Convergence of IRGNM type methods under a tangential cone condition - PowerPoint PPT Presentation

Setting Continuous version Discretized version Error estimate Model Example Convergence of IRGNM type methods under a tangential cone condition in Banach space Mario Luiz Previatti de Souza Barbara Kaltenbacher Alpen-Adria-Universit¨ at Klagenfurt mario.previatti@aau.at barbara.kaltenbacher@aau.at Col ´ oquio UFSC, 6 de Outubro, 2017 Regularization and Discretization of Inverse Problems for PDEs in Banach Spaces

Setting Continuous version Discretized version Error estimate Model Example Overview Setting Continuous version Discretized version Error estimate Model Example D-A-CH project Regularization and Discretization of Inverse Problems for PDEs in Banach Spaces

Setting Continuous version Discretized version Error estimate Model Example Iteratively Regularized Gauss Newton Method (IRGNM) Nonlinear ill-posed operator equation F ( x ) = y δ , � y − y δ � ≤ δ

Setting Continuous version Discretized version Error estimate Model Example Iteratively Regularized Gauss Newton Method (IRGNM) Nonlinear ill-posed operator equation F ( x ) = y δ , � y − y δ � ≤ δ IRGNM-Tikhonov k ) − ( y δ − F ( x δ k )) � p + α k R ( x ) ′ ( x δ x δ k )( x − x δ k +1 ∈ argmin x ∈D ( F ) � F k = 0 , 1 , . . . p ∈ [1 , ∞ ) IRGNM-Ivanov ′ ( x δ x δ k )( x − x δ k ) − ( y δ − F ( x δ k +1 ∈ argmin x ∈D ( F ) � F k )) � s.t. R ( x ) ≤ ρ k k = 0 , 1 , . . .

Setting Continuous version Discretized version Error estimate Model Example Setting • F : D ( F ) ⊂ X − → Y , X and Y real Banach spaces • x † ∈ D ( F ) exists, i.e., F ( x † ) = y • R ( x ) is proper, convex and l.s.c. with R ( x † ) < ∞ • discrepancy principle k ∗ = min { k ∈ N 0 : � F ( x δ k ) − y δ � ≤ τδ } , τ > 1 fixed

Setting Continuous version Discretized version Error estimate Model Example Setting • F : D ( F ) ⊂ X − → Y , X and Y real Banach spaces • x † ∈ D ( F ) exists, i.e., F ( x † ) = y • R ( x ) is proper, convex and l.s.c. with R ( x † ) < ∞ • discrepancy principle k ∗ = min { k ∈ N 0 : � F ( x δ k ) − y δ � ≤ τδ } , τ > 1 fixed • tangential cone condition c tc < 1 / 3 ′ ( x )(˜ � F (˜ x ) − F ( x ) − F x − x ) � ≤ c tc � F (˜ x ) − F ( x ) � , ∀ x ∈ B R ⊂ D ( F ) • α k , ρ k a priori �� � p � c tc α k = α 0 θ k for some θ ∈ and ρ k ≡ ρ ≥ R ( x † ) 2 , 1 1 − c tc

Setting Continuous version Discretized version Error estimate Model Example Convergence result (continuous version) Theorem Let for all r ≥ R ( x † ), the sublevel set B r = { x ∈ D ( F ) : R ( x ) ≤ r } be compact with respect to some topology τ on X . Let for all ′ ( x ) and F be τ -to-norm continuous, for some chosen x ∈ B R , F R > R ( x † ). Then, for fixed δ and y δ , the iterations are well defined and remain in B R and the stopping index k ∗ is finite. Moreover, we have τ -convergence as δ → 0. If the solution x † of F ( x ) = y is unique in B R , then x δ k ∗ ( δ, y δ ) → x † as δ → 0. Additionally, k ∗ satisfies the asymptotics k ∗ = O ( | log 1 /δ | ).

Setting Continuous version Discretized version Error estimate Model Example Proof idea

Setting Continuous version Discretized version Error estimate Model Example Proof idea 1. Existence of minimizer: direct method of calculus of variations

Setting Continuous version Discretized version Error estimate Model Example Proof idea 1. Existence of minimizer: direct method of calculus of variations 2. Find a recursive estimate formula for � F ( x δ k ) − y δ � p

Setting Continuous version Discretized version Error estimate Model Example Proof idea 1. Existence of minimizer: direct method of calculus of variations 2. Find a recursive estimate formula for � F ( x δ k ) − y δ � p 3. Find upper bound for k ∗ ( δ, y δ ) ≤ ¯ k ( δ ) = O (log 1 /δ ) � � � � �� d 0 + α 0 C θ − q R ( x † ) p log(1 /δ )+log − log τ − ˜ k ∗ ( δ, y δ ) ≤ ¯ 1 − q k ( δ ) = − 1 log 1 /θ

Setting Continuous version Discretized version Error estimate Model Example Proof idea 1. Existence of minimizer: direct method of calculus of variations 2. Find a recursive estimate formula for � F ( x δ k ) − y δ � p 3. Find upper bound for k ∗ ( δ, y δ ) ≤ ¯ k ( δ ) = O (log 1 /δ ) � � � � �� d 0 + α 0 C θ − q R ( x † ) p log(1 /δ )+log − log τ − ˜ k ∗ ( δ, y δ ) ≤ ¯ 1 − q k ( δ ) = − 1 log 1 /θ 4. Find upper bound for R ( x δ k ) ≤ R for k ∈ { 1 , . . . , k ∗ ( δ, y δ ) } � � � − 1 � � � α 0 + R ( x † ) d 0 C C R := θ 1 + τ − ˜ θ − q 1 − q 1 − q

Setting Continuous version Discretized version Error estimate Model Example Proof idea 1. Existence of minimizer: direct method of calculus of variations 2. Find a recursive estimate formula for � F ( x δ k ) − y δ � p 3. Find upper bound for k ∗ ( δ, y δ ) ≤ ¯ k ( δ ) = O (log 1 /δ ) � � � � �� d 0 + α 0 C θ − q R ( x † ) p log(1 /δ )+log − log τ − ˜ k ∗ ( δ, y δ ) ≤ ¯ 1 − q k ( δ ) = − 1 log 1 /θ 4. Find upper bound for R ( x δ k ) ≤ R for k ∈ { 1 , . . . , k ∗ ( δ, y δ ) } � � � − 1 � � � α 0 + R ( x † ) d 0 C C R := θ 1 + τ − ˜ θ − q 1 − q 1 − q 5. Repeat for IRGNM-Ivanov

Setting Continuous version Discretized version Error estimate Model Example Discretized version IRGNM-Tikhonov ′ x δ h � F k h ( x δ k , h )( x − x δ k , h ) − ( y δ − F k h ( x δ k , h )) � p + α k R ( x ) k +1 , h ∈ argmin x ∈D ( F ) ∩ X k k = 0 , 1 , . . . p ∈ [1 , ∞ ) IRGNM-Ivanov ′ x δ h � F k h ( x δ k , h )( x − x δ k , h ) − ( y δ − F k h ( x δ k +1 , h ∈ argmin x ∈D ( F ) ∩ X k k , h )) � s.t. R ( x ) ≤ ρ k k = 0 , 1 , . . .

Setting Continuous version Discretized version Error estimate Model Example Approach Auxiliary sequence

Setting Continuous version Discretized version Error estimate Model Example Approach Auxiliary sequence • ✚ x 4 ✚✚✚✚✚ h 4 ⋄ ✟ x 4 ✟✟✟✟✟ • x 3 � � h 3 � ⋄ � ✟ x 3 ✟✟✟✟✟ � ✚� • x 2 ✟✚✚✚✚✚ h 2 ⋄ ✟ ✑✟✟✟✟✟ x 2 x 1 ⋄ ✑✑✑✑✑ • ✟✟✟✟✟ x 1 h 1 • x 0 ✲ • • • • • k = 0 k = 1 k = 2 k = 3 k = 4

Setting Continuous version Discretized version Error estimate Model Example Convergence result (discretized version) Corollary Assume the error estimates � F ( x δ k , h ) − y δ � − � F ( x δ k ) − y δ � ≤ η k � F k h ( x δ k , h ) − y δ � − � F k h ( x δ k ) − y δ � ≤ ξ k R ( x δ k , h ) − R ( x δ k ) ≤ ζ k hold with ζ k ≤ ¯ η k ≤ ¯ τδ, ξ k ≤ ˆ τδ, ζ (tolerance) for all k ≤ k ∗ ( δ, y δ ). Then, the Theorem remains valid for x δ k ∗ ( δ, y δ ) , h in place of x δ k ∗ ( δ, y δ ) .

Setting Continuous version Discretized version Error estimate Model Example Goal oriented error estimators � A ( x , u ) = 0 model equation F ( x ) = y ⇔ C ( u ) = y observation operator for F = C ◦ S such that A ( x , S ( x )) = 0, S parameter-to-state map

Setting Continuous version Discretized version Error estimate Model Example Goal oriented error estimators � A ( x , u ) = 0 model equation F ( x ) = y ⇔ C ( u ) = y observation operator for F = C ◦ S such that A ( x , S ( x )) = 0, S parameter-to-state map IRGNM-Tikhonov in a decoupled way ( x δ k +1 , h , v δ k , h , u δ k +1 , h , u δ k , h ) solves u ) − y δ � p + α k R ( x ) ′ (˜ min u ) � C u ) v + C (˜ ( x , v , u , ˜ ′ ′ x ( x δ u )( x − x δ u ( x δ s.t. A k , h , ˜ k , h ) + A k , h , ˜ u ) v = 0 , A ( x δ k , h , ˜ u ) = 0 , A ( x , u ) = 0

Setting Continuous version Discretized version Error estimate Model Example Goal oriented error estimators � A ( x , u ) = 0 model equation F ( x ) = y ⇔ C ( u ) = y observation operator for F = C ◦ S such that A ( x , S ( x )) = 0, S parameter-to-state map IRGNM-Tikhonov in a decoupled way ( x δ k +1 , h , v δ k , h , u δ k +1 , h , u δ k , h ) solves u ) − y δ � p + α k R ( x ) ′ (˜ min u ) � C u ) v + C (˜ ( x , v , u , ˜ ′ ′ x ( x δ u )( x − x δ u ( x δ s.t. A k , h , ˜ k , h ) + A k , h , ˜ u ) v = 0 , A ( x δ k , h , ˜ u ) = 0 , A ( x , u ) = 0 Quantities of interest u ) − y δ � I 1 ( x , v , u , ˜ u ) = � C (˜ u ) = � C ( u ) − y δ � � just in theory I 2 ( x , v , u , ˜ I 3 ( x , v , u , ˜ u ) = R ( x )

Setting Continuous version Discretized version Error estimate Model Example Computing η k , ξ k , ζ k for IRGNM-Tikhonov • Lagrange functional L ( z ), z its stationary point, ′ ( z )( dz ) = 0 , ∀ dz L ′ ( z )(¯ • Auxiliary functional M i ( z , ¯ z ) = I i ( z ) + L z ), i = 1 , 2 , 3, ˜ z = ( z , ¯ z ) its stationary point z h , ˜ z h are the discrete stationary points of L and M