The inverse conductivity problem with power densities in dimension n - PowerPoint PPT Presentation

The inverse conductivity problem with power densities in dimension n ≥ 2 Fran¸ cois Monard Guillaume Bal Dept. of Applied Physics and Applied Mathematics, Columbia University. June 19th, 2012 UC Irvine Conference in honor of Gunther Uhlmann

Outline 1 Preliminaries 2 Local reconstructions Scalar factor Anisotropic structure 3 Admissible sets and global reconstruction schemes

Preliminaries The inverse conductivity (diffusion) problem The model: X ⊂ R n bounded domain, n ≥ 2. • Calder´ on’s problem: u | ∂ X = g (prescribed) Does Λ γ determine γ uniquely ? stably ? � � ∇ · ( γ ∇ u ) ≡ � n γ ij ∂ j u i , j =1 ∂ i = 0 [Calder´ on ’80] X • Power density problem: Does H γ determine γ H γ ( g ) = ∇ u · γ ∇ u (power density) uniquely ? stably ? Application: EIT or OT coupled with acoustic waves. Λ γ ( g ) := ν · γ ∇ u | ∂ X (Dirichlet-to Neumann) γ is uniformly elliptic . 2 / 19

Preliminaries Derivation of power densities - 1/2 By ultrasound modulation focused wave "on" c = δ ( x − x 0 ) Physical focusing [Ammari et al. ’08] ∇ · ( σ (1 + ǫ c ) ∇ u ǫ ) = 0 Synthetic focusing X [Kuchment-Kunyansky ’10] u ǫ | ∂ X = g [Bal-Bonnetier-M.-Triki ’11] M ǫ = σ (1 + ǫ c ) ∂ u ǫ ∂ n | ∂ X Small perturbation model: ( M ǫ −M 0 ) gives an approximation of ∇ u 0 · γ ∇ u 0 at x 0 . ǫ 3 / 19

Preliminaries Derivation of power densities - 2/2 By thermoelastic effects (Impedance-Acoustic CT) 1: voltage is prescribed at ∂ X u | ∂ X = g ( x ) δ ( t ) 2: currents are generated inside the domain ∇ · ( γ ∇ u ) = 0 X 3: the energy absorbed generates elastic waves ∂ 2 p 1 ∂ t 2 − ∆ p = 0 v 2 s p | t =0 = Γ H γ [ g ] ∂ t p | t =0 = 0 4: waves are measured at ∂ X by ultrasound transducers One reconstructs Γ H γ = Γ ∇ u · γ ∇ u over X (Γ: Gr¨ uneisen coefficient) [Gebauer-Scherzer ’09] 4 / 19

Preliminaries Power density measurements - References Resolution of the power density problem: 2D isotropic ’09] . [Capdeboscq et al. 2D-3D isotropic linearized [Kuchment-Kunyansky ’11] . 2D-3D isotropic [Bal-Bonnetier-M.-Triki,IPI ’12] . n -D isotropic and measurements of the form H ij = σ 2 α ∇ u i · ∇ u j [M.-Bal,IPI ’12] . 2D anisotropic: reconstruction formulas, stability and numerical implementation [M.-Bal,IP ’12] . Pseudodifferential calculus on the linearized isotropic case [Kuchment-Steinhauer,’12] . n -D anisotropic [M., Ph.D. thesis ’12] 5 / 19

Preliminaries Power density measurements - References The zero-Laplacian problem: Reconstruct a scalar conductivity γ from knowledge of one power density H = γ |∇ u | 2 . This yields the non-linear PDE ∇ · ( H |∇ u | − 2 ∇ u ) = 0 u | ∂ X = g . ( X ) , Hyperbolic equation nicknamed the zero-Laplacian . References: Newton-based numerical methods to recover ( u , γ ) ’08, Gebauer-Scherzer ’09] . [Ammari et al. Theoretical work on the Cauchy problem [Bal ’11] . 6 / 19

Preliminaries Resolution - Overview Problem: Reconstruct γ from H ij = ∇ u i · γ ∇ u j with ∇ · γ ∇ u i = 0 ( X ) , u i | ∂ X = g i , 1 ≤ i ≤ m . 1 n ˜ Decompose γ = (det γ ) γ with det ˜ γ = 1. We accept redundancies of data (no limitation on m a priori). Outline: Local reconstruction algorithms (and their conditions of validity) of det γ from known anisotropic structure ˜ γ of the anisotropic structure ˜ γ Global questions: study of admissible boundary conditions study of reconstructible tensors 7 / 19

Outline 1 Preliminaries 2 Local reconstructions Scalar factor Anisotropic structure 3 Admissible sets and global reconstruction schemes

Local reconstructions Scalar factor The frame approach, local reconstruction of det γ Differential geometric setup: Euclidean metric and connection ∇ . Frame condition: Let n conductivity solutions such that ( ∇ u 1 , . . . , ∇ u n ) is a frame over some Ω ⊂ X . 1 1 n � A with det � 2 = (det A ) Def: A := γ A = 1. Set S i := A ∇ u i . Data is H ij = ∇ u i · γ ∇ u j = S i · S j and S i solves: A − 1 S i ) ♭ = F ♭ ∧ ( � 1 ∇· ( � AS i ) = − F · � d ( � A − 1 S i ) ♭ , n . AS i , F := ∇ log(det A ) � � 1 1 2 H ij ) · � A − 1 S j by studying � We first derive F = ∇ ( | H | AS i 1 n | H | 2 the behavior of the dual frame to ( � A − 1 S 1 , . . . , � A − 1 S n ). Legend: known data, unknown, anisotropic structure (known here). 8 / 19

Local reconstructions Scalar factor Local reconstruction of det γ A first-order quasi-linear system is then derived for the frame S ∇ S i = H kq H jp ( ∇ � AS q S i · S p ) S j ⊗ ( � A − 1 S k ) ♭ , where AS q S i · S p = � AS q · ∇ H ip + � AS p · ∇ H iq − � AS i · ∇ H pq + 2 H pq F · � AS i − 2 H qi F · � 2 ∇ � AS p − A � A ( S q , S p ) · S i − A � A ( S i , S p ) · S q + A � A ( S q , S i ) · S p . In short, ∇ S i = S i ( S , � A , d � A , H , dH ) , 1 ≤ i ≤ n , where S i is Lipschitz w.r.t. ( S 1 , . . . , S n ). Then, ∇ log det γ = F ( S , � A , H , dH ) . ◮ Overdetermined PDEs, solvable for S and log det γ over Ω ⊂ X via ODE’s along any characteristic curves. 9 / 19

Local reconstructions Scalar factor Local reconstruction of det γ Theorem (Uniqueness and Lipschitz stability in W 1 , ∞ (Ω)) Over Ω ⊂ X where the frame condition is satisfied, det γ is uniquely determined up to a (multiplicative) constant. Moreover, � log det γ − log det γ ′ � W 1 , ∞ ≤ ε 0 + C ( � H − H ′ � W 1 , ∞ + � � A − � A ′ � W 1 , ∞ ) , where ε 0 is the error commited at some x 0 ∈ Ω . [Capdeboscq et al. ’09], [Bal-Bonnetier-M.-Triki, ’12], [M.-Bal, IP ’12], [M.-Bal, IPI ’12] ◮ Well-posed problem if the anisotropy is known. ◮ No loss of derivative/resolution on | γ | . 10 / 19

Local reconstructions Anisotropic structure Anisotropy reconstruction - derivation - 1/2 Goal: Reconstruct ˜ γ from enough functionals H ij = ∇ u i · γ ∇ u j . • Start from a frame of conductivity solutions ( ∇ u 1 , . . . , ∇ u n ) and consider an additional solution v . • Key fact: the decomposition of A ∇ v in the basis ( S 1 , . . . , S n ) is known from the power densities : A ∇ v = µ i S i , with µ i ( H ) known . • Using ∇ · ( AS i ) = 0 and d ( A − 1 S i ) ♭ = 0 , we obtain A − 1 S i ) ♭ = 0 , Z i · � i ∧ ( � Z ♭ Z i = ∇ µ i . AS i = 0 and Writing Z = [ Z 1 | . . . | Z n ], this is equivalent to ( � A , B � := tr ( AB T )) � � � � AS , Z � = 0 and AS , ZH Ω � = 0 , Ω ∈ A n ( R ) . 2 ) linear constraints on � This is 1 + r ( n − r +1 AS , where r = rank Z . 1 Equations: ∇ · ( γ ∇ u i ) = 0 ( X ) , u i | ∂ X = g i , A := γ S i = A ∇ u i 2 , 11 / 19

Local reconstructions Anisotropic structure Anisotropy reconstruction - derivation - 2/2 • Hyperplane condition: Assume that ( v 1 , . . . , v ℓ ) are so that Z (1) , . . . , Z ( ℓ ) yield n 2 − 1 independent constraints on � AS . • Reconstruct B = � AS via a generalization of the cross-product in M n ( R ). A 2 = BH − 1 B T , then S = ˜ γ − 1 γ = � 2 B (then det γ ). • Reconstruct ˜ Theorem (Uniqueness and stability for ˜ γ ) Over Ω ⊂ X where the frame condition and the hyperplane condition are satisfied, ˜ γ is uniquely determined, with stability γ ′ � L ∞ (Ω) ≤ C � H − H ′ � W 1 , ∞ ( X ) . � ˜ γ − ˜ [M.-Bal, IP ’12] in 2D. ◮ Explicit reconstruction . Loss of one derivative on ˜ γ . 12 / 19

Local reconstructions Anisotropic structure Anisotropy reconstruction - remark In the linearized case, one full-rank matrix Z (i.e. one well-chosen additional solution) yields a Fredholm inversion (requires the inversion of a strongly coupled elliptic system whose invertibility cannot always be established), although this is only 1 + n ( n − 1) 2 constraints. [Bal-M.-Guo ’12], in progress. 13 / 19

Outline 1 Preliminaries 2 Local reconstructions Scalar factor Anisotropic structure 3 Admissible sets and global reconstruction schemes

Admissible sets and global reconstruction schemes Admissible sets - the frame condition Question: How to fulfill the frame condition globally ? • Admissibility sets G m γ , m ≥ n : ( g 1 , . . . , g m ) ∈ G m γ if one can cover X with open sets Ω p with a frame made of ∇ u i ’s on each Ω p . expressible in terms of continuous functionals of the data ∇ u i · γ ∇ u j . ◮ det γ is reconstructible if G m γ � = ∅ for some m ≥ n . • Patching local ODE-based reconstructions: ∇ log det γ = F ( S , H , dH , � A ) , ∇ S i = S i ( S , H , dH , � A , d � A ) , 1 ≤ i ≤ n . 14 / 19

Admissible sets and global reconstruction schemes Admissible sets - the hyperplane condition Question: How to fulfill the hyperplane condition globally ? • The admissibility sets A m ,ℓ ( g ) for g ∈ G m γ : γ g provides a support basis throughout X . ( h 1 , . . . , h ℓ ) ∈ A m ,ℓ ( g ) if the hyperplane condition (expressible γ in terms of H ij and dH ij ) is satisfied throughout X . γ is reconstructible if A m ,ℓ ◮ ˜ ( g ) � = ∅ for some ℓ ≥ 1. γ 15 / 19