Regularizations for an all-at-once formulation of the electrical impedance tomography problem

Continuum model.

The equations of the continuum model for the electric potential u are

with Neumann boundary condition

if a current f is applied, or with Dirichlet boundary condition

if a voltage g is applied. Consider the closed subspaces ( 0 ≤ s ≤ 1 )

Formulation with applied current. Suppose that the current f ∈ H ⋄ - 1 / 2 ⁢ ( ∂ ⁡ Ω ) is applied at the boundary ∂ ⁡ Ω . Then the electric potential u ¯ ∈ H ⋄ 1 ⁢ ( Ω ) is obtained ( u ¯ is solution of (A.1) and (A.2)). The resulting voltage on ∂ ⁡ Ω is given by γ ⁢ u ¯ ∈ H ⋄ 1 / 2 ⁢ ( ∂ ⁡ Ω ) . In this case, the all-at-once formulation of the EIT inverse problem can be cast into the form of ($\boldsymbol{I}$) with the spaces X , Z = H ⋄ 1 ⁢ ( Ω ) , X ~ = H ⋄ 1 + s ⁢ ( Ω ) , the map A : L ∞ ⁢ ( Ω ) × X → Z ⋆ defined by

and the following choices for each type of observable data (recall that C : L ∞ ⁢ ( Ω ) × X → Y ):

1. Y = H 1 / 2 ⁢ ( ∂ ⁡ Ω ) , C ⁢ ( σ , u ) = γ ⁢ u , and y ¯ = γ ⁢ u ¯ ∈ H ⋄ 1 / 2 ⁢ ( ∂ ⁡ Ω ) ,

2. Y = L 2 ⁢ ( Ω ) , C ⁢ ( σ , u ) = σ ⁢ | ∇ ⁡ u | , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | ∈ L 2 ⁢ ( Ω ) ,

3. Y = L 1 ⁢ ( Ω ) , C ⁢ ( σ , u ) = σ ⁢ | ∇ ⁡ u | 2 , y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | 2 ∈ L 1 ⁢ ( Ω ) .

Formulation with applied voltage. Suppose that the voltage g ∈ H 1 / 2 ⁢ ( ∂ ⁡ Ω ) is applied at the boundary ∂ ⁡ Ω . Then, the electric potential u ¯ ∈ H 1 ⁢ ( Ω ) is obtained ( u ¯ is solution of (A.1) and (A.3)). The resulting current on ∂ ⁡ Ω is given by σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν ∈ H ⋄ - 1 / 2 ⁢ ( ∂ ⁡ Ω ) . Let w g ∈ H 1 ⁢ ( Ω ) with the property γ ⁢ w g = g in H 1 / 2 ⁢ ( ∂ ⁡ Ω ) . In this case, the all-at-once formulation of the EIT inverse problem can be cast into the form of ($\boldsymbol{I}$) with the spaces X , Z = H 0 1 ⁢ ( Ω ) , X ~ = H 0 1 + s ⁢ ( Ω ) , the map A : L ∞ ⁢ ( Ω ) × X → Z ⋆ defined by

and the following choices for each type of observable data (recall that C : L ∞ ⁢ ( Ω ) × X → Y ):

1. Y = ( H 1 ⁢ ( Ω ) ) * , C ⁢ ( σ , u ) ⁢ ( w ) ≔ ∫ Ω σ ⁢ ∇ ⁡ ( u + w g ) ⋅ ∇ ⁡ w ⁢ d ⁢ 𝐱 , and y ¯ = σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν ∘ γ ∈ { ϕ ∈ Y : ϕ ⁢ ( 1 ) = 0 } ,

2. Y = L 2 ⁢ ( Ω ) , C ⁢ ( σ , u ) = σ ⁢ | ∇ ⁡ ( u + w g ) | , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | ∈ L 2 ⁢ ( Ω ) ,

3. Y = L 1 ⁢ ( Ω ) , C ⁢ ( σ , u ) = σ ⁢ | ∇ ⁡ ( u + w g ) | 2 , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | 2 ∈ L 1 ⁢ ( Ω ) .

Alternative formulation. From the previous formulations, we have either the voltage-current pair ( γ ⁢ u ¯ , f ) ∈ H ⋄ 1 / 2 ⁢ ( ∂ ⁡ Ω ) × H ⋄ - 1 / 2 ⁢ ( ∂ ⁡ Ω ) (with u ¯ being solution of (A.1) and (A.2)) or ( g , σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν ) ∈ H 1 / 2 ⁢ ( ∂ ⁡ Ω ) × H ⋄ - 1 / 2 ⁢ ( ∂ ⁡ Ω ) (with u ¯ being solution of (A.1) and (A.3)). An alternative all-at-once formulation of the EIT inverse problem with the weak form of (A.1) as model equation is given in the form of ($\boldsymbol{I}$) with the spaces X = H 1 ⁢ ( Ω ) , Z = H 0 1 ⁢ ( Ω ) , Y = H 1 / 2 ⁢ ( ∂ ⁡ Ω ) × ( H 1 ⁢ ( Ω ) ) ⋆ , X ~ = H 1 + s ⁢ ( Ω ) , and the maps

Here there are two possibilities for the exact observation y ¯ :

1. y ¯ = ( γ ⁢ u ¯ , f ∘ γ ) ∈ H ⋄ 1 / 2 ⁢ ( ∂ ⁡ Ω ) × { ϕ ∈ ( H 1 ⁢ ( Ω ) ) * : ϕ ⁢ ( 1 ) = 0 } ( u ¯ is solution to (A.1) and (A.2)),

2. y ¯ = ( g , σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν ∘ γ ) ∈ H 1 / 2 ⁢ ( ∂ ⁡ Ω ) × { ϕ ∈ ( H 1 ⁢ ( Ω ) ) * : ϕ ⁢ ( 1 ) = 0 } ( u ¯ is solution to (A.1) and (A.3));

(i) for measured voltage-applied current and (ii) for applied voltage-measured current.

Shunt model.

The equations of the shunt model for the electric potential ( u , U ) are

with

if a current pattern 𝒞 = ( 𝒞 1 , … , 𝒞 M ) is applied, or with

if a voltage pattern 𝒱 = ( 𝒱 1 , … , 𝒱 M ) is applied. Consider the closed subspaces ( 0 ≤ s ≤ 1 )

Formulation with applied current. Suppose that the current pattern 𝒞 ∈ ℝ ⋄ M is applied through electrodes Γ 1 , … , Γ M . Then, the electric potential ( u ¯ , U ¯ ) ∈ ℋ ⋄ 1 is obtained ( ( u ¯ , U ¯ ) is solution of (A.4)–(A.6), and (A.7)). The resulting voltage on the electrodes is given by U ¯ ∈ ℝ ⋄ M . In this case, the all-at-once formulation of the EIT inverse problem can be cast into the form of ($\boldsymbol{I}$) with the spaces X , Z = ℋ ⋄ 1 , X ~ = ℋ ⋄ 1 + s , the map A : L ∞ ⁢ ( Ω ) × X → Z ⋆ defined by

and the following choices for each type of observable data (recall that C : L ∞ ⁢ ( Ω ) × X → Y ):

1. Y = ℝ M , C ⁢ ( σ , ( u , U ) ) = U , and y ¯ = U ¯ ∈ ℝ ⋄ M ,

2. Y = L 2 ⁢ ( Ω ) , C ⁢ ( σ , ( u , U ) ) = σ ⁢ | ∇ ⁡ u | , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | ∈ L 2 ⁢ ( Ω ) ,

3. Y = L 1 ⁢ ( Ω ) , C ⁢ ( σ , ( u , U ) ) = σ ⁢ | ∇ ⁡ u | 2 , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | 2 ∈ L 1 ⁢ ( Ω ) .

Formulation with applied voltage. Suppose that the voltage pattern V ∈ ℝ M is applied through electrodes Γ 1 , … , Γ M . Then, the electric potential u ¯ ∈ H 1 ⁢ ( Ω ) is obtained ( ( u ¯ , V ) is solution of (A.4)–(A.6), and (A.8)). The resulting current on the electrodes is given by ( 〈 σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν , γ ⁢ e m 〉 ) m = 1 M ∈ ℝ ⋄ M , where e 1 , … , e M are functions in H 1 ⁢ ( Ω ) with the property γ m ⁢ e m = 1 and γ m ′ ⁢ e m = 0 for m ′ ≠ m . Let w 𝒱 ∈ H 1 ⁢ ( Ω ) with the property ( γ m ⁢ w 𝒱 ) m = 1 M = 𝒱 . In this case, the all-at-once formulation of the EIT inverse problem can be cast into the form of ($\boldsymbol{I}$) with the spaces X , Z = ℋ 0 1 , X ~ = ℋ 0 1 + s , the map A : L ∞ ⁢ ( Ω ) × X → Z ⋆ defined by

and the following choices for each type of observable data (recall that C : L ∞ ⁢ ( Ω ) × X → Y ):

1. Y = ℝ M , C ⁢ ( σ , u ) = ( ∫ Ω σ ⁢ ∇ ⁡ ( u + w 𝒱 ) ⋅ ∇ ⁡ e m ⁢ d ⁢ 𝐱 ) m = 1 M , and y ¯ = ( 〈 σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν , γ ⁢ e m 〉 ) m = 1 M ∈ ℝ ⋄ M ,

2. Y = L 2 ⁢ ( Ω ) , C ⁢ ( σ , u ) = σ ⁢ | ∇ ⁡ ( u + w 𝒱 ) | , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | ∈ L 2 ⁢ ( Ω ) ,

3. Y = L 1 ⁢ ( Ω ) , C ⁢ ( σ , u ) = | ∇ ⁡ ( u + w 𝒱 ) | 2 , and y ¯ = σ ¯ ⁢ | ∇ ⁡ u ¯ | 2 ∈ L 1 ⁢ ( Ω ) .

Alternative formulation. From the previous formulations, we have either the voltage-current pair ( U ¯ , 𝒞 ) ∈ ℝ ⋄ M × ℝ ⋄ M (with ( u ¯ , U ¯ ) being solution of (A.4)–(A.6), and (A.7)) or ( 𝒱 , ( 〈 σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν , γ ⁢ e m 〉 ) m = 1 M ) ∈ ℝ M × ℝ ⋄ M (with u ¯ being solution of (A.4)–(A.6), and (A.8)). An alternative all-at-once formulation of the EIT inverse problem with the weak form of (A.4)–(A.6) as model equation is given in the form of ($\boldsymbol{I}$) with the spaces X = ℋ 1 , Z = ℋ 0 1 , Y = ℝ M × ℝ M , X ~ = ℋ 1 + s , and the maps

Here there are two possibilities for the exact observation y ¯ :

1. y ¯ = ( U ¯ , 𝒞 ) ∈ ℝ ⋄ M × ℝ ⋄ M ( ( u ¯ , U ¯ ) is solution of (A.4)–(A.6), and (A.7)),

2. y ¯ = ( 𝒱 , ( 〈 σ ¯ ⁢ ∂ ⁡ u ¯ ∂ ⁡ ν , γ ⁢ e m 〉 ) m = 1 M ) ∈ ℝ M × ℝ ⋄ M ( u ¯ is solution of (A.4)–(A.6), and (A.8));

(i) for measured voltage-applied current and (ii) for applied voltage-measured current.

Gap model.

The equations of the gap model for the electric potential ( u , U ) are

with σ ¯ ⁢ ∂ ⁡ u ∂ ⁡ ν | Γ m = 𝒞 m | Γ m | for m = 1 , … , M if a current pattern 𝒞 = ( 𝒞 1 , … , 𝒞 M ) is applied, or with U m = 𝒱 m for m = 1 , … , M if a voltage pattern 𝒱 = ( 𝒱 1 , … , 𝒱 M ) is applied. The same instances presented for the shunt model work here, but with the subspaces

the functions e 1 , … , e M with the property ∫ Γ m γ m ⁢ e m ⁢ d 𝐬 = | Γ m | and γ m ′ ⁢ e m = 0 for m ′ ≠ m , and a function w 𝒱 with the property ( 1 | Γ m | ⁢ ∫ Γ m γ m ⁢ w 𝒱 ⁢ d 𝐬 ) m = 1 M = 𝒱 .

Smoothened complete electrode model.

In [19] was proposed the smoothened complete electrode model, which replaces the contact impedances of the complete electrode model with contact admittance functions capable to vanish on some subsets of the electrodes. It can be said that the contact admittances are represented by functions ζ 1 , … , ζ M satisfying ζ m ∈ L ∞ ⁢ ( ℰ m ) , ζ m ≥ 0 a.e. on ℰ m , and ζ m ≢ 0 . Therefore, it suffices to replace the contact impedances z 1 , … , z M by ζ 1 , … , ζ M in the instances that were proposed for the complete electrode model.

Model map with applied current.

In this case, x = ( u , U ) and X = Z = H 1 ⁢ ( Ω ) × ℝ ⋄ M . The first order approximation A k : L ∞ ⁢ ( Ω ) × X → Z ⋆ of A at ( σ k , ( u k , U k ) ) is given by

with Q : X → Z ⋆ , S : L ∞ ⁢ ( Ω ) → Z ⋆ , φ ∈ Z ⋆ defined by

Model map with applied voltage.

In this case, x = u and X = Z = H 1 ⁢ ( Ω ) . The first order approximation A k : L ∞ ⁢ ( Ω ) × X → Z ⋆ of A at ( σ k , u k ) is given by

with Q : X → Z ⋆ , S : L ∞ ⁢ ( Ω ) → Z ⋆ , φ ∈ Z ⋆ defined by

Alternative model map.

In this case, x = ( u , U ) , X = H 1 ⁢ ( Ω ) × ℝ ⋄ M , and Z = H 1 ⁢ ( Ω ) . The first order approximation A k : L ∞ ⁢ ( Ω ) × X → Z ⋆ of A at ( σ k , ( u k , U k ) ) is given by

with Q : X → Z ⋆ , S : L ∞ ⁢ ( Ω ) → Z ⋆ , φ ∈ Z ⋆ defined by

For magnitudes of current density field and interior power density data the first order approximations are function of ( σ , u ) in all cases: σ ⁢ | ∇ ⁡ u | is approximated with ( σ , u ) ↦ σ ⁢ | ∇ ⁡ u k ⁢ | + σ k | ⁢ ∇ ⁡ u k | - 1 ⁢ ∇ ⁡ u k ⋅ ∇ ⁡ u - σ k ⁢ | ∇ ⁡ u k | , and σ ⁢ | ∇ ⁡ u | 2 is approximated with ( σ , u ) ↦ σ ⁢ | ∇ ⁡ u k | 2 + 2 ⁢ σ k ⁢ ∇ ⁡ u k ⋅ ∇ ⁡ u - 2 ⁢ σ k ⁢ | ∇ ⁡ u k | 2 .