Complex Electrical Impedance Tomography in Geoelectrics Andreas - PowerPoint PPT Presentation

Introduction The Ddirect Problem Inverse Problem Numerical Examples Complex Electrical Impedance Tomography in Geoelectrics Andreas Helfrich-Schkarbanenko July 17, 2009 Institute for Algebra and Geometry University of Karlsruhe Germany Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Complex Electrical Impedance Tomography in Geoelectrics Introduction 1 Problem Setting The Ddirect Problem 2 Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems Inverse Problem 3 Operators Implementation Numerical Examples 4 Synthetic Examples Real Example Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Problem Setting Inverse Problem Numerical Examples Model Ω ⊂ R d . d ∈ { 2 , 3 } ; X 2 u : Ω → C is electrical potential which � � is induced by current pattern f and solves -R R 0 X 1 −∇ · ( γ ∇ u ) = f in Ω (1a) ν · ∇ u = 0 on Γ 1 (1b) � R � ν · ∇ u + α u = 0 on Γ 2 . (1c) � � Figure: Problem setting Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Weak Formulation of the Problem Let B γ,α : H 1 (Ω) × H 1 (Ω) → C defined by Z Z u , v ∈ H 1 (Ω) B γ,α ( u , v ) := γ �∇ u , ∇ v � dx + γα uvds , (2) Ω Γ 2 and the linear form L f : H 1 (Ω) → C defined by Z v ∈ H 1 (Ω) . L f ( v ) := f vdx , (3) Ω Problem For given f ∈ H − 1 (Ω) , α ∈ L ∞ (Γ 2 , R + ) and γ ∈ L ∞ (Ω , R + ) find u ∈ H 1 (Ω) with: ∀ v ∈ H 1 (Ω) . B γ,α ( u , v ) = L f ( v ) , (4) Is this problem uniquely solvable? Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Theorem (Existence and uniqueness of weak solution) Let Ω be a Lipschitz domain, f ∈ H − 1 (Ω) , α ∈ L ∞ (Γ 2 , R + ) and γ ( x , ω ) = σ ( x ) + i ωǫ ( x ) ∈ L ∞ (Ω × R + , C + ) , where C + := { x ∈ C : Re ( x ) > 0 } . Then the variational problem (4) has a unique solution in the Sobolev space H 1 (Ω) . Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Theorem (Existence and uniqueness of weak solution) Let Ω be a Lipschitz domain, f ∈ H − 1 (Ω) , α ∈ L ∞ (Γ 2 , R + ) and γ ( x , ω ) = σ ( x ) + i ωǫ ( x ) ∈ L ∞ (Ω × R + , C + ) , where C + := { x ∈ C : Re ( x ) > 0 } . Then the variational problem (4) has a unique solution in the Sobolev space H 1 (Ω) . Proof idea. Apply the on B γ,α , L f and use in P.Monk Finite Element Lax-Milgram theorem lemma Methods for Maxwell’s Equations , 2003. Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Theorem (Existence and uniqueness of weak solution) Let Ω be a Lipschitz domain, f ∈ H − 1 (Ω) , α ∈ L ∞ (Γ 2 , R + ) and γ ( x , ω ) = σ ( x ) + i ωǫ ( x ) ∈ L ∞ (Ω × R + , C + ) , where C + := { x ∈ C : Re ( x ) > 0 } . Then the variational problem (4) has a unique solution in the Sobolev space H 1 (Ω) . Proof idea. Apply the on B γ,α , L f and use in P.Monk Finite Element Lax-Milgram theorem lemma Methods for Maxwell’s Equations , 2003. FE method: Find u ∈ H h = span { ψ k } N p k =1 ⊂ H 1 (Ω). Does the unique solution exist in H h , too? Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Theorem (Existence and uniqueness of weak solution) Let Ω be a Lipschitz domain, f ∈ H − 1 (Ω) , α ∈ L ∞ (Γ 2 , R + ) and γ ( x , ω ) = σ ( x ) + i ωǫ ( x ) ∈ L ∞ (Ω × R + , C + ) , where C + := { x ∈ C : Re ( x ) > 0 } . Then the variational problem (4) has a unique solution in the Sobolev space H 1 (Ω) . Proof idea. Apply the on B γ,α , L f and use in P.Monk Finite Element Lax-Milgram theorem lemma Methods for Maxwell’s Equations , 2003. FE method: Find u ∈ H h = span { ψ k } N p k =1 ⊂ H 1 (Ω). Does the unique solution exist in H h , too? Yes! By means of C´ ea lemma . Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Further results For the case of ν · ∇ u = g on Γ 1 the problem has a unique solution as well (provided some properties on f , g and u | Γ 1 ). Prove it as above. Stability: � u � H 1 (Ω) ≤ C ( � f � H − 1 (Ω) ) + � g � H − 1 / 2 (Γ 1 ) ), C > 0. u depends continuously on γ = γ ( x , t ), if γ ∈ C 1 (Ω , R ). Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Im{u} für f 1 −3 Im{u} für f 2 −3 Im{u} für f 3 −3 Im{u} für f 4 −3 x 10 x 10 x 10 x 10 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 −2 −2 −2 −2 −2 −2 −5 −4 −2 −4 −4 −4 −4 −6 −4 −10 −4 −6 −8 −6 −6 −6 −6 −5 0 5 −5 0 5 −5 0 5 −5 0 5 Im{u} für f 5 −3 Im{u} für f 6 −3 Im{u} für f 7 −3 Im{u} für f 8 −3 x 10 x 10 x 10 x 10 2 2 2 4 2 2 2 0 0 0 0 0 0 0 0 −5 −2 −2 −2 −5 −2 −2 −2 −4 −10 −4 −4 −4 −4 −6 −4 −10 −8 −15 −6 −6 −6 −6 −6 −5 0 5 −5 0 5 −5 0 5 −5 0 5 −3 x 10 3 5 0 2 } } 1 1 Re{u Γ Im{u Γ −5 1 −10 0 −15 −5 0 5 −5 0 5 −3 x 10 x x 3 5 0 2 } } k k Re{u E Im{u E −5 1 −10 0 −15 −2 −1 0 1 2 −2 −1 0 1 2 x x Figure: Solution of the forward problem, Im { u } and the corresponding measurements Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems Solving 2.5D and 2D direct problems 1 Solving 3D problem is time- and memory-consuming 2 Assume γ = γ ( x 1 , x 2 ) and apply the Fourier-transform with respect to x 3 , [DM76] 3 We get 2D Helmholtz-equation with a wave number 4 Solve the obtained problem in image space and transform it back + X 2 � � X 3 + � � 0 X 1 � � 2 - - � � � 3 � � Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction Weak Formulation of the Problem The Ddirect Problem Existence and uniqueness Inverse Problem Numerical Solution Numerical Examples Solving 2.5D and 2D direct problems 3D 2.5D 2D Ω 3 ⊂ R 3 Ω 3 ⊂ R 3 Ω 2 ⊂ R 2 γ = γ ( x 1 , x 2 , x 3 ) assume γ = γ ( x 1 , x 2 ) γ = γ ( x 1 , x 2 ) ✲ ❄ ❄ ❄ BVP in Ω 3 BVP in Ω 3 BVP in Ω 2 F cos ❄ Helmholtz BVP FEM 3D FEM 2D FEM 2D ❄ { e u ω } ω ∈ R F − 1 cos for x 3 = 0 on Ω 2 ❄ ❄ ❄ u 3 u 3 | Ω 2 u 2 = if γ = γ ( x 1 , x 2 ) (a) (b) (c) Figure: Comparison of 3D, 2.5D and 2D forward problem. Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Operators Inverse Problem Implementation Numerical Examples The inverse problem consists of the reconstruction of γ in Ω from current and voltage measurement data on Γ 1 ⊂ ∂ Ω, i.e. we have to find γ for the given Neumann-to-Dirichlet operator Λ γ : f → u | Γ 1 . In addition, we consider a nonlinear operator with respect to γ  L ∞ (Ω) H 1 (Ω) → F f := (5) γ �→ u = Λ γ f so the inverse problem is nonlinear, too. For the numerical implementation of Newton-like algorithms the differentiation of F f is essential. Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics