Dynamic Inverse Problems: Schmitt Efficient Algorithms and - PowerPoint PPT Presentation

Inverse Problems A.K. Louis, U. Dynamic Inverse Problems: Schmitt Efficient Algorithms and Approximate Inverse Problems Definitions Inverse Two Constraints Semi - discrete Case Examples A.K. Louis 1 U. Schmitt 2 Approximate Inverse 1 Institut für Angewandte Mathematik Universität des Saarlandes http://www.num.uni-sb.de 2 Minewave GmbH Saarbrücken Vancouver, 28.06.2007

Content Inverse Problems A.K. Louis, U. Schmitt Static and Dynamic Inverse Problems 1 Inverse Problems Definitions Definitions Two Constraints Two Constraints Semi - discrete Case Examples 2 Semi - discrete Case Approximate Inverse Examples 3 4 Approximate Inverse

Definitions Inverse Problems Static Inverse Problems Continuous and semi-discrete A.K. Louis, U. Schmitt versions Inverse Problems � Definitions Af ( t , x ) = k ( t , x , y ) f ( y ) dy = g ( t , x ) Two Constraints Semi - discrete Case A ℓ f = g ℓ Examples Approximate Inverse

Definitions Inverse Problems Static Inverse Problems Continuous and semi-discrete A.K. Louis, U. Schmitt versions Inverse Problems � Definitions Af ( t , x ) = k ( t , x , y ) f ( y ) dy = g ( t , x ) Two Constraints Semi - discrete Case A ℓ f = g ℓ Examples Approximate Dynamic Inverse Problems Inverse � � Af ( t , x ) = k ( t , τ, x , y ) f ( τ, y ) dyd τ A ℓ f ℓ = g ℓ N EED OF REGULARIZATION BOTH IN TIME AND SPACE

Temporal Regularization Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse Activation curves without and with temporal regularization

Undertermined Problems Inverse Problems Consider A.K. Louis, U. Schmitt A is matrix which has more colums than rows Inverse Problems Definitions A maps from an infintite-dimensional Hilbert space to Two Constraints Semi - discrete finitely many data Case A maps from function spaces of different dimensions Examples Approximate Inverse

Undertermined Problems Inverse Problems Consider A.K. Louis, U. Schmitt A is matrix which has more colums than rows Inverse Problems Definitions A maps from an infintite-dimensional Hilbert space to Two Constraints Semi - discrete finitely many data Case A maps from function spaces of different dimensions Examples Approximate ⇒ Prefer to solve Af = g as Inverse AA ∗ u = g and put f = A ∗ u or its Tikhonov - Phillips variants.

Content Inverse Problems A.K. Louis, U. Schmitt Static and Dynamic Inverse Problems 1 Inverse Problems Definitions Definitions Two Constraints Two Constraints Semi - discrete Case Examples 2 Semi - discrete Case Approximate Inverse Examples 3 4 Approximate Inverse

Two Constraints Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Minimize Two Constraints � Af − g � 2 + γ 2 � f � 2 + µ 2 � Bf � 2 Semi - discrete Case where in our application one of the terms is used as Examples Approximate spatial the other as temporal smoothness condition. Inverse

Two Constraints Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Minimize Two Constraints � Af − g � 2 + γ 2 � f � 2 + µ 2 � Bf � 2 Semi - discrete Case where in our application one of the terms is used as Examples Approximate spatial the other as temporal smoothness condition. Inverse Only when A and B commute then this can be written in the above mentioned form.

Formulation with slack variable Inverse Problems A.K. Louis, U. Minimize with respect to f and d Schmitt � Af − g � 2 + γ 2 � f � 2 + µ 2 � d � 2 + α 2 � Bf − d � 2 Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse

Formulation with slack variable Inverse Problems A.K. Louis, U. Minimize with respect to f and d Schmitt � Af − g � 2 + γ 2 � f � 2 + µ 2 � d � 2 + α 2 � Bf − d � 2 Inverse Problems Definitions Two Constraints Change of variables d := γ µ y Then this is Semi - discrete Case Examples � Af − g � 2 + γ 2 � f � 2 + γ 2 � y � 2 + � α Bf − αγ µ y � 2 Approximate Inverse

Formulation with slack variable Inverse Problems A.K. Louis, U. Minimize with respect to f and d Schmitt � Af − g � 2 + γ 2 � f � 2 + µ 2 � d � 2 + α 2 � Bf − d � 2 Inverse Problems Definitions Two Constraints Change of variables d := γ µ y Then this is Semi - discrete Case Examples � Af − g � 2 + γ 2 � f � 2 + γ 2 � y � 2 + � α Bf − αγ µ y � 2 Approximate Inverse With the new variables � f � g � � ξ = and h = y 0 this is equivalent to

Compact Formulation Inverse Problems A.K. Louis, U. Minimze Schmitt � T α ξ − h � 2 + γ 2 � ξ � 2 Inverse Problems Definitions with Two Constraints A 0 � � Semi - discrete T α = − α γ Case α B µ I Examples The solution of Approximate � u Inverse � T α T ∗ = h α v with � u � ξ = T ∗ α v leads to the desired result.

Efficient Minimization with two constraints Inverse Problems A.K. Louis, U. Schmitt Theorem (L., Schmitt, 2002, 2007) Inverse Problems The minimzation of Definitions Two Constraints � Af − g � 2 + γ 2 � f � 2 + µ 2 � Bf � 2 Semi - discrete Case Examples is computed in two steps. First we solve for u with ω = γ 2 Approximate µ 2 Inverse I − B ∗ ( BB ∗ + ω I ) − 1 B A ∗ u + γ 2 u = g � � A and put I − B ∗ ( BB ∗ + ω I ) − 1 B A ∗ u � � f γ,µ =

Semi - discrete Case Inverse Problems A.K. Louis, U. Schmitt A ℓ : H → G ℓ , ℓ = 1 , . . . , L Inverse Problems Definitions A ℓ F ℓ = G ℓ Two Constraints Semi - discrete Case Minimize Examples L L L − 1 � f ℓ + 1 − f ℓ � 2 Approximate � A ℓ f ℓ − g ℓ � 2 + γ 2 � f ℓ � 2 + µ 2 � � � Inverse J ( f ) = ( t ℓ + 1 − t ℓ ) 2 ℓ = 1 ℓ = 1 ℓ = 1 Define A = diag ( A ℓ ) ∈ L ( H L , G 1 ⊕ · · · ⊕ G L )

Inverse Problems A.K. Louis, U. Schmitt Inverse Problems f = ( f 1 , · · · f L ) Definitions Two Constraints B = D ⊕ I H ∈ L ( H L , H L − 1 Semi - discrete Case � τ 1 − τ 1 Examples � − τ 2 τ 2 Approximate ∈ R L × ( L − 1 ) D = Inverse · · − τ L − 1 τ L − 1 τ i = ( t i + 1 − t i ) − 1 Generalized Sylvester Equations

Dynamic Computerized Tomography Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse � f ( s ω + t ω ⊥ ) dt R ℓ f ( s ) = Rf ( ω ℓ , s ) =

Dynamic Computerized Tomography Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse

Current Density Reconstruction Inverse Problems Application: study of neurological activity in the brain A.K. Louis, U. Forward Model Schmitt Inverse Problems div ( σ ∇ Φ) = div j in Ω Definitions Two Constraints Semi - discrete � σ ∇ Φ , n � = 0 at δ Ω Case Examples σ conductivity tensor Approximate Φ electrical potential Inverse Data Φ at the boundary

Current Density Reconstruction Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse Reconstruction without temporal smoothness constrains

Current Density Reconstruction Inverse Problems A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse Reconstruction with temporal smoothness constrains

Approximate Inverse Inverse Problems L., Inverse Problems,1996, 1999 A.K. Louis, U. Schmitt Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Inverse Problems Given : Definitions A : L 2 (Ω 1 , µ 1 ) → L 2 (Ω 2 , µ 2 ) linear, continuous Two Constraints Semi - discrete Mollifier δ x ≈ e γ ( x , · ) or δ ′ x ≈ e γ ( x , · ) Case Examples Approximate Inverse

Approximate Inverse Inverse Problems L., Inverse Problems,1996, 1999 A.K. Louis, U. Schmitt Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Inverse Problems Given : Definitions A : L 2 (Ω 1 , µ 1 ) → L 2 (Ω 2 , µ 2 ) linear, continuous Two Constraints Semi - discrete Mollifier δ x ≈ e γ ( x , · ) or δ ′ x ≈ e γ ( x , · ) Case Examples Approximate Compute Inverse f γ ( x ) = � f , e γ ( x , · ) � L 2 (Ω 1 ,µ 1 ) = � Af , ψ γ ( x ) � L 2 (Ω 2 ,µ 2 )

Approximate Inverse Inverse Problems L., Inverse Problems,1996, 1999 A.K. Louis, U. Schmitt Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Inverse Problems Given : Definitions A : L 2 (Ω 1 , µ 1 ) → L 2 (Ω 2 , µ 2 ) linear, continuous Two Constraints Semi - discrete Mollifier δ x ≈ e γ ( x , · ) or δ ′ x ≈ e γ ( x , · ) Case Examples Approximate Compute Inverse f γ ( x ) = � f , e γ ( x , · ) � L 2 (Ω 1 ,µ 1 ) = � Af , ψ γ ( x ) � L 2 (Ω 2 ,µ 2 ) Idea : Solve A ∗ ψ γ ( x ) = e γ ( x , · )

Approximate Inverse in L 2 -Spaces Inverse Problems Data g given A.K. Louis, U. Schmitt Inverse Problems Definitions Two Constraints Semi - discrete Case Examples Approximate Inverse