Variational discretization of PDE constrained optimal control - PowerPoint PPT Presentation

Variational discretization of PDE constrained optimal control problems with measure controls Evelyn Herberg joint work with Michael Hinze and Henrik Schumacher Workshop: New trends in PDE constrained optimization October 14, 2019 - RICAM Linz, Austria

Motivation Goal: preserve sparsity on the discrete level Herberg Variational discretization October 14, 2019 1

Problem formulation Parabolic control problem (from [2]) J ( u 0 , u ) = 1 q � y − y d � q min L q ( Q ) + α � u � M ( ¯ Q c ) + β � u 0 � M ( ¯ Ω c ) , ( u 0 , u ) ∈M ( ¯ Ω c ) ×M ( ¯ Q c ) ( P ) where y solves the state equation:  ∂ t y − △ y = u in Q = Ω × (0 , T )    in Ω ⊂ R d y ( x , 0) = u 0   y ( x , t ) = 0 on Σ = Γ × (0 , T )  Theorem ([2, Theorem 2.7]) u 0 ) ∈ M (¯ Q c ) × M ( ¯ Problem (P) has at least one solution (¯ u , ¯ Ω c ) for � � 2 , d +2 every 1 ≤ q < min . If q > 1 the solution is unique. d ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 2

Problem formulation Optimality conditions Theorem ([2, Theorem 3.1]) Let (¯ u , ¯ u 0 ) denote a solution to (P) with associated state ¯ y. Then there 0 ( Ω )) ∩ C (¯ exists an element w ∈ L 2 (0 , T ; H 1 Q ) satisfying  − ∂ t w − △ w = ¯ g in Q ,    w ( x , T ) = 0 in Ω,   w ( x , t ) = 0 on Σ ,  where y − y d | q − 2 (¯  � � 2 , d +2 = | ¯ y − y d ) , if 1 < q < min ,  d g ¯ ∈ sign (¯ y − y d ) , if q = 1 .  ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 3

Problem formulation Optimality conditions Theorem (continued) w also fulfills:  � Q c w d ¯ u + α � ¯ u � M ( ¯ Q c ) = 0 , ¯    � = α , if ¯ u � = 0 , � w � C ( ¯  Q c )  ≤ α , if ¯ u = 0   � Ω c w (0) d ¯ u 0 + α � ¯ u 0 � M ( ¯ Ω c ) = 0 , ¯    � = β , if ¯ u 0 � = 0 , � w (0) � C ( ¯  Ω c )  ≤ β , if ¯ u 0 = 0  Furthermore w is unique if q > 1 . ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 4

Problem formulation Sparsity structure Remark ([2, Corollary 3.2]) Let (¯ u , ¯ u 0 ) denote a solution to ( P ) with associated state ¯ y. Then we have the following sparsity structure: � � u + ) ⊂ ( x , t ) ∈ ¯ Supp (¯ Q c : w ( x , t ) = − α � � u − ) ⊂ ( x , t ) ∈ ¯ Supp (¯ Q c : w ( x , t ) = + α � � u + x ∈ ¯ Supp (¯ 0 ) ⊂ Ω c : w ( x , 0) = − β � � u − x ∈ ¯ Supp (¯ 0 ) ⊂ Ω c : w ( x , 0) = + β u + − ¯ u − and ¯ u + u − where ¯ u = ¯ u 0 = ¯ 0 − ¯ 0 are the Jordan decompositions. ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 5

Fenchel predual (based on [3]) • W := { w ∈ L p ( Q ) : ∂ t w , ∂ x w , ∂ 2 x w ∈ L p ( Q ) , w | ¯ Ω ×{ T } = w | Γ × [0 , T ] = 0 } → C (¯ Q ), Φ : W → C ( ¯ Ω c ) × C (¯ Q c ) embedding and restriction, 1 q + 1 p = 1, • W ֒ • L := − ( ∂ t + ∆) : W → L p ( Q ) the adjoint operator of the state equation The Fenchel predual of (P) is: p � Lw � p w ∈ W q K ( w ) := 1 ( P ∗ ) min L p ( Q ) + � Lw , y d � L p ( Q ) , L q ( Q ) + ℓ α,β (Φ w ) , with ℓ α,β : C ( ¯ Ω c ) × C (¯ Q c ) → ¯ R � 0 , if � f � C ( ¯ Q c ) ≤ α and � f 0 � C ( ¯ Ω c ) ≤ β, ℓ α,β ( f 0 , f ) := ∞ , else . − → existence and uniqueness (for q > 1) of solutions ——————————————————————————————— [3] Clason, C., & Kunisch, K. (2011) Herberg Variational discretization October 14, 2019 6

Variational discretization (based on [6]) q � y σ ( u 0 , u ) − y d � q J σ ( u 0 , u ) := 1 min ( P σ ) L q ( Q h ) ( u 0 , u ) ∈ M ( ¯ Ω c ) ×M ( ¯ Q c ) + α � u � M ( ¯ Q c ) + β � u 0 � M ( ¯ Ω c ) Petrov-Galerkin method • discrete state space: • piecewise linear and continuous finite elements in space • piecewise constant functions w.r.t. time • discrete test space: • piecewise linear and continuous finite elements in space • piecewise linear and continuous functions w.r.t. time • control space is not discretized • yields Crank-Nicholson scheme with implicit Euler step [4] , [5] ——————————————————————————————— [4] Daniels, N. von, Hinze, M., & Vierling, M. (2015) [5] Goll, C., Rannacher, R., & Wollner, W. (2015) [6] Hinze, M. (2005) Herberg Variational discretization October 14, 2019 7

Variational discretization discrete optimality system delivers: � ¯ w σ � ∞ ≤ α, � ¯ w 0 , h � ∞ ≤ β, � � u + ) ⊂ ( x , t ) ∈ ¯ supp(¯ Q c : ¯ w σ ( x , t ) = − α , � � u − ) ⊂ ( x , t ) ∈ ¯ supp(¯ Q c : ¯ w σ ( x , t ) = + α , � � u + x ∈ ¯ supp(¯ 0 ) ⊂ Ω c : ¯ w 0 , h ( x ) = − β , � � u − x ∈ ¯ supp(¯ 0 ) ⊂ Ω c : ¯ w 0 , h ( x ) = + β . Herberg Variational discretization October 14, 2019 8

Variational discretization discrete optimality system delivers: � ¯ w σ � ∞ ≤ α, � ¯ w 0 , h � ∞ ≤ β, � � u + ) ⊂ ( x , t ) ∈ ¯ supp(¯ Q c : ¯ w σ ( x , t ) = − α , � � u − ) ⊂ ( x , t ) ∈ ¯ supp(¯ Q c : ¯ w σ ( x , t ) = + α , � � u + x ∈ ¯ supp(¯ 0 ) ⊂ Ω c : ¯ w 0 , h ( x ) = − β , � � u − x ∈ ¯ supp(¯ 0 ) ⊂ Ω c : ¯ w 0 , h ( x ) = + β . in the generic setting: supp(¯ u ) ⊂ { ( x j , t k ) } and supp(¯ u 0 ) ⊂ { ( x j ) } . induced discrete control space: dirac measures in space and time U h × U σ Herberg Variational discretization October 14, 2019 8

Variational discretization q � y σ ( u 0 , u ) − y d � q J σ ( u 0 , u ) := 1 min ( P σ ) L q ( Q h ) ( u 0 , u ) ∈ M ( ¯ Ω c ) ×M ( ¯ Q c ) + α � u � M ( ¯ Q c ) + β � u 0 � M ( ¯ Ω c ) , where y σ ( u 0 , u ) = L −∗ σ (Φ ∗ h Υ h u 0 + Φ ∗ σ Υ σ u ). (Φ h ⊕ Φ σ ) ∗ : U h × U σ → W ∗ ( Υ h ⊕ Υ σ ) : M ( ¯ Ω c ) × M ( ¯ L −∗ : W ∗ Q c ) → U h × U σ , σ , σ → Y σ σ Herberg Variational discretization October 14, 2019 9

Variational discretization q � y σ ( u 0 , u ) − y d � q J σ ( u 0 , u ) := 1 min ( P σ ) L q ( Q h ) ( u 0 , u ) ∈ M ( ¯ Ω c ) ×M ( ¯ Q c ) + α � u � M ( ¯ Q c ) + β � u 0 � M ( ¯ Ω c ) , where y σ ( u 0 , u ) = L −∗ σ (Φ ∗ h Υ h u 0 + Φ ∗ σ Υ σ u ). (Φ h ⊕ Φ σ ) ∗ : U h × U σ → W ∗ ( Υ h ⊕ Υ σ ) : M ( ¯ Ω c ) × M ( ¯ L −∗ : W ∗ Q c ) → U h × U σ , σ , σ → Y σ σ u ) ∈ M ( ¯ Ω c ) × M (¯ • multiple solutions (ˆ u 0 , ˆ Q c ) • unique solution (¯ u 0 , h , ¯ u σ ) ∈ U h × U σ for q > 1 u ) ∈ M ( ¯ Ω c ) × M (¯ • ( Υ h ˆ u 0 , Υ σ ˆ u ) = (¯ u 0 , h , ¯ u σ ) for all solutions (ˆ u 0 , ˆ Q c ) (similar to [1]) ——————————————————————————————— [1] Casas, E., Clason, C., & Kunisch, K. (2013) Herberg Variational discretization October 14, 2019 9

Variational discretization - Convergence Theorem Let (¯ u 0 , h , ¯ u σ ) be the unique solution of ( P σ ) belonging to U h × U σ , Γ of class C 1 , 1 , 1 < q < min { 2 , d +2 d } and σ = ( τ, h ) the discretization parameter consisting of temporal ( τ ) and spatial (h) gridsizes . If { (¯ u 0 , h , ¯ u σ ) } σ is a sequence of such solutions with associated states { ¯ y σ } σ the following convergence properties hold: | σ |→ 0 � ¯ lim y − ¯ y σ � L q ( Q ) = 0 , u σ ) ∗ u ) as | σ | → 0 in M ( ¯ Ω c ) × M (¯ (¯ u 0 , h , ¯ ⇀ (¯ u 0 , ¯ Q c ) , | σ |→ 0 ( � ¯ lim u 0 h � M ( ¯ Ω c ) , � ¯ u σ � M ( ¯ Q c ) ) = ( � ¯ u 0 � M ( ¯ Ω c ) , � ¯ u � M ( ¯ Q c ) ) , where (¯ u 0 , ¯ u ) is the unique solution of ( P σ ) and ¯ y associated state. proof similar to the convergence proof in [2] ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 10

Full Discretization (from [2]) J DG ( u 0 h , u σ ) := 1 q � y σ ( u 0 h , u σ ) − y d � q min ( P σ ) L q ( Q h ) ( u 0 h , u σ ) ∈ U σ × U h + α � u σ � M ( ¯ Q c ) + β � u 0 h � M ( ¯ Ω c ) Discontinuous Galerkin method • discrete state and test space: • piecewise linear and continuous finite elements in space • piecewise constant functions w.r.t. time • discrete control space: • dirac measures concentrated in the finite element nodes in space • piecewise constant functions w.r.t. time • yields Euler time stepping scheme − → similar convergence properties ——————————————————————————————— [2] Casas, E., & Kunisch, K. (2016) Herberg Variational discretization October 14, 2019 11