Nonlinear Integral Equations for the Inverse Problem in Corrosion - PowerPoint PPT Presentation

Intoduction Uniqueness Reconstruction Literature Nonlinear Integral Equations for the Inverse Problem in Corrosion Detection from Partial Cauchy Data Fioralba Cakoni Department of Mathematical Sciences, University of Delaware email: cakoni@math.udel.edu Jointly with R. Kress and C. Schuft Research supported by grants from AFOSR and NSF

Intoduction Uniqueness Reconstruction Literature Formulation of the Problem ∆ u = 0 in D u = f on Γ a ∂ u ∂ν + λ u = 0 on Γ c We assume that D has Lipshitz boundary ∂ D such that ∂ D = Γ a ∪ Γ c and λ ( x ) ≥ 0 is in L ∞ (Γ c ) . If f ∈ H 1 / 2 (Γ a ) this problem has a unique solution u ∈ H 1 ( D )

Intoduction Uniqueness Reconstruction Literature The Inverse Problem The inverse problem is: given the Dirich- let data f ∈ H 1 / 2 (Γ a ) and the (measured) Neumann data g := ∂ u g ∈ H − 1 / 2 (Γ a ) on Γ a ∂ν determine the shape of the portion Γ c of the boundary and the impedance function λ ( x ) . In particular, λ = 0 corresponds to homogeneous Neumann boundary condition on Γ c and λ = ∞ corresponds to homogeneous Dirichlet boundary condition on Γ c .

Intoduction Uniqueness Reconstruction Literature Uniqueness of the Inverse Problem Does one pair of Cauchy data u | Γ a = f ∈ H 1 / 2 (Γ a ) and � � ∂ u � = g ∈ H − 1 / 2 (Γ a ) uniquely determine Γ c ? � ∂ν Γ a Consider first the Dirichlet case, i.e. λ = ∞ Let D 1 , D 2 be such that ∂ D 1 = Γ a ∪ Γ 1 c and ∂ D 2 = Γ a ∪ Γ 2 c ∆ u i = 0 in D i , i = 1 , 2 u i = 0 on Γ i c , u 1 = u 2 = f and ∂ u 1 /∂ν = ∂ u 2 /∂ν = g on Γ a . Holmgren’s theorem = ⇒ u 1 = u 2 in D 1 ∩ D 2 . ∆ u 2 = 0 in Ω and u 2 = 0 on ∂ Ω = ⇒ u 2 = 0 and thus f = 0.

Intoduction Uniqueness Reconstruction Literature Uniqueness of the Inverse Problem This idea does not work in the case of impedance boundary condition. Indeed by the same reasoning we arrive at the following problem for w := u 2 in Ω ∆ w = 0 in Ω ∂ w ∂ν + λ 2 w = 0 on ∂ Ω 2 ∂ w ∂ν − λ 1 w = 0 ∂ Ω 1 on where ν is the normal outward to Ω . This is not a coercive problem!

Intoduction Uniqueness Reconstruction Literature Examples of Non-Uniqueness One pair of Cauchy data does not uniquely determine Γ c in the case of impedance boundary condition even for known impedance λ . Example 1: Cakoni-Kress, Inverse Problems and Imaging (2007) . � � ( x , y ) ∈ R 2 : 0 < x < π/ 2 , − α < y < 1 D = Take λ = 1 and consider the harmonic function u ( x , y ) = ( cos x + sin x ) e y . Then ∂ u ∂ν + u = 0 on Γ c . For the data f := u | Γ a and g := ∂ u /∂ν | Γ a we have infinitely many solu- tions by changing α .

Intoduction Uniqueness Reconstruction Literature Examples of Non-Uniqueness Example 2: Γ 1 c consists of two arcs of the form ( x − c ) 2 + y 2 = 1 λ 2 joined by y = 1 /λ . Γ 2 c consists of arcs of the above form with different c . Pagani-Pieroti, Inverse Problems u ( x , y ) = y , f := y | Γ a , (2009) g := ∂ y /∂ν | Γ a Examples of non-uniqueness for the case of impedance obstacle surrounded by the measurement surface are given in Haddar-Kress, J. Inverse Ill-Posed Problems, (2006) and Rundell, Inverse Problems, (2008) .

Intoduction Uniqueness Reconstruction Literature Uniqueness Question: What is the optimal measurements that uniquely determine Γ c ? This was first answered in Bacchelli, Inverse Problems, (2009) with improvement in Pagani-Pieroti, Inverse Problems (2009) . Theorem Assume that Γ i c , i = 1 , 2 , are C 1 , 1 -smooth curves such that ∂ D i := Γ a ∪ Γ i c are C 1 , 1 -curvilinear polygons and λ i ∈ L ∞ (Γ i c ) . Let f 1 , f 2 ∈ H 3 / 2 (Γ a ) be such that f 1 and f 2 are linearly independent, and f 1 > 0 and u i , i = 1 , 2 , be the harmonic functions in D i corresponding to λ i , f i . If ∂ u 1 ∂ν = ∂ u 2 on some open arc of Γ a ∂ν Γ 1 c = Γ 2 then and λ 1 = λ 2 . c

Intoduction Uniqueness Reconstruction Literature Remarks The uniqueness result is valid in R 2 or R 3 . If Γ c is known then one pair of Cauchy data uniquely determines λ ∈ L ∞ (Γ c ) . This is a simple consequence of Holmgren’s Theorem. In the case of Neumann boundary condition (i.e. λ = 0) one pair of Cauchy data uniquely determines Γ c . The proof follows the idea of the Dirichlet case with more care to handle irregular ∂ Ω (could have cusps); in R 2 one can use the conjugate harmonic of the solution. Logarithmic stability estimates for both Γ c and λ with two Cauchy data pairs is proven in Sincich, SIAM J. Math. Anal. (2010) .

Intoduction Uniqueness Reconstruction Literature Nonlinear Integral Equation Cauchy Problem: Given the pair f ∈ H 1 / 2 (Γ a ) and g ∈ H − 1 / 2 (Γ a ) find α ∈ H 1 / 2 (Γ c ) and β ∈ H − 1 / 2 (Γ c ) such that there exists a harmonic function u ∈ H 1 ( D ) satisfying � � � � ∂ u ∂ u � � u | Γ a = f , = g , u | Γ c = α, = β. � � ∂ν ∂ν Γ a Γ c Let us focus in R 2 and make the ansatz � Φ( x , y ) ϕ ( y ) ds ( y ) , x ∈ D , ϕ ∈ H − 1 / 2 ( ∂ D ) u ( x ) := ( S ϕ )( x ) = ∂ where Φ( x , y ) := 2 π ln | x − y | − 1 , and for x ∈ ∂ D define � ( S ϕ )( x ) := Φ( x , y ) ϕ ( y ) ds ( y ) ∂ D � ∂ Φ( x , y ) ( K ′ ϕ )( x ) := ∂ν ( x ) ϕ ( y ) ds ( y ) . ∂ D

Intoduction Uniqueness Reconstruction Literature Determination of λ Inverse Impedance Problem: ∂ D is known – determine λ from a knowledge of one pair of Cauchy data ( f , g ) on Γ a . This problem is related to completion of Cauchy data. Theorem � � α := u | Γ c , β = ∂ u � is a solution of the Cauchy if and only if � ∂ν Γ a u := ( S ϕ )( x ) where ϕ ∈ H − 1 / 2 ( ∂ D ) is a solution of the ill-posed equation � � � f � S ϕ K ′ ϕ + ϕ A ϕ := = . g 2 Γ a

Intoduction Uniqueness Reconstruction Literature Determination of λ We can prove Theorem The operator A : L 2 ( ∂ D ) → L 2 (Γ a ) × L 2 (Γ a ) is compact, injective and has dense range. To reconstruct λ ( x ) ∈ L ∞ (Γ c ) Solve A ϕ = ( f , g ) for ϕ using Tikhonov regularization. Compute u , α and β . Find impedance λ ( x ) as least square solution of α + λβ = 0

Intoduction Uniqueness Reconstruction Literature Example of Reconstruction of λ D is the ellipse z ( t ) = ( 0 . 3 cos t , 0 . 2 sin t ) , t ∈ [ 0 , 2 π ] and λ ( t ) = sin 4 t , t ∈ [ π, 2 π ] . 0.25 Γ m exact Γ c reconstructed 0.2 3% noise 1 0.15 0.8 0.1 0.05 0.6 0 0.4 − 0.05 0.2 − 0.1 − 0.15 0 − 0.2 − 0.2 − 0.25 0 − 0.3 − 0.2 − 0.1 0.1 0.2 0.3 0 0.5 1 1.5 2 2.5 3 (a) Geometry of the boundary (b) Reconstruction

Intoduction Uniqueness Reconstruction Literature Nonlinear Integral Equations Inverse Shape and Impedance Problem: Determine both Γ c and λ from a knowledge of two pairs of Cauchy data ( f , g ) on Γ a . Theorem The inverse shape and impedance problem is equivalent to solving S ϕ i = f i on Γ a K ′ ϕ i + ϕ i 2 = g i on Γ a and K ′ ϕ i + ϕ i 2 + λ S ϕ i = 0 on Γ c i = 1 , 2 , for Γ c , ϕ 1 , ϕ 2 and λ .

Intoduction Uniqueness Reconstruction Literature Remarks It is possible to obtain a different system of nonlinear integral equations equivalent to the inverse shape and impedance problem by staring with a different ansatz for u . In particular, � � � ϕ ( y ) ∂ Φ( x , y ) u ( x ) := − ψ ( y )Φ( x , y ) ds ( y ) , x ∈ D ∂ν ∂ D Here by Green’s representation theorem � � ψ = ∂ u � ϕ = u | ∂ D . � ∂ν ∂ D Cakoni, Kress and Schuft, Inverse Problems, (2010) .

Intoduction Uniqueness Reconstruction Literature Newton Iterative Method Assume now that ∂ D := { z ( t ) : 0 ≤ t ≤ 2 π } , Γ a := { z ( t ) : π ≤ t ≤ 2 π } , Γ c := { z ( t ) : 0 ≤ t ≤ π } . Setting ψ ( t ) = | z ( t ) ′ | ϕ ( z ( t )) we have � 2 π S ψ )( t ) = 1 1 ( � ln | z ( t ) − z ( τ ) | ψ ( τ ) d τ 2 π 0 and � 2 π [ z ′ ( t )] ⊥ · [ z ( t ) − z ( τ )] 1 ψ ( τ ) d τ + ψ ( t ) ( � K ′ ψ )( t ) = − 2 π | z ′ ( t ) | | z ( t ) − z ( τ ) | 2 2 | z ′ ( t ) | 0 for t ∈ [ 0 , 2 π ] .

Intoduction Uniqueness Reconstruction Literature Newton Iterative Method Then the system of nonlinear integral equations we need to solve reads: � S ψ i = f i on [ π, 2 π ] , � K ′ ψ i = g i on [ π, 2 π ] and K ′ ψ i + λ � � S ψ i = 0 on [ 0 , π ] for i = 1 , 2, where λ = λ ◦ z on [ 0 , π ] , f i = f i ◦ z and g i = g i ◦ z on [ π, 2 π ] . We linearize the system with respect ψ i , λ and z c ( t ) , t ∈ [ 0 , π ] .

Intoduction Uniqueness Reconstruction Literature Newton Iterative Method ψ i + χ i , λ + µ , z c + ζ (w.l.o.g. we assume ζ = q [ z ′ ] ⊥ ) S ( ψ i , z ) + � � S ( χ i , z ) + d � S ( ψ i , z ; ζ ) = on [ π, 2 π ] , f i K ′ ( ψ i , z ) + � � K ′ ( χ i , z ) + d � K ′ ( ψ i , z ; ζ ) = g i on [ π, 2 π ] , and K ′ ( ψ i , z ) + � � K ′ ( χ i , z ) + d � K ′ ( ψ i , z ; ζ ) � � S ( ψ i , z ) + � � S ( χ i , z ) + d � + µ � + λ S ( ψ i , z ; ζ ) S ( ψ i , z ) = 0 on [ 0 , π ] for i = 1 , 2. K ′ and d � Here, the operators d � S denote the Fréchet derivatives K ′ and � with respect to z in direction ζ of the operators � S , respectively.