Some issues on Electrical Impedance Tomography with complex - PowerPoint PPT Presentation

Some issues on Electrical Impedance Tomography with complex coefficient Elisa Francini (Universit` a di Firenze) in collaboration with Elena Beretta and Sergio Vessella E. Francini (Universit` a di Firenze) EIT 1 / 42

Outline 1 Electrical Impedance Tomography with complex coefficient 2 Reconstruction from the DN map: a stability result. 3 Size estimate from a single boundary measurement. E. Francini (Universit` a di Firenze) EIT 2 / 42

Electrical Impedance Tomography Electrical Impedance Tomography: Determine electrical conductivity and permittivity in the interior of a body by measurements of electric currents and voltages at the boundary of the body. Medical diagnostics: detection of pulmonary emboli, breast tumours, screening of organs in transplant surgery Nondestructive testing of materials: detection of corrosion or cracks Geophysical prospection: detection of underground mineral deposits E. Francini (Universit` a di Firenze) EIT 3 / 42

Mathematical model γ : Ω ⊂ R n → C complex admittivity function γ = σ + i ωǫ electrical conductivity frequency electrical permittivity σ ≥ λ − 1 > 0 (dissipation of energy) , | γ | ≤ λ The potential u ∈ H 1 (Ω , C ) is a weak solution to equation div ( γ ∇ u ) = 0 in Ω . Boundary data: u | ∂ Ω and γ ∂ u ∂ν | ∂ Ω . E. Francini (Universit` a di Firenze) EIT 4 / 42

Mathematical model γ : Ω ⊂ R n → C complex admittivity function γ = σ + i ωǫ electrical conductivity frequency electrical permittivity σ ≥ λ − 1 > 0 (dissipation of energy) , | γ | ≤ λ The potential u ∈ H 1 (Ω , C ) is a weak solution to equation div ( γ ∇ u ) = 0 in Ω . Boundary data: u | ∂ Ω and γ ∂ u ∂ν | ∂ Ω . E. Francini (Universit` a di Firenze) EIT 4 / 42

Mathematical model γ : Ω ⊂ R n → C complex admittivity function γ = σ + i ωǫ electrical conductivity frequency electrical permittivity σ ≥ λ − 1 > 0 (dissipation of energy) , | γ | ≤ λ The potential u ∈ H 1 (Ω , C ) is a weak solution to equation div ( γ ∇ u ) = 0 in Ω . Boundary data: u | ∂ Ω and γ ∂ u ∂ν | ∂ Ω . E. Francini (Universit` a di Firenze) EIT 4 / 42

Part I Reconstruction from infinite boundary measurements E. Francini (Universit` a di Firenze) EIT 5 / 42

DN map Given f ∈ H 1 / 2 ( ∂ Ω) there exists a unique weak solution u ∈ H 1 (Ω) of the problem � div ( γ ∇ u ) = 0 in Ω = on ∂ Ω . u f Consider the Dirichlet to Neumann map Λ γ : H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) given by Λ γ f = γ ∂ u , ∂ν | ∂ Ω where ν is the exterior unit normal vector to ∂ Ω. EIT Inverse Problem: Determine γ from the knowledge of the Dirichlet-to Neumann map Λ γ E. Francini (Universit` a di Firenze) EIT 6 / 42

DN map Given f ∈ H 1 / 2 ( ∂ Ω) there exists a unique weak solution u ∈ H 1 (Ω) of the problem � div ( γ ∇ u ) = 0 in Ω = on ∂ Ω . u f Consider the Dirichlet to Neumann map Λ γ : H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) given by Λ γ f = γ ∂ u , ∂ν | ∂ Ω where ν is the exterior unit normal vector to ∂ Ω. EIT Inverse Problem: Determine γ from the knowledge of the Dirichlet-to Neumann map Λ γ E. Francini (Universit` a di Firenze) EIT 6 / 42

DN map Given f ∈ H 1 / 2 ( ∂ Ω) there exists a unique weak solution u ∈ H 1 (Ω) of the problem � div ( γ ∇ u ) = 0 in Ω = on ∂ Ω . u f Consider the Dirichlet to Neumann map Λ γ : H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) given by Λ γ f = γ ∂ u , ∂ν | ∂ Ω where ν is the exterior unit normal vector to ∂ Ω. EIT Inverse Problem: Determine γ from the knowledge of the Dirichlet-to Neumann map Λ γ E. Francini (Universit` a di Firenze) EIT 6 / 42

Known results Uniqueness n ≥ 3 If γ is sufficiently smooth the results obtained by Sylvester and Uhlmann (1987, Ann. of Math.) for the conductivity case can be extended to the complex case. (Λ γ 1 = Λ γ 2 implies γ 1 = γ 2 ) n = 2 F.(2000, Inverse Problems) for small frequencies ω , Bukhgeim for arbitrary frequencies and smooth coefficients (2008, Inv. Ill Posed Problems). n ≥ 3 , γ ∈ L ∞ (Ω) completely open arinta for real conductivities σ ∈ L ∞ (Ω) ( 2006, n = 2 Astala and P¨ aiv¨ Ann. of Math.) E. Francini (Universit` a di Firenze) EIT 7 / 42

Known results Stability n ≥ 3 Alessandrini (1988, Appl. Anal.) has proved for the conductivity case that if � σ j � W 2 , ∞ (Ω) ≤ E for j = 1 , 2, then � σ 1 − σ 2 � L ∞ (Ω) ≤ ω ( � Λ σ 1 − Λ σ 2 � ) where ω ( t ) = C | log t | − η . n = 2 Barcelo, Faraco and Ruiz (2007, Jour. Math. Pures Appl.) have proved for the conductivity case an analogous logarithmic estimate under the a priori assumption � σ � C α (Ω) ≤ E . Clop, Faraco, Ruiz (2009, IPI) for σ j ∈ W α, p , α > 0, j = 1 , 2, and 1 < p < ∞ then � σ 1 − σ 2 � L 2 (Ω) ≤ ω ( � Λ σ 1 − Λ σ 2 � ) where ω ( t ) = C | log t | − η . E. Francini (Universit` a di Firenze) EIT 8 / 42

A priori assumptions Mandache (2001, Inverse Problems) has proved that logarithmic stability is the best possible stability also using as a-priori assumption � σ � C k (Ω) ≤ E , ∀ k = 0 , 1 , 2 .... E. Francini (Universit` a di Firenze) EIT 9 / 42

Strategy: Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability Assume: γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.) E. Francini (Universit` a di Firenze) EIT 10 / 42

Strategy: Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability Assume: γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.) E. Francini (Universit` a di Firenze) EIT 10 / 42

Strategy: Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability Assume: γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.) E. Francini (Universit` a di Firenze) EIT 10 / 42

Main assumptions (H1) Ω ⊂ R n , bounded, | Ω | ≤ A , Lipschitz boundary (H2) γ ( x ) = � N j =1 γ j ✶ D j ( x ) unknown complex numbers known open sets D i ∩ D j = ∅ , i � = j , Lipschitz boundaries and ∪ N j =1 D j = Ω (H3) ∂ D 1 ∩ ∂ Ω contains an open flat portion Σ 1 . For every j ∈ { 2 , . . . , N } there exists a chain of subdomains connecting D 1 to D j so that consecutive domains share a flat portion of boundary E. Francini (Universit` a di Firenze) EIT 11 / 42

A possible configuration D j D 1 Σ 1 E. Francini (Universit` a di Firenze) EIT 12 / 42

Chain D j D 1 Σ 1 E. Francini (Universit` a di Firenze) EIT 13 / 42

Main result Theorem Let Ω satisfy assumption (H1) . Let γ ( k ) , k = 1 , 2 be two complex piecewise constant functions of the form N γ ( k ) γ ( k ) ( x ) = � ✶ D j ( x ) , j j =1 where γ ( k ) satisfy for k = 1 , 2 assumption (H2) and D j , j = 1 , . . . , N satisfy assumption (H3) . Then there exists a positive constant C such that � γ (1) − γ (2) � L ∞ (Ω) ≤ C � Λ γ (1) − Λ γ (2) � L ( H 1 / 2 ( ∂ Ω) , H − 1 / 2 ( ∂ Ω)) . The constant C depends on the Lipschitz constant of the boundaries, on the volume of Ω , on n, N and λ . [Beretta, F., Comm. PDE, 2011] E. Francini (Universit` a di Firenze) EIT 14 / 42

Main idea of the proof (uniqueness) Assume Λ γ (1) = Λ γ (2) � � � � γ (1) ∇ u 1 γ (2) ∇ u 2 div = 0 div = 0 in Ω ⇓ � � γ (1) − γ (2) � 0 = < (Λ γ (1) − Λ γ (2) ) u 1 , u 2 > = ∇ u 1 ∇ u 2 dx Ω We want to use this identity for ”singular” solutions E. Francini (Universit` a di Firenze) EIT 15 / 42

Main idea of the proof (uniqueness) Assume Λ γ (1) = Λ γ (2) � � � � γ (1) ∇ u 1 γ (2) ∇ u 2 div = 0 div = 0 in Ω ⇓ � � γ (1) − γ (2) � 0 = < (Λ γ (1) − Λ γ (2) ) u 1 , u 2 > = ∇ u 1 ∇ u 2 dx Ω We want to use this identity for ”singular” solutions E. Francini (Universit` a di Firenze) EIT 15 / 42

Main idea of the proof (uniqueness) Assume Λ γ (1) = Λ γ (2) � � � � γ (1) ∇ u 1 γ (2) ∇ u 2 div = 0 div = 0 in Ω ⇓ � � γ (1) − γ (2) � 0 = < (Λ γ (1) − Λ γ (2) ) u 1 , u 2 > = ∇ u 1 ∇ u 2 dx Ω We want to use this identity for ”singular” solutions E. Francini (Universit` a di Firenze) EIT 15 / 42

Main idea of the proof (uniqueness) Assume Λ γ (1) = Λ γ (2) � � � � γ (1) ∇ u 1 γ (2) ∇ u 2 div = 0 div = 0 in Ω ⇓ � � γ (1) − γ (2) � 0 = < (Λ γ (1) − Λ γ (2) ) u 1 , u 2 > = ∇ u 1 ∇ u 2 dx Ω We want to use this identity for ”singular” solutions E. Francini (Universit` a di Firenze) EIT 15 / 42