Bang Bang control of elliptic PDEs M. Hinze Fachbereich Mathematik - PowerPoint PPT Presentation

Bang Bang control M. Hinze 1 Bang Bang control of elliptic PDEs M. Hinze Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg (joint work with Klaus Deckelnick) Mariatrost, October 11, 2011

Bang Bang control M. Hinze 2 ESF Summerschool and Workshop Adaptivity and Model Order Reduction in PDE Constrained Optimization Organisers Michael Hinze, Kunibert G. Siebert, and Winnifried Wollner Hamburg, July 23-27

Bang Bang control M. Hinze 3 Model problem  J ( u ) = 1 � | y − y 0 | 2 + α 2 � u � 2 min  ( P ) α L 2 2 u ∈ U ad Ω subject to y = G ( u ) .  Here, α ≥ 0 and we are interested in the solution for α = 0. U ad := { v ∈ L 2 (Ω); a ≤ u ≤ b } ⊆ L 2 (Ω) with a < b constants, and y = G ( u ) iff − ∆ y = u in Ω , and y = 0 on ∂ Ω . More general elliptic operators may be considered, and also control operators which map abstract controls to feasible right-hand sides of the elliptic equation.

Bang Bang control M. Hinze 4 Existence and uniqueness, optimality conditions The optimal control problem admits a unique solution. The function u ∈ U ad is a solution of the optimal control problem iff there exists an adjoint state p such that y = G ( u ) , p = G ( y − y 0 ) and ( α u + p , v − u ) ≥ 0 for all v ∈ U ad . � � − 1 There holds u = P U ad for α > 0, α p   a , α u + p > 0 , = a , p > 0 ,   − 1 u = α p , α u + p = 0 , if α > 0 , and u ∈ [ a , b ] p = 0 , if α = 0 . α u + p < 0 , = b , p < 0 , b ,  

Bang Bang control M. Hinze 5 Variational discretization Discrete optimal control problem:  J ( u ) = 1 � | y h − y 0 | 2 + α 2 � u � 2 min  ( P ) α L 2 2 u ∈ U ad h Ω subject to y h = G h ( u ) .  Here, G h ( u ) denotes the piecewise linear and continuous finite element approximation to y ( u ) , i.e. a ( y h , v h ) := ( ∇ y h , ∇ v h ) = ( u , v h ) for all v h ∈ X h , where with the triangulation T h X h := { w ∈ C 0 (¯ Ω); w | ∂ Ω = 0 , w | T linear for all T ∈ T h } . This problem is still ∞− dimensional. Ritz projection R h : H 1 0 (Ω) → X h , a ( R h w , v h ) = a ( w , v h ) for all v h ∈ X h

Bang Bang control M. Hinze 6 Existence and uniqueness, optimality conditions for discrete problem The variational-discrete optimal control problems admits a solution u h ∈ U ad , which is unique in the case α > 0. The state y h is unique (also in the case α = 0). Let u h ∈ U ad be a solution of the optimal control problem. Then there exists a unique adjoint state p h such that y h = G h ( u h ) , p h = G h ( y h − y 0 ) and ( α u h + p h , v − u h ) ≥ 0 for all v ∈ U ad . � � − 1 There holds u h = P U ad for α > 0, α p h   a , α u h + p h > 0 , = a , p h > 0 ,   − 1 u h = α p h , α u h + p h = 0 , if α > 0 , and u h ∈ [ a , b ] p h = 0 , if α = 0 . α u h + p h < 0 , = b , p h < 0 , b ,  

Bang Bang control M. Hinze 7 Error estimates It is well known that � y − y h � + α � u − u h � ∼ � y − y h ( u ) � + � p − p h ( y ( u )) � So one expects estimates for y − y h also in the case α = 0. Estimates for � u − u h � ? Estimate for the states ( S := { x ∈ Ω | p ( x ) � = 0 } ⊂ ¯ Ω ) � h 2 + ( b − a ) � p − R h p � L 1 (Ω \ S ) + � p − R h p � L ∞ � u − u h � L 1 ( S ) � � y − y h � ≤ C , � p − p h � L ∞ ≤ C � y − y h � + � p − R h p � L ∞ , follow from 0 ≤ ( p − p h , u h − u ) = ( R h p − p h , u h − u ) + ( p − R h p , u h − u ) ≡ I + II . 2 � y − y h � 2 + 1 I ≤ − 1 2 � y − R h y � 2 � � II = Ω \ S ( p − R h p )( u h − u ) + S ( p − R h p )( u h − u ) .

Bang Bang control M. Hinze 8 Error estimates Structural assumption ∃ C > 0 ∀ ǫ > 0 : L ( { x ∈ ¯ Ω; | p ( x ) | ≤ ǫ } ) ≤ C ǫ β for the solution u at α = 0 with some β ∈ ( 0 , 1 ] yields 1 � � h 2 + � p − R h p � 2 − β � y − y h � + � p − p h � L ∞ ≤ C ; L ∞ � � β h 2 β + � p − R h p � 2 − β � u − u h � L 1 ≤ C . L ∞

Bang Bang control M. Hinze 9 Sketch of proof for β = 1 � h 2 + � p − R h p � L ∞ � � u − u h � L 1 , � y − y h � , � p − p h � L ∞ ≤ C Sketch of proof: 0 ≤ ( p − p h , u h − u ) = ( R h p − p h , u h − u ) + ( p − R h p , u h − u ) ≡ I + II . 2 � y − y h � 2 + 1 I ≤ − 1 2 � y − R h y � 2 � II = S ( p − R h p )( u h − u ) . Combine now � u − u h � L 1 ≤ ( b − a ) L ( { p > 0 , p h ≤ 0 } ∪ { p < 0 , p h ≥ 0 } ) { p > 0 , p h ≤ 0 } ∪ { p < 0 , p h ≥ 0 } ⊆ {| p ( x ) | ≤ � p − p h � ∞ } ⇒ L ( {| p ( x ) | ≤ � p − p h � ∞ } ) ≤ C � p − p h � ∞ � u − u h � L 1 ≤ C � p − p h � ∞ � p − p h � ∞ ≤ � p − R h p � ∞ + � R h p − p h � ∞ � R h p − p h � ∞ ≤ C � y − y h � to estimate II .

Bang Bang control M. Hinze 10 Special cases 1. u 0 ∈ U ad exists such that y 0 = G ( u 0 ) . Then � y − y h � + � p − p h � L ∞ ≤ Ch 2 . 2. If p ∈ C 1 (¯ Ω) satisfies where K = { x ∈ ¯ min x ∈ K |∇ p ( x ) | > 0 , Ω | p ( x ) = 0 } . Then, the structural assumption is satisfied with β = 1. 3. If p ∈ W 2 , ∞ (Ω) and satisfies the structural assumption, then � y − y h � + � p − p h � L ∞ + � u − u h � L 1 ≤ Ch 2 | log h | γ ( d ) .

Bang Bang control M. Hinze 11 Algorithms for P α h Define � − 1 � G h ( u ) = u − P U ad α p h ( y h ( u )) . The optimality condition reads G h ( u ) = 0 and motivates the fix–point iteration u given, do until convergence � − 1 � u + = P U ad u = u + . α p h ( y h ( u )) , 1. Is this algorithm numerically implementable? Yes, whenever for given u it is possible to numerically evaluate the expression � − 1 � α p h ( y h ( u )) P U ad in the i − th iteration, with an numerical overhead which is independent of the iteration counter of the algorithm.

Bang Bang control M. Hinze 12 Semi–smooth Newton algorithm for α > 0 2. Does the fix–point algorithm converge? Yes, if α > � RB ∗ S ∗ h S h B � L ( U ) , since P U ad is non–expansive. Condition too restrictive for our purpose → semi–smooth Newton method applied to G h ( u ) = 0: u given, solve until convergence h ( u ) u + = − G h ( u ) + G ′ G ′ u = u + . h ( u ) u , 1. This algorithm is implementable whenever the fix–point iteration is, since − G h ( u ) + G ′ h ( u ) u = � − 1 � − 1 � − 1 � α P ′ S ∗ = − P U ad α p h ( u ) α p h ( u ) h S h u . U ad 2. For every α > 0 this algorithm is locally fast convergent (H. (COAP 2005), Vierling).

Bang Bang control M. Hinze 13 Numerical example with 2 switching points, fix-point iteration 1 exact exact 0.2 discrete discrete 0.8 FE grid FE grid 0.15 0.6 0.1 0.4 0.05 0.2 0 0 −0.05 −0.2 −0.1 −0.4 −0.15 −0.6 −0.2 −0.8 −0.25 −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25 0.3 0.35 0.4 0.45 Experimental order of convergence: � u − u h � L 1 : 3.00077834 Function values 1.99966106 � p − p h � L ∞ : 1.99979367 � y − y h � L ∞ : 1.9997965 � p − p h � L 2 : 1.99945711

Bang Bang control M. Hinze 14 Homotopy in α with semi–smooth Newton, Tröltzsch checkerboard D. & G. Wachsmuth (ESAIM: COCV 2011 (Preprint 2009)), von Daniels (Diploma Thesis 2010): � u 0 − u α � ∼ √ α , � u α − u α, h � ∼ h 2 α − 1 , thus 2 � u 0 − u α, h � ∼ h 3 1 u ( x ) = − sign p ( x ) , p ( x ) = − 128 π 2 sin ( 8 π x 1 ) sin ( 8 π x 2 ) , y ( x ) = sin ( π x 1 ) sin ( π x 2 ) . Loop i � u − u h � L 1 � u − u h � L 2 EOC L 1 ( u ) EOC L 2 ( u ) Nit 3 2.5008e-001 4.7416e-001 1.10 0.61 4 4 1.2045e-001 3.4864e-001 1.05 0.44 5 5 3.6487e-002 1.9368e-001 1.72 0.85 4 6 5.8124e-003 6.2070e-002 1.33 0.82 3 7 2.1287e-003 3.7590e-002 1.45 0.72 3 mean 1.33 0.69 Numerical example by Nicolaus von Daniels

Bang Bang control M. Hinze 15 Checkerboard example, plots

Bang Bang control M. Hinze 16 Related approaches, next steps Related approaches In a recent talk Walter Alt for linear–quadratic optimal control problems with ODEs proposed to use the zeros of the discrete switching function to define the control → This relates to post–processing of Meyer/Rösch combined with piecewise constant control approximations in the present situation. Structural assumptions on p imply the required regularity of the discrete active set. Next steps: Parabolic problems Thank you very much for your attention!

Bang Bang control M. Hinze 16 Related approaches, next steps Related approaches In a recent talk Walter Alt for linear–quadratic optimal control problems with ODEs proposed to use the zeros of the discrete switching function to define the control → This relates to post–processing of Meyer/Rösch combined with piecewise constant control approximations in the present situation. Structural assumptions on p imply the required regularity of the discrete active set. Next steps: Parabolic problems Thank you very much for your attention!