On Augmented Lagrangian approach for inverse problems Adriano De - PowerPoint PPT Presentation

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments On Augmented Lagrangian approach for inverse problems Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai Inverse problems: recent developments and applications Florian´ opolis - 2014 Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments The Inverse Problem 1 Inverse Problem Piecewise constant solution 2 Level set approaches Level set formulation Piecewise constant level set approach (PCLS) 3 PCLS formulation (PCLS)-regularization approaches Numerical experiments 4 Inverse potential problem - IPP Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments The Inverse Problem 1 Inverse Problem Piecewise constant solution 2 Level set approaches Level set formulation Piecewise constant level set approach (PCLS) 3 PCLS formulation (PCLS)-regularization approaches Numerical experiments 4 Inverse potential problem - IPP Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Inverse Problem Recover u : Ω → R from the ”nonlinear”ill-posed equation F ( u ) = y δ (1) F : D ( F ) ⊂ X → Y s.t. � y − y δ � Y ≤ δ . (2) Assumption (A1): F : D ( F ) ⊂ X → Y is continuous w.r.t. the L 1 ( Ω ) - topology. Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Piecewise constant solution Assumption u is piecewise constant in Ω c 1 , c 2 constant. w.l.g. u ∈ { c 1 , c 2 } Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Piecewise constant solution Assumption u is piecewise constant in Ω c 1 , c 2 constant. w.l.g. u ∈ { c 1 , c 2 } ∃ D 1 ⊂ Ω | D 1 | > 0 s.t . c 1 , � x ∈ D 1 u ( x ) = c 2 , x ∈ D 2 : = Ω − D 1 . Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Piecewise constant solution Assumption u is piecewise constant in Ω c 1 , c 2 constant. w.l.g. u ∈ { c 1 , c 2 } ∃ D 1 ⊂ Ω | D 1 | > 0 s.t . c 1 , � x ∈ D 1 u ( x ) = c 2 , x ∈ D 2 : = Ω − D 1 . Remark: u as above appears in many applications!!! Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Piecewise constant solution Assumption u is piecewise constant in Ω c 1 , c 2 constant. w.l.g. u ∈ { c 1 , c 2 } ∃ D 1 ⊂ Ω | D 1 | > 0 s.t . c 1 , � x ∈ D 1 u ( x ) = c 2 , x ∈ D 2 : = Ω − D 1 . Remark: u as above appears in many applications!!! Under this framework the Inverse Problem consist in recover χ D 1 and the values { c 1 , c 2 } . Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments The Inverse Problem 1 Inverse Problem Piecewise constant solution 2 Level set approaches Level set formulation Piecewise constant level set approach (PCLS) 3 PCLS formulation (PCLS)-regularization approaches Numerical experiments 4 Inverse potential problem - IPP Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Level set formulation The level set idea parameterize u using a (smooth) level set function φ : Ω → R s.t. D 1 : { x ∈ Ω : φ ( x ) ≥ 0 } D 2 : { x ∈ Ω : φ ( x ) < 0 } u = P ls ( φ , c j ) . (3) where P ls ( φ , c j ) = c 1 H ( φ )+ c 2 ( 1 − H ( φ )) . Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Level set formulation The level set idea parameterize u using a (smooth) level set function φ : Ω → R s.t. D 1 : { x ∈ Ω : φ ( x ) ≥ 0 } D 2 : { x ∈ Ω : φ ( x ) < 0 } u = P ls ( φ , c j ) . (3) where P ls ( φ , c j ) = c 1 H ( φ )+ c 2 ( 1 − H ( φ )) . standard level set approach!!! Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments Level set formulation The level set idea parameterize u using a (smooth) level set function φ : Ω → R s.t. D 1 : { x ∈ Ω : φ ( x ) ≥ 0 } D 2 : { x ∈ Ω : φ ( x ) < 0 } u = P ls ( φ , c j ) . (3) where P ls ( φ , c j ) = c 1 H ( φ )+ c 2 ( 1 − H ( φ )) . standard level set approach!!! in this presentation: piecewise constant level set approach (PCLS) Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments The Inverse Problem 1 Inverse Problem Piecewise constant solution 2 Level set approaches Level set formulation Piecewise constant level set approach (PCLS) 3 PCLS formulation (PCLS)-regularization approaches Numerical experiments 4 Inverse potential problem - IPP Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments PCLS formulation (PCLS) φ ∈ L 2 ( Ω ) – (non-smooth) such that φ ( x ) = i − 1 x ∈ D i rewritten u as u = c 1 ψ 1 ( φ )+ c 2 ψ 2 ( φ ) : = P ( φ , c j ) . (4) where ψ 1 ( t ) = 1 − t and ψ 2 ( t ) = t . Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments PCLS formulation (PCLS) The inverse problem: can be rewritten as: find φ ∈ L 2 ( Ω ) (”and c j ”) s.t. F ( P ( φ , c j )) = y δ . (5) the piecewise constant assumption of φ correspond to the constraint K ( φ ) = φ ( φ − 1 ) = 0 , smooth or K ( φ ) : = | φ || φ − 1 | = 0 , � non-smooth Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments PCLS formulation (PCLS) The inverse problem: can be rewritten as: find φ ∈ L 2 ( Ω ) (”and c j ”) s.t. F ( P ( φ , c j )) = y δ . (5) the piecewise constant assumption of φ correspond to the constraint K ( φ ) = φ ( φ − 1 ) = 0 , smooth or K ( φ ) : = | φ || φ − 1 | = 0 , � non-smooth Assumption (A2): ∃ φ ∗ ∈ L 2 ( Ω ) and c j ∗ ∈ R s.t. P ( φ ∗ , c j ∗ ) = u ∗ F ( u ∗ ) = y and K ( φ ∗ ) = 0 . Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems

Plan The Inverse Problem Level set approaches Piecewise constant level set approach (PCLS) Numerical experiments (PCLS)-regularization approaches penalty method Tikhonov regularization + penalty method minimize G α ( φ , c j ) : = � F ( P ( φ , c j )) − y δ � 2 Y + µ � K ( φ ) � L 1 (6) + α | P ( φ , c j ) | BV + � c j � 2 � � . R 2 where µ > 0 plays the role of a scaling factor. Adriano De Cezaro- FURG in collaboration with Antonio Leit˜ ao & X-C. Tai On Augmented Lagrangian approach for inverse problems