. the limit exists.

27 Dynamical probe method Reflected solution. Lemma 5.. y D, 0 < s < T, {v j (y,s) } W (Ω T ) : Runge s approximation functions, u j (y,s) := u(f, Av j (y,s) Γ N ), wj T (y,s) := uj (y,s) vj (y,s) Then, w j (y,s) has a limit w (y,s) W (Ω T ) satisfying P D w (y,s) = div x ((Ã A)χ D x V (y,s) ) in Ω T, w (y,s) = 0 on Γ D T, Aw (y,s) = 0 on Γ N T. w (y,s) (x, 0) = 0 for x Ω. w (y,s) : reflected solution

28 Dynamical probe method Representation formula. Theorem 6.. For y, y D, 0 < s, s < T such that (y, s) (y, s ), the pre-indicator function I(y, s ; y, s) has the representation formula in terms of the reflected solution w (y,s) :. I(y, s ; y, s) = w (y,s) (y, s ) Ω T w (y,s) A Γ (, ; y, s )dσdt

29 Seperated inclusions case result Main result (indicator function). Definition 7.. C := {c(λ) ; 0 λ 1} : non-selfintersecting C 1 curve in Ω, c(0), c(1) Ω (We call this C a needle.) Then, for each c(λ) Ω and each fixed s (0, T ), indicator function (mathematical testing machine) J(c(λ), s) := lim lim sup I(c(λ δ), s + ϵ 2 ; c(λ δ), s) ϵ 0. whenever the limit exists. δ 0

30 Seperated inclusions case result c(0) c(λ) C c(λ δ) Ω D c(1) Figure 1: Domains Ω, D, and a curve C

31 Seperated inclusions case result Seperated inclusions case result (theorem). Theorem 8.. Let D consist of separated inclusions, and C, c(λ) be as in the definition above. Fix s (0, T ). (i) C Ω \ D except c(0) and c(1) = J(c(λ), s) < for all λ, 0 λ 1 (ii) C D λ s (0 < λ s < 1) s.t. c(λ s ) D, c(λ) Ω \ D (0 < λ < λ s ) =. λ s = sup{ 0 < λ < 1 ; J(c(λ ), s) < for any 0 < λ < λ }.

32 Seperated inclusions case result Remark : (i) A numerical realization of this reconstruction scheme has been done for isotropic conductivities. (ii) If Γ D and f(, t) = 0 = g(, t) (t > T ) with 0 < T < T, then u(f, g) has the decaying property. That is u(f, g) decays exponentially after t = T. Hence, in this case, we can guarantee the exponential decay of the temperature after the experiment.

33 Outline of the proof Proof of Theorem 6: Consider only the case n = 3 in the rest of the arguments. First, we recall the previous two facts. (i) w (y,s) W (Ω T ) : solution to P D w (y,s) = div x ((Ã A)χ D x V (y,s) ) in Ω T, w (y,s) = 0 on Γ D T, Aw (y,s) = 0 on Γ N T w (y,s) (x, 0) = 0 for x Ω. (ii) I(y, s ; y, s) = w (y,s) (y, s ) w (y,s) A Γ (, ; y, s )dσdt Ω T If y = c(λ) is away from D, it is easy to see the indicator function is finite at y. So, let s consider the case y is close to D.

34 Outline of the proof Setup Note that P D w (y,s) = div x ((Ã A)χ D x V (y,s) ) in Ω T Hence, E(x, t; y, s) := w (y,s) (x, t) + V (y,s) (x, t) ( fundamental solution for P D.) Let P = c(λ 0 ) D for some λ 0 x = y = c(λ 0 δ) C \ D for δ > 0. Φ : R 3 R 3 with Φ(P ) = O (C 1,α diffeomorphism, 0 < α 1), Φ(D) R 3 = {ξ = (ξ 1, ξ 2, ξ 3 ) R 3 ; ξ 3 < 0}, Jacobi matrix of Φ at P = identity matrix.

35 Outline of the proof Let E : t ((A(x) + (Ã(x) A(x))χ D) ) Γ P Γ Γ 0 Γ 0 : t ((A(x) + (Ã(P ) A(x))χ D) ) : t ((A(Φ 1 (ξ)) + (Ã(P ) A(Φ 1 (ξ)))χ ) ) : t ((A(P ) + (Ã(P ) A(P ))χ ) ) : t (A(P ) ) Γ : t (A(x) ). be the fund. sol. and corresponding operators, where χ is the characteristic function of the space R 3.

36 Outline of the proof Main part of the proof Decompose w (y,s) as follows: w (y,s) (x, t) = E(x, t; y, s) Γ(x, t; y, s) = {E(x, t; y, s) Γ P (x, t; y, s)} + {Γ P (x, t; y, s) Γ (Φ(x), t; Φ(y), s)} + {Γ (Φ(x), t; Φ(y), s) Γ 0 (Φ(x), t; Φ(y), s)} + {Γ 0 (Φ(x), t; Φ(y), s) Γ 0 (Φ(x), t; Φ(y), s)} + {Γ 0 (Φ(x), t; Φ(y), s) Γ 0 (x, t; y, s)} + {Γ 0 (x, t; y, s) Γ(x, t; y, s)} + +{Γ(x, t; y, s) V (y,s) (x, t)}, To show : w (y,s) (y, s ) as s s, y D Let ξ = η = Φ(x) = Φ(y) O (δ 0) and consider the case, for example n = 3.

37 Outline of the proof Behavior of each term 1. lim sup E(x, s + ε 2 ; y, s) Γ P (x, s + ε 2 ; y, s) = O(ε µ 3 ), δ 0 as ε lim sup ( Γ p Γ )(ξ, s + ε 2 ; η, s) = O(ε α 3 ) as ε 0. δ 0 3. lim sup Γ (ξ, t + ε 2 ; η, s) Γ 0 (ξ, t + ε 2 ; η, s) = O(ε µ 3 ) as ε 0. δ 0 (In 1,2,3, we used a pointwise space gradient estimate for a fundamental solution of parabolic equation with disconti. coeff..)

38 Outline of the proof 4. Put W (ξ, t; η, s) := Γ 0 (ξ, t; η, s) Γ 0 (ξ, t; η, s) (dominant) Denote W (ξ, t; η, s) for ±ξ n > 0 by W ± (ξ, t; η, s). Then, there exist a constant C > 0 such that lim W + (η, s + ε 2 ; η, s) Cε 3 as ε 0. δ 0 5. lim sup Γ 0 (Φ(x), t; Φ(y), s) Γ 0 (x, t; y, s) = 0. δ 0 6. Let G(x, t; y, s) = Γ 0 (x, t; y, s) Γ(x, t; y, s). Then, lim sup G(y, s + ε 2 ; y, s) = O(ε 2 ) as ε 0. δ 0

39 Outline of the proof 7. It follows from the definitions of Γ and v that Γ(x, t; y, s) V (y,s) (x, t) is C 1 t C 2 x at (y, s) and so bounded in some closed neighborhood of (y, s).

40 Remark for non-separated inclusions case Remark for non-separated inclusions (open question) The previous proof for the separated inclusions case works well except the estimate for W (ξ, t; η, s).

41 Identifying isotropic conductivity Identifying isotropic conductivities Joint work with H.Sasayama

42 Identifying isotropic conductivity Let the conductivity γ be homogeneous and isotropic: γ = 1 + (k 1)χ D (1) with 0 < k 1 (constant). Then, the dominant term of w (y,s) (y, s + ε 2 ) (y D) is W + (η, s + ε; η, s) (δ 0 or η 0), (2) where W + (ξ, t; η, s) = W (ξ, t; η, s) (ξ n > 0) and W (ξ, t; η, s) = Γ 0 (ξ, t; η, s) Γ 0 (ξ, t; η, s) (3) which satisfies t W ξ (1 + (k 1)χ ) ξ W = (k 1) ξ (χ ξ Γ 0 ) (4) with χ is the characteristic function of R n := {ξ n < 0}.

43 Identifying isotropic conductivity Let n = 2, 3. Case k > 1 where k 1 lim W + (η, s + ε 2 ; η, s) = δ 0 2 n π H n(k), (5) 3 n/2 H n (k) = 1 0 (k + 1)r k dr (6) r(r + k(1 r))(kr r + 1) n/2 Case 0 < k < 1 k(1 k) lim W + (η, s + ε 2 ; η, s) = δ 0 2 n 3 π H n(k), (7) 3 n/2 where H n (k) = 1 0 r(1 r) dr (8) (r + k(1 r))(kr r + 1) n/2

44 Identifying isotropic conductivity. Theorem 1.. For the case k > 1 we can prove that k 1H n (k) is monotone increasing. Hence, in this case, we can recover k from the measured data. However, for the case 0 < k < 1, we can only recover k from the measured data for those k in certain intervals. If n = 2, then. k(1 k)h2 (k) is convex.

45 Identifying isotropic conductivity H k k Figure 2: 2-dim, k > 1, H[k]

46 Identifying isotropic conductivity H k k Figure 3: 3-dim, k > 1, H[k]

47 Identifying isotropic conductivity H k k Figure 4: 2-dim, 0 < k < 1, H[k]

48 Identifying isotropic conductivity 1 k H k k Figure 5: 2-dim, k > 1, d dk k 1H[k]

49 Identifying isotropic conductivity 1 k k H k k Figure 6: 2-dim, 0 < k < 1, d dk k(1 k)h[k]

50 Identifying isotropic conductivity 1 k k H k k Figure 7: 2-dim, 0 < k < 1, d 2 dk 2 k(1 k)h[k]

51 Identifying isotropic conductivity 1 k k H k k Figure 8: 2-dim, 0 < k < 1, d 2 dk 2 k(1 k)h[k]

52 Identifying isotropic conductivity 1 k H k k Figure 9: 3-dim, k > 1, d dk k 1H[k]

53 Identifying isotropic conductivity 1 k k H k k Figure 10: 3-dim, 0 < k < 1, d dk k(1 k)h[k]

54 Identifying isotropic conductivity k k H k k Figure 11: 3-dim, 0.95 < k < 1, d dk k(1 k)h[k]

55 Identifying isotropic conductivity 1 k k H k k Figure 12: 3-dim, 0 < k < 1, d 2 dk 2 k(1 k)h[k]

56 Identifying isotropic conductivity 1 k k H k k Figure 13: 3-dim, 0.2 < k < 0.8, d 2 dk 2 k(1 k)h[k]

57 Identifying isotropic conductivity Thank you for your attention.