A Double Regularization Approach for Inverse Problems with Noisy Data and Inexact Operator Ismael Rodrigo Bleyer Prof. Dr. Ronny Ramlau Johannes Kepler Universit¨ at - Linz Florian´ opolis - September, 2011. supported by Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation Bleyer, Ramlau JKU Linz 1 / 27
Overview Introduction � Proposed method: DBL-RTLS � Computational aspects � Numerical illustration � Outline and future work � Bleyer, Ramlau JKU Linz 2 / 27
Introduction Overview Introduction � Proposed method: DBL-RTLS � Computational aspects � Numerical illustration � Outline and future work � Bleyer, Ramlau JKU Linz 2 / 27
Introduction Inverse problems “ Inverse problems are concerned with determining causes for a desired or an observed effect” [Engl, Hanke, and Neubauer, 2000] Consider a linear operator equation Ax = y. Inverse problems most oft do not fulfill Hadamard ’s postulate [1902] of well posedness ( existence, uniqueness and stability ). Computational issues: observed effect has measurement errors or perturbations caused by noise . Bleyer, Ramlau JKU Linz 3 / 27
Introduction 1st Case: noisy data � ≤ δ . � � Solve Ax = y 0 out of the measurement y δ with � y 0 − y δ Need apply some regularization technique � 2 + α � 2 . � � � � minimize � Ax − y δ � Lx x Tikhonov regularization fidelity term (based on LS); regularization parameter α ; stabilization term (quadratic). [Tikhonov, 1963, Phillips, 1962] Bleyer, Ramlau JKU Linz 4 / 27
Introduction 1st Case: noisy data � ≤ δ . � � Solve Ax = y 0 out of the measurement y δ with � y 0 − y δ Need apply some regularization technique � 2 + α R ( x ) . � � minimize � Ax − y δ x Tikhonov-type regularization fidelity term (based on LS); regularization parameter α ; R is a proper , convex and weakly lower semicontinuous functional . [Burger and Osher, 2004, Resmerita, 2005] Bleyer, Ramlau JKU Linz 4 / 27
Introduction Subgradient The Fenchel subdifferential of a functional R : U → [0 , + ∞ ] at u ∈ U is the set ¯ ∂ F R (¯ u ) = { ξ ∈ U ∗ | R ( v ) − R (¯ � � u ) ≥ ξ , v − ¯ u ∀ v ∈ U } . First in 1960 by Moreau & Rockafellar and extended by Clark 1973. Optimality condition: If ¯ u minimizes R then 0 ∈ ∂ F R (¯ u ) Bleyer, Ramlau JKU Linz 5 / 27
Introduction Example Consider the function R ( u ) = | u | 1 0 − 1 Figure: Function (left) and its subdifferential (right). Bleyer, Ramlau JKU Linz 6 / 27
Introduction 2nd Case: inexact operator and noisy data Solve A 0 x = y 0 under the assumptions � ≤ δ . � � (i) noisy data � y 0 − y δ � ≤ ǫ . � � (ii) inexact operator � A 0 − A ǫ What have been done so far? Linear case - based on TLS [Golub and Van Loan, 1980]: R-TLS : Regularized TLS [Golub et al., 1999]; D-RTLS : Dual R-TLS [Lu et al., 2007]. Nonlinear case : no publication (?) LS : y δ and A 0 TLS : y δ and A ǫ � � � � minimize y � y − y δ minimize � [ A, y ] − [ A ǫ , y δ ] � � 2 F subject to y ∈ R ( A 0 ) subject to y ∈ R ( A ) Bleyer, Ramlau JKU Linz 7 / 27
Introduction 2nd Case: inexact operator and noisy data Solve A 0 x = y 0 under the assumptions � ≤ δ . � � (i) noisy data � y 0 − y δ � ≤ ǫ . � � (ii) inexact operator � A 0 − A ǫ What have been done so far? Linear case - based on TLS [Golub and Van Loan, 1980]: R-TLS : Regularized TLS [Golub et al., 1999]; D-RTLS : Dual R-TLS [Lu et al., 2007]. Nonlinear case : no publication (?) LS : y δ and A 0 TLS : y δ and A ǫ � � � � minimize y � y − y δ minimize � [ A, y ] − [ A ǫ , y δ ] � � 2 F subject to y ∈ R ( A 0 ) subject to y ∈ R ( A ) Bleyer, Ramlau JKU Linz 7 / 27
Introduction Illustration Solve 1D problem: am = b , find the slope m . TLS vs LS 3 2.5 Given: 2 1. b δ , a ǫ (red) 1.5 1 Solution: 0.5 1. LS solution (blue) 0 LS solution 2. TLS solution (green) TLS solution −0.5 noisy data true data −1 −1 −0.5 0 0.5 1 1.5 2 2.5 slope 45.9078 Example: arctan(1) = 45 o [Van Huffel and Vandewalle, 1991] Bleyer, Ramlau JKU Linz 8 / 27
Introduction R-TLS The R-TLS method [Golub, Hansen, and O’leary, 1999] � 2 + � 2 � � � � minimize � A − A ǫ � y − y δ � Ax = y subject to � 2 ≤ M . � � � Lx If the inequality constraint is active, then ǫ A ǫ + αL T L + βI A T x = A T � = M � � � � ˆ ǫ y δ and � L ˆ x � 2 � � � A ǫ ˆ x − y δ � 2 ) , β = − � � with α = µ (1 + � ˆ and µ > 0 is the Lagrange x � 2 � � 1 + � ˆ x multiplier. � 2 . � � Lx † � Difficulty: requires a reliable bound M for the norm Bleyer, Ramlau JKU Linz 9 / 27
Proposed method: DBL-RTLS Overview Introduction � Proposed method: DBL-RTLS � Computational aspects � Numerical illustration � Outline and future work � Bleyer, Ramlau JKU Linz 9 / 27
Proposed method: DBL-RTLS Consider the operator equation B ( k, f ) = g 0 where B is a bilinear operator (nonlinear) B : U × V − → H ( k, f ) �− → B ( k, f ) and B is characterized by a function k 0 . K · = B (˜ k, · ) compact linear operator for a fixed ˜ k ∈ U F · = B ( · , ˜ f ) linear operator for a fixed ˜ f ∈ V � � � � � B ( k 0 , · ) V → H ≤ C � k 0 U ; � � � � � � � � � B ( k, f ) H ≤ C V ; � k � f � � � U Example: � B ( k, f )( s ) := k ( s, t ) f ( t ) dt . Ω Bleyer, Ramlau JKU Linz 10 / 27
Proposed method: DBL-RTLS Consider the operator equation B ( k, f ) = g 0 where B is a bilinear operator (nonlinear) B : U × V − → H ( k, f ) �− → B ( k, f ) and B is characterized by a function k 0 . K · = B (˜ k, · ) compact linear operator for a fixed ˜ k ∈ U F · = B ( · , ˜ f ) linear operator for a fixed ˜ f ∈ V � � � � � B ( k 0 , · ) V → H ≤ C � k 0 U ; � � � � � � � � � B ( k, f ) H ≤ C V ; � k � f � � � U Example: � B ( k, f )( s ) := k ( s, t ) f ( t ) dt . Ω Bleyer, Ramlau JKU Linz 10 / 27
Proposed method: DBL-RTLS We want to solve B ( k 0 , f ) = g 0 out of the measurements k ǫ and g δ with � � (i) noisy data � g 0 − g δ H ≤ δ . � � � (ii) inexact operator � k 0 − k ǫ U ≤ ǫ . � We introduce the DBL-RTLS minimize J ( k, f ) := T ( k, f, k ǫ , g δ ) + R ( k, f ) k,f where T measures of accuracy (closeness/discrepancy) R promotes stability. Bleyer, Ramlau JKU Linz 11 / 27
Proposed method: DBL-RTLS DBL-RTLS minimize J ( k, f ) := T ( k, f, k ǫ , g δ ) + R ( k, f ) (1) k,f where T ( k, f, k ǫ , g δ ) = 1 H + γ � 2 � 2 � � � � � B ( k, f ) − g δ � k − k ǫ U 2 2 R ( k, f ) = α � 2 � � V + β R ( k ) � Lf 2 T is based on TLS method, measures the discrepancy on both data and operator; L : V → V is a linear bounded operator; α , β are the regularization parameters and γ is a scaling parameter; double regularization [You and Kaveh, 1996], R : U → [0 , + ∞ ] is proper convex function and w-lsc . Bleyer, Ramlau JKU Linz 12 / 27
Proposed method: DBL-RTLS Main theoretical results Assumption: (A1) B is strongly continuous, ie, if ( k n , f n ) ⇀ (¯ k, ¯ f ) then B ( k n , f n ) → B (¯ k, ¯ f ) Proposition Let J be the functional defined on (1) and L be a bounded and positive operator. Then J is positive , weak lower semi-continuous and coercive functional. Theorem (existence) Let the assumptions of Proposition 1 hold. Then there exists a global minimum of minimize J ( k, f ) . Bleyer, Ramlau JKU Linz 13 / 27
Proposed method: DBL-RTLS Theorem (stability) δ j → δ and ǫ j → ǫ g δ j → g δ and k ǫ j → k ǫ α, β > 0 ( k j , f j ) is a minimizer of J with g δ j and k ǫ j Then there exists a convergent subsequence of ( k j , f j ) j ( k j m , f j m ) − → (¯ k, ¯ f ) where (¯ k, ¯ f ) is a minimizer of J with g δ , k ǫ , α and β . Bleyer, Ramlau JKU Linz 14 / 27
Proposed method: DBL-RTLS Theorem (stability) δ j → δ and ǫ j → ǫ g δ j → g δ and k ǫ j → k ǫ α, β > 0 ( k j , f j ) is a minimizer of J with g δ j and k ǫ j Then there exists a convergent subsequence of ( k j , f j ) j ( k j m , f j m ) − → (¯ k, ¯ f ) where (¯ k, ¯ f ) is a minimizer of J with g δ , k ǫ , α and β . Bleyer, Ramlau JKU Linz 14 / 27
Proposed method: DBL-RTLS Consider the convex functional Φ( k, f ) := 1 � 2 + η R ( k ) � � � Lf 2 where the parameter η represents the different scaling of f and k . For convergence results we need to define Definition We call ( k † , f † ) a Φ - minimizing solution if ( k † , f † ) = arg min { Φ( k, f ) | B ( k, f ) = g 0 } . ( k,f ) Bleyer, Ramlau JKU Linz 15 / 27
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