Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Inverse problems with L 1 data fitting Christian Clason, Bangti JIN, Karl Kunisch Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Graz Applied Inverse Problems 2009 Wien, July 24, 2009 Problem formulation Solution of optimality system Parameter choice method Numerical results 1 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Inverse problem Find x such that Kx = y δ K : L 2 (Ω) → L 2 (Ω) bounded linear operator Ω ⊂ R n bounded domain y δ ∈ L 2 (Ω) noisy measurement Impulsive noise (e.g., salt & pepper, random valued) Problem formulation Solution of optimality system Parameter choice method Numerical results 2 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Inverse problem L 1 data fitting x ∈ L 2 � Kx − y δ � L 1 + α 2 � x � 2 ( P ) min L 2 More robust in the presence of outliers Applicable in image processing, signal processing (single pixel failure) Challenge: Nondifferentiable functional, noise level unknown Regularization assumes smooth solution; alternative: TV Problem formulation Solution of optimality system Parameter choice method Numerical results 3 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Fenchel duality V , Y Banach spaces, topological duals V ∗ , Y ∗ Λ ∈ L ( V , Y ) , F : V → R ∪ {∞} , G : Y → R ∪ {∞} Fenchel conjugate of F : F ∗ : V ∗ → R ∪ {∞} F ∗ ( v ∗ ) = sup � v ∗ , v � V ∗ , V − F ( v ) v ∈ V Problem formulation Solution of optimality system Parameter choice method Numerical results 4 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Fenchel duality Fenchel duality theorem F , G convex and lower semicontinuous ∃ v 0 ∈ V : F ( v 0 ) < ∞ , G (Λ v 0 ) < ∞ , G continuous at Λ v 0 : q ∈ Y ∗ −F ∗ (Λ ∗ q ) − G ∗ ( − q ) v ∈ V F ( v ) + G (Λ v ) = sup (FD) inf Extremality relations: v , q solutions of ( FD ) iff � Λ ∗ q ∈ ∂ F ( v ) , (ER) − q ∈ ∂ G (Λ v ) , Problem formulation Solution of optimality system Parameter choice method Numerical results 4 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Dual problem Define F ( v ) = α F : L 2 → R , 2 � v � 2 L 2 , G : L 2 → R , G ( v ) = � v − y δ � L 1 , Λ : L 2 → L 2 , Λ v = Kv . Fenchel conjugates F ∗ ( q ) = 1 F ∗ : L 2 → R , 2 α � q � 2 L 2 , � � q , y δ � L 2 if � q � L ∞ ≤ 1 , G ∗ : L 2 → R ∪ {∞} , G ∗ ( q ) = ∞ if � q � L ∞ > 1 . Problem formulation Solution of optimality system Parameter choice method Numerical results 5 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Dual problem Dual problem 1 2 α � K ∗ p � 2 L 2 − � p , y δ � L 2 min ( P ∗ ) p ∈ L 2 s.t. � p � L ∞ ≤ 1 , Fenchel duality theorem: ( P ∗ ) has solution p α Solution not unique if ker K ∗ � = { 0 } ! Problem formulation Solution of optimality system Parameter choice method Numerical results 6 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Dual problem Solutions x α , p α related by K ∗ p α = α x α , � 0 ≤ � Kx α − y δ , p − p α � L 2 , for all p ∈ L 2 with � p � L ∞ ≤ 1. Given a solution p α , unique solution of ( P ) : x α = 1 α K ∗ p α Problem formulation Solution of optimality system Parameter choice method Numerical results 7 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Characterization of minimizer For all p ∈ L 2 , p ≥ 0: � Kx α − y δ , p � L 2 = if supp p ⊂ { x : | p α ( x ) | < 1 } , 0 � Kx α − y δ , p � L 2 if supp p ⊂ { x : p α ( x ) = 1 } , ≥ 0 � Kx α − y δ , p � L 2 ≤ if supp p ⊂ { x : p α ( x ) = − 1 } . 0 Interpretation: Box constraint on p α active where data is not attained by x α Sign of p α gives sign of noise Problem formulation Solution of optimality system Parameter choice method Numerical results 8 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Regularization of dual problem 1 2 α � K ∗ p � 2 L 2 − � p , y δ � L 2 min ( P ∗ ) p ∈ L 2 s.t. � p � L ∞ ≤ 1 , Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0 Problem formulation Solution of optimality system Parameter choice method Numerical results 9 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Regularization of dual problem 1 2 α � K ∗ p � 2 L 2 − � p , y δ � L 2 min p ∈ H 1 ( P ∗ c ) + 1 L 2 + 1 2 c � max ( 0 , c ( p − 1 )) � 2 2 c � min ( 0 , c ( p + 1 )) � 2 L 2 Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0 Problem formulation Solution of optimality system Parameter choice method Numerical results 9 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Regularization of dual problem 1 L 2 + β 2 α � K ∗ p � 2 2 �∇ p � 2 L 2 − � p , y δ � L 2 min p ∈ H 1 ( P ∗ β, c ) + 1 L 2 + 1 2 c � max ( 0 , c ( p − 1 )) � 2 2 c � min ( 0 , c ( p + 1 )) � 2 L 2 Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0 Problem formulation Solution of optimality system Parameter choice method Numerical results 9 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Existence and convergence of minimizers Regularized problem 1 L 2 + β 2 α � K ∗ p � 2 2 �∇ p � 2 L 2 − � p , y δ � L 2 min p ∈ H 1 + 1 L 2 + 1 2 c � max ( 0 , c ( p − 1 )) � 2 2 c � min ( 0 , c ( p + 1 )) � 2 L 2 Strictly convex if ker K ∗ ∩ ker ∇ = { 0 } ⇒ Existence of unique minimizer p β, c Theorem (Convergence) H 1 L 2 L 2 p β, c − c →∞ p β − − → − β → 0 p α , − − ⇀ p β, c − β → 0 p c − − ⇀ Problem formulation Solution of optimality system Parameter choice method Numerical results 10 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Optimality system (regularized) 1 α KK ∗ p c − β ∆ p c − y δ + λ c = 0 , λ c = max ( 0 , c ( p c − 1 )) + min ( 0 , c ( p c + 1 )) Nonlinear equation for p c Pointwise max , min semismooth ⇒ solution by generalized Newton method Problem formulation Solution of optimality system Parameter choice method Numerical results 11 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Semismoothness in function spaces X , Y Banach spaces, D ⊂ X open Definition F : D ⊂ X → Y Newton differentiable at x ∈ D , if there is neighborhood N ( x ) , G : N ( x ) → L ( X , Z ) � F ( x + h ) − F ( x ) − G ( x + h ) h � = o ( � h � ) Set { G ( s ) : s ∈ N ( x ) } Newton derivative of F at x. Definition F semismooth if N-differentiable and G ( s ) − 1 uniformly bounded. F semismooth ⇒ generalized Newton method G ( s k ) δ x = − F ( x k ) , s k ∈ N ( x k ) , converges locally superlinearly. Problem formulation Solution of optimality system Parameter choice method Numerical results 12 / 35
Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Semismoothness of projection operator Projection operator P ( p ) := max ( 0 , ( p − 1 )) + min ( 0 , ( p + 1 )) is semismooth from L q to L p , if and only if q > p , Newton derivative � h ( x ) if | p ( x ) | > 1 , D N P ( p ) h = h χ {| p | > 1 } := 0 if | p ( x ) | ≤ 1 . Problem formulation Solution of optimality system Parameter choice method Numerical results 13 / 35
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