Asymptotic Spectral Imaging

Transcription

1 Rennes p. /4 Asymptotic Spectral Imaging Habib Ammari CNRS & Ecole Polytechnique

2 Rennes p. 2/4 Motivation Image internal defects from eigenvalue and eigenvector boundary measurements (or spectral measurements). Vibration testing: valuable addition to other nondestructive testing methods. Describe the effects of damages (like small inclusions, cracks, corrosion) on eigenvalues and eigenvectors. Design robust and accurate algorithms based on this asymptotic formalism. Potential applications: nondestructive testing of nuclear plants, pipelines,...

3 Rennes p. 3/4 Contents Asymptotic theory for spectral perturbations: (i) due to the presence of inclusions, cracks or internal corrosion; (ii) under inclusion shape deformations. Derivations based on combining Gohberg-Sigal theory for meromorphic operator-valued functions + layer potential techniques. General and unified boundary integral approach. Design new imaging algorithms based on a duality approach which are stable and accurate.

4 Rennes p. 4/4 References Results from: (i) with H. Kang, E. Kim, and H. Lee, MMAS 09; (ii) with H. Kang, E. Kim, and K. Louati, Stud. Appl. Math 09; (iii) with H. Kang, M. Lim and H. Zribi, Trans. AMS 09; (iv) with E. Beretta, E. Francini, H. Kang, and M. Lim, Math. of Comp. 09.

5 Rennes p. 5/4 References For more details: with H. Kang and H. Lee, Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs Series, Vol. 53, Amer. Math. Soc., Providence, 2009)

6 Rennes p. 6/4 Gohberg-Sigal Theory Generalization of the argument principle to meromorphic operator-valued functions. Argument principle: Let f(z) be a meromorphic function and g(z) be a holomorphic function. Let z 0 be either a pole or a zero of f. If V z0 a complex neighborhood of z 0 does not contain any pole or zero other than z 0, then 2π V z0 g(z) f (z) f(z) dz = ±(order of z 0) g(z 0 ). zero characteristic value of A (φ(z) holomorphic at z 0,φ(z 0 ) 0,A(z)φ(z) holomorphic at z 0 and vanishes at this point; φ: root function)

7 Gohberg-Sigal Theory Let G, H be Banach spaces, and L(G, H) be the set of all bounded linear operators. A : V z0 L(G, H) is normal at z 0 if A is invertible and holomorphic in V z0 \ {z 0 } and finitely meromorphic of Fredholm type: A(z) = (z z 0 ) j B j + A 0 +(z z }{{} 0 )A + j=s,..., }{{} Fredholm finite dimensional V : simply connected bounded domain with rectifiable boundary V. A is normal with respect to V if A is invertible in V, except for a finite number of points of V which are normal points of A. Rennes p. 7/4

8 Generalized Argument Principle Theorem (Ghoberg-Sigal, 7). A(z): operator-valued function which is normal with respect to V ; g(z): scalar function which is holomorphic in V and continuous in V. Then 2π tr V g(z)a (z) d σ dz A(z) dz = M(A,z j )g(z j ), where z j, j =,,σ, are all the points in V which are either poles or characteristic values of A(z) and M(A,z j ) is the multiplicity of z j. j= tr: trace of a finite-dimensional operator; tr A(z)B(z)dz = tr B(z)A(z)dz. V V Rennes p. 8/4

9 Rennes p. 9/4 Layer Potentials Γ ω (x): fundamental solution to the Helmholtz operator + ω 2 (ω > 0) in R d,d = 2, 3, 4 H() 0 (ω x ), d = 2, Γ ω (x) = e ω x, d = 3, 4π x for x 0; H () 0 : Hankel function of the first kind of order 0. H () 0 : logarithmic behavior in ω.

10 Layer Potentials Ω: bounded Lipschitz domain in R d. The single- and the double-layer potentials: SΩ[ϕ](x) ω = Γ ω (x y)ϕ(y)dσ(y), x R d, Ω DΩ[ϕ](x) ω Γ ω (x y) = ϕ(y)dσ(y), x R d \ Ω, ν(y) for ϕ L 2 ( Ω). Ω KΩ ω: the singular integral operator defined for ϕ L2 ( Ω) by KΩ[ϕ](x) ω Γ ω (x y) = p.v. ϕ(y)dσ(y). ν(y) Ω Rennes p. 0/4

11 General Framework If ( + ω 2 )u = 0 in Ω u = DΩ(u ω Ω ) SΩ( ω u ) in Ω. ν If u is an eigenfunction of on Ω with Neumann boundary conditions (using trace formulas) ( 2 I + Kω Ω)(u Ω ) = 0 on Ω. Characterization of the eigenvalues as characteristic values of an operator-valued function. Gohberg-Sigal theory for meromorphic operator-valued functions + layer potential techniques asymptotic theory of eigenvalue problems. Rennes p. /4

12 General Framework { u + ω 2 u = 0 in Ω R 3, u = 0 on Ω. Suppose that Ω contains a small hole D = z + ɛb. Impose Dirichlet boundary conditions on D. How can we calculate the eigenvalue perturbations due to the small hole? (with rigorous proofs) Rennes p. 2/4

13 General Framework ω 2 j 0 : simple eigenvalue (with associated normalized eigenfunction u j0 ). Due to the small hole D, ω j0 ω ɛ. ω ɛ corresponds exactly to a characteristic value of a family of operator-valued functions A ɛ (ω) in some complex neighborhood V(ω j0 ), where ω A ɛ (ω) is a holomorphic operator. Use the generalized argument principle for A ɛ (ω) in ɛ to get an exact formula for the perturbed eigenvalue. Rennes p. 3/4

14 General Framework Exact formula: ω ɛ ω j0 = 2π tr V(ω j0 ) (ω ω j0 ) A ɛ (ω) (ω) d dω A ɛ(ω) dω Expand A ɛ (ω) in ɛ uniformly in ω V(ω j0 ) (difficult part). Derive a full asymptotic expansion of ω ɛ as ɛ 0. Asymptotic formula valid for all j 0 such that ω j0 ɛ. Rennes p. 4/4

15 Perturbations due to a hole Theorem 2. (d = 3) Assume that ωj 2 0 is simple. Up to O(ɛ 3 ): ωɛ 2 ωj 2 0 = ɛc B u j0 (z) 2 ɛ 2 α u j0 (z)u j0 (z)mb+c α Bu 2 j0 (z) 2 w j0 (z), C B = MB α = B B w j0 explicit function: α = (S 0 B) ()dσ (capacity), y α (S 0 B) () + (S 0 B) (y α ) dσ w j0 (z) = lim x z ( ) u i (z)u i (x) ω 2 i j 0 i + Γ ωj0 (x z) ω2 j 0 (polarization tensor),. Rennes p. 5/4

16 Perturbations due to a hole In connection with this formula: Ozawa (leading-order terms for a circular or spherical inclusion), Besson (existence of a complete asymptotic in 2D), Courtois (perturbation theory for the Dirichlet spectrum in a compactly perturbed domain in terms of the capacity of the compact perturbation),... Hadamard s formula: Kato, Garabedian-Schiffer, Sanchez Hubert-Sanchez Palencia, Kozlov (nonsmooth boundaries),... Rennes p. 6/4

17 Perturbations due to an inclusion Suppose that Ω contains a small inclusion D = z + ɛb with material parameter 0 < k < +. Consider the eigenvalue problem: ( + (k )χ(d)) u ɛ + ωɛu 2 ɛ = 0 in Ω, u ɛ ν = 0 Asymptotic formula: on Ω. ω 2 ɛ ω 2 j 0 = ɛ d u j0 (z)m(k,b) u j0 (z) + o(ɛ d ). M(k, B): polarization tensor. Perturbation of order the volume (while it is of order the characteristic size for a hole): well-known fact in low-frequency electromagnetics. Rennes p. 7/4

18 Perturbations due to an inclusion Asymptotic formulas for u ɛ u j0 : inner and outer expansions. The inner expansion holds for x near z: u ɛ (x) = u j0 (z) + ɛ d l= l u j0 (z)ψ l ( x z ɛ ψ l defined by { ( + (k )χ(b)) ψl = 0 in R d, ) + o(ɛ), ψ l (x) x l = O( x d ) as x +. Rennes p. 8/4

19 Perturbations due to an inclusion The outer expansion holds uniformly for x Ω: (u ɛ u j0 )(x) = ɛ d u j0 (z) M(k,B) N ω j 0 Ω (x,z) + o(ɛd ), The Neumann function N ω j 0 Ω given ( x + ωj 2 0 )N ω j 0 Ω (x,y) = δ y + u j0 (x)u j0 (y) in Ω, N ω j 0 Ω = 0 on Ω, ν N ω j 0 Ω u j 0 = 0. Ω Entries of M(k, B) given by (k ) B ψ l ν x l dσ. Rennes p. 9/4

20 Perturbations due to an inclusion If one wants to reconstruct a small inclusion D = z + ɛb from variations of the spectral parameters (ω ɛ ω j0, (u ɛ u j0 ) ) Ω, then only the location z and the matrix M(k,B) can be reconstructed. M(k,B): characterizes all the information about the anomaly that can be learned from spectral measurements (eigenvalue expansions + outer expansion of the eigenvectors). The use of the inner expansion would yield to better reconstruction: it contains high-frequency information. Rennes p. 20/4

21 Perturbations due to an inclusion M(k,B): mixture of the material parameter k and low-frequency geometric information. Separate material parameter/size: Higher-order terms in the asymptotic expansion. Separate material parameter/size: Requires very sensitive measurement system. Imaging motivates the study of the concept of polarization tensors. Rennes p. 2/4

22 Polarization Tensor Properties of the polarization tensor: (i) M is symmetric. (ii) If k >, then M is positive definite, and it is negative definite if 0 < k <. (iii) Hashin-Shtrikman bounds: k trace(m) (d + k ) B, (k ) trace(m ) d + k B Optimal size estimates; Thickness estimates; Pólya Szegö conjecture. Lipton, Capdeboscq-Vogelius, Capdeboscq-Kang, Kang-Milton.. Rennes p. 22/4

23 Polarization Tensor Visualization of PT (k,k )=(.5,.5) (k,k )=(.5,3.0) (k,k )=(.5,5.0) 2 Figure : When the two disks have the same radius and the conductivity of the one on the right-hand side is increasing, the equivalent ellipse is moving toward the right anomaly. (E. Kim) Rennes p. 23/4

24 Polarization Tensor Visualization of PT (r,r 2 )=(0.2,0.2) (r,r 2 )=(0.2,0.4) (r,r 2 )=(0.2,0.8) Figure 2: When the conductivities of the two disks is the same and the radius of the disk on the right-hand side is increasing, the equivalent ellipse is moving toward the right anomaly. (E. Kim) Rennes p. 24/4

25 Polarization Tensor - with H. Kang, Polarization and Moment Tensors: With Applications in Imaging and Effective Medium Theory, Applied Mathematical Sciences, Vol. 62, Springer, New York, Rennes p. 25/4

26 Perturbations due to a Crack Consider a perfectly conducting linear crack of size 2ɛ located at z. Use Finite-Hilbert transform. Asymptotic formula: [ ωɛ 2 ωj 2 0 = 2π ln ɛ + C u j 0 (z) 2 + 2ɛ 2 u ] j 0 T 2 (z) + o(ɛ 2 ), C depends on Ω and ω j0. T : tangential vector to the crack. Harder to find the direction of a crack than its location. Rennes p. 26/4

27 Perturbations due to Corrosion Consider D Ω. D is corroded, where the corroded part I is of size ɛ and located at z. The internal corrosion is modelled by a change of boundary conditions on D (from Neumann to Robin boundary conditions). Asymptotic formula: γ: the corrosion coefficient. ω 2 ɛ ω 2 j 0 = ɛγ u j0 (z) 2 + O(ɛ 2 ). One can not separate the corrosion coefficient from the size of the corroded part. Rennes p. 27/4

28 Other Extensions Multiple eigenvalue: splitting problem (with F. Triki, JDE 04) Linear elasticity (with H. Kang and H. Lee, CPDE 07 & E. Beretta, E. Francini, H. Kang, and M. Lim, preprint 09) Band-Gap calculations in photonic and phononic crystals: sensitivity analysis of the band-gaps with respect to geometry and material parameters (with H. Kang, S. Soussi, and H. Zribi, SIAM MMS 06 & H. Kang and H. Lee, ARMA 09) Numerical methods based on integral equations and Muller s method for finding complex roots of a function for bang-gap calculations (L. Greengard) Rennes p. 28/4

29 Applications to Imaging Characterize all the defect s features we can reconstruct. "Stable reconstruction: problem reduction + principle component analysis". "Remove the ill-posedness of the inverse problem". Rennes p. 29/4

30 Applications to Imaging Minimize the difference between the measured and the computed data. Minimize the difference between the measured data and the asymptotic expansion. Duality approach. Rennes p. 30/4

31 Imaging of Inclusions To reconstruct the inclusion D, a standard approach: min x Ω,M the residual ω ɛ ω j0 l.o.t. 2 + u ɛ u j0 l.o.t 2 Duality approach: introduce w g to be a solution to ( + ωj 2 0 )w g = 0 in Ω, w g ν = g on Ω, ( Ω gu j 0 = 0). Asymptotics of the eigenvalue perturbations M u j0 (z) w g (z) M u j0 (z) u j0 (z) = g(u ωɛ 2 ωj 2 ɛ u j0 ) + 0 Ω Ω Ω u j0 w g + o(ɛ) Rennes p. 3/4

32 Imaging of Inclusions Duality approach for reconstructing D: Finite number of linearly independent functions g,...,g l, on Ω ( g Ω ju j0 dσ = 0). The method for detecting the inclusion: l min J(x,M) := x Ω,M j= M u j0 (x) w gj (x) M u j0 (x) u j0 (x) leading-order term 2, over symmetric M satisfying the Hashin-Shtrikman optimal bounds. Rennes p. 32/4

33 Numerical Examples k= k = Figure 3: Reconstruction of a small inclusion. (E. Kim) Rennes p. 33/4

34 Imaging of Shape Deformations ωj 2 0 eigenvalue (associated with normalized eigenfunction u j0 ): u + ω 2 u = 0 in Ω \ D, u + ω2 u = 0 in D, k u + u = 0, u k u = 0 on D, ν + ν u ν = 0 on Ω. D ɛ (ɛ-perturbation of D): { D ɛ = x : x = x + ɛh(x)ν(x), x D }. Rennes p. 34/4

35 Imaging of Shape Deformations Eigenvalue perturbations: [ ωɛ 2 ωj 2 0 = ɛ(k ) h k( u j 0 ν ) 2 + ( u ] j 0 T )2 dσ + O(ɛ 2 ). D Eigenvector boundary perturbations: u ɛ u j0 = ɛ(k )v j0 + o(ɛ), uniformly on Ω, where [ ω j0 uj0 v j0 (x) = h(y) T (y) N Ω,D T (x,y) D +k u j 0 ν (y) N ωj0 Ω,D ν ] (x,y) dσ(y) Rennes p. 35/4

36 Imaging of Shape Deformations The Neumann function N ω j 0 Ω,D given by ) ( ( + (k )χ(d)) + ω 2j0 N ω j 0 Ω,D (x,y) = δ y +u j0 (x)u j0 (y) in Ω, N ω j 0 Ω,D Ω ν = 0 on Ω, N ω j 0 Ω,D u j 0 = 0. Rennes p. 36/4

37 Imaging of Shape Deformations Reconstruct a small deformation D ɛ of D from variations of the spectral parameters (ω ɛ ω j0, (u ɛ u j0 ) ) Ω : ( + (k )χ(d ɛ )) u ɛ + ωɛu 2 ɛ = 0 in Ω, u ɛ ν = 0 on Ω. Duality approach: g L 2 ( Ω) ( gu Ω j 0 dσ = 0), w g : a solution to ( (χ(ω \ D) + kχ(d)) + ωj 2 0 )w g = 0 in Ω, w g ν = g on Ω, w g u j0 0. Ω Rennes p. 37/4

38 Imaging of Shape Deformations Standard approach: min x Ω,M the residual ω ɛ ω j0 l.o.t. 2 + u ɛ u j0 l.o.t 2 Minimization of quadratic energies isn t optimal: [ h k( u j 0 ν ) 2 + ( u ] j 0 T )2 dσ. D Duality approach: g(u ɛ u j0 ) + (ωj 2 0 ωɛ) 2 Ω = ɛ( k) D ( uj0 h T Ω Ω w g u j0 w g T + k u j 0 ν ) w g ν + O(ɛ 2 ). Rennes p. 38/4

39 Imaging of Shape Deformations To have less filtering: Duality approach for reconstructing a shape deformation: Asymptotics of the eigenvalue perturbations minimize over h: l J[h] := g j (u ɛ u j0 ) + (ωj 2 0 ωɛ) 2 w gj u j0 j= Ω ɛ( k) D ( uj0 h T ) w gj T + k u j 0 ν w gj 2 ν. Functionals J[h] and J(x, M): "quadratic forms" Minimization of "projections on subspaces" or "quadratic forms" and not "quadratic energies". Ω Rennes p. 39/4

40 Imaging of Shape Deformations Higher the oscillations in ( ) uj0 w gj p j := T T + k u j 0 ν w gj ν better the reconstruction. on D Rennes p. 40/4

41 Imaging of Shape Deformations Optimal choice of the functions (= g,...,g l )? Consider Λ(D) : g L 2 ( Ω) p L 2 ( D). Choose basis of the image space of Λ(D). Singular-value decomposition of (Λ (D)Λ(D)). w g : solution to a Helmholtz equation (with frequency ω j0 ) filtering effect "kills" all the high-frequency oscillations in p. (SNR: finite) Trade-off between resolution and stability. Rennes p. 4/4

42 Imaging of Shape Deformations Optimal representation of the changes h? Write h = l α j p j j= on D for some coefficients α j, where (p j ) is the optimal basis. To reconstruct the shape deformation h, compute α j. Resolution limit? Use Shannon s sampling theorem: δres 2π max j p D j 2 D p j T 2. Rennes p. 42/4

43 Imaging of Shape Deformations δres measures the maximum of oscillations in the basis functions p j. δres is function of ω j0 and the distance between D and Ω. Incomplete eigenvector boundary measurements: boundary measurements of the eigenvector on Γ 2 = Ω \ Γ. Introduce Λ local : g L 2 ( Ω),g = 0 on Γ p L 2 ( D). Similar approach to the case with complete eigenvector boundary measurements. Rennes p. 43/4

44 Numerical Examples Figure 4: Reconstruction of shape deformations. (M. Lim) Rennes p. 44/4

45 Conclusion Layer potential techniques: powerful for eigenvalue perturbation problems. Asymptotic theory of imaging: demystifies regularization. Provide optimal representation of unknown shape deformations Important issues: quantify the resolution limit and understand the trade-off between resolution and stability. Rennes p. 45/4