Time-Parallel optimal control solver for parabolic equations - PowerPoint PPT Presentation

Time-Parallel optimal control solver for parabolic equations Mohamed Kamal RIAHI, joint work with Yvon MADAY & Julien SALOMON 22 mai 2011 MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 1 / 21

Optimal control’s tools The cost quadratic functional considered is as below � T J ( v ) := 1 2 � y ( T ) − y target � 2 + α � v � 2 dt (1) 2 0 The primal state variable y ( t , x ; v ) depends linearly on the control v through the heat equation starting form the initial data y 0 The first derivative of the Euler-Lagrange equations gives the following optimality system ;  ∂ t y − µ ∆ y = Bv Primal (2) y ( t = 0 ) = y 0  ∂ t p + µ ∆ p = 0 Dual (3) p ( t = T ) = y ( T ) − y target ∇ J ( v ) = α v + t Bp . gradient (4) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 2 / 21

Summary 1 Introduction Tools of : Optimal Control Solver 2 SITPOC : Parallel optimal control solver Parallelization setting 3 Coupling SITPOC & Parareal Parareal algorithm PITPOC algorithm 4 Numerical experiments 5 Further reading 6 Conclusion MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 3 / 21

Parallelization of the optimal control solver Now we aim to : Divide the global system into series of dependent time problems Solve iteratively sub problem independently, so that in the limit the solution of the global problem is obtained. The key ingredient is the introduction of the intermediate targets and initial conditions as follows. ∀ n ≥ 0 ◮ λ n := y ( t n ; v n ) , γ n := p ( t n ; v n ) on [ 0 , T ] × Ω (5) ◮ ξ ( t n ; v n ) := y ( t n ; v n ) − p ( t n ; v n ) , on [ 0 , T ] × Ω (6) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 4 / 21

Parallelization of the optimal control solver Now we aim to : Divide the global system into series of dependent time problems Solve iteratively sub problem independently, so that in the limit the solution of the global problem is obtained. The key ingredient is the introduction of the intermediate targets and initial conditions as follows. ∀ n ≥ 0 ◮ λ n := y ( t n ; v n ) , γ n := p ( t n ; v n ) on [ 0 , T ] × Ω (5) ◮ ξ ( t n ; v n ) := y ( t n ; v n ) − p ( t n ; v n ) , on [ 0 , T ] × Ω (6) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 4 / 21

Parallelization of the optimal control solver Now we aim to : Divide the global system into series of dependent time problems Solve iteratively sub problem independently, so that in the limit the solution of the global problem is obtained. The key ingredient is the introduction of the intermediate targets and initial conditions as follows. ∀ n ≥ 0 ◮ λ n := y ( t n ; v n ) , γ n := p ( t n ; v n ) on [ 0 , T ] × Ω (5) ◮ ξ ( t n ; v n ) := y ( t n ; v n ) − p ( t n ; v n ) , on [ 0 , T ] × Ω (6) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 4 / 21

Parallelization of the optimal control solver Now we aim to : Divide the global system into series of dependent time problems Solve iteratively sub problem independently, so that in the limit the solution of the global problem is obtained. The key ingredient is the introduction of the intermediate targets and initial conditions as follows. ∀ n ≥ 0 ◮ λ n := y ( t n ; v n ) , γ n := p ( t n ; v n ) on [ 0 , T ] × Ω (5) ◮ ξ ( t n ; v n ) := y ( t n ; v n ) − p ( t n ; v n ) , on [ 0 , T ] × Ω (6) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 4 / 21

Parallelization setting That definition allows us to define sub cost functional as follows ; � T n + 1 J n ( w n , λ n , ξ n + 1 ) := 1 n + 1 ) − ξ n + 1 � 2 + α � w n � 2 dt 2 � y n ( T − (7) 2 T n where y n ( T − n + 1 ) is the solution at time t = T n + 1 evolved from the initial data y n ( T + n ) = λ n according to the PDE : ∂ t y n + µ ∆ y n = Bw n . Here we note that the local (on I n := [ T n , T n + 1 ] ) optimality system is : � ∂ t y n − µ ∆ y n = Bw n on I n × Ω (8) y n ( t = n ) = λ n � ∂ t p n + µ ∆ p n = 0 on I n × Ω (9) p n ( t − n + 1 ) = y n ( t − n + 1 ) − ξ n + 1 ∇ J n ( w n ) = α w n + t Bp n . (10) MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 5 / 21

Theoretical support Lemma (Consistence Lemma) Let τ ∈ ] 0 , T [ , and the optimal control problem : Find w ⋆ τ ∈ H such that w ⋆ τ := argmin w ∈H J τ ( w ) where Z T n + 1 J τ ( w ) := 1 2 � y ( τ ) − ξ ⋆ ( τ ) � 2 + α � w � 2 (11) 2 T n with y ( τ ) the solution of Equation (2) . We have w ⋆ τ = v ⋆ I [ 0 ,τ ] Intermediate targets : With the notations above, denote by ξ ⋆ the target trajectory defined by Equation (6) with y = y ⋆ and p = p ⋆ and by y ⋆ n , p ⋆ n , v ⋆ n the solutions of Equations (8–10) associated with v ⋆ . One has : v ⋆ n = v ⋆ | In . With an arbitrary subinterval index n MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 6 / 21

Play SITPOC algorithm MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 7 / 21

Over view of the Parareal algorithm The parareal algorithm is an iterative preconditioned scheme that ensure convergence at its n th iteration, this happens thanks to the pascal triangle behavior. λ k + 1 n + 1 = G ∆ T ( λ k + 1 ) + F ∆ T ( λ k n ) − G ∆ T ( λ k n ) (12) n Compatibility with parallel architecture. No sleeping process (with some particular implementation). Fast convergence if it holds (stability question). For instance we get : when the error is about 1 . E − 4 � � Nb processor 4 # 8 # 16 # � � � � Nb itrations 3 3 3 � � � � Wallclock mn : s 2 : 23 1 : 15 0 : 49 � � MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 8 / 21

Over view of the Parareal algorithm The parareal algorithm is an iterative preconditioned scheme that ensure convergence at its n th iteration, this happens thanks to the pascal triangle behavior. λ k + 1 n + 1 = G ∆ T ( λ k + 1 ) + F ∆ T ( λ k n ) − G ∆ T ( λ k n ) (12) n Compatibility with parallel architecture. No sleeping process (with some particular implementation). Fast convergence if it holds (stability question). For instance we get : when the error is about 1 . E − 4 � � Nb processor 4 # 8 # 16 # � � � � Nb itrations 3 3 3 � � � � Wallclock mn : s 2 : 23 1 : 15 0 : 49 � � MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 8 / 21

Over view of the Parareal algorithm The parareal algorithm is an iterative preconditioned scheme that ensure convergence at its n th iteration, this happens thanks to the pascal triangle behavior. λ k + 1 n + 1 = G ∆ T ( λ k + 1 ) + F ∆ T ( λ k n ) − G ∆ T ( λ k n ) (12) n Compatibility with parallel architecture. No sleeping process (with some particular implementation). Fast convergence if it holds (stability question). For instance we get : when the error is about 1 . E − 4 � � Nb processor 4 # 8 # 16 # � � � � Nb itrations 3 3 3 � � � � Wallclock mn : s 2 : 23 1 : 15 0 : 49 � � MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 8 / 21

Over view of the Parareal algorithm The parareal algorithm is an iterative preconditioned scheme that ensure convergence at its n th iteration, this happens thanks to the pascal triangle behavior. λ k + 1 n + 1 = G ∆ T ( λ k + 1 ) + F ∆ T ( λ k n ) − G ∆ T ( λ k n ) (12) n Compatibility with parallel architecture. No sleeping process (with some particular implementation). Fast convergence if it holds (stability question). For instance we get : when the error is about 1 . E − 4 � � Nb processor 4 # 8 # 16 # � � � � Nb itrations 3 3 3 � � � � Wallclock mn : s 2 : 23 1 : 15 0 : 49 � � MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 8 / 21

PITPOC algorithm In this algorithm, our aim is to Reduce complexity by applying the coarse operator G instead of the fine operator F Replace y ( T )( v ) by λ N on the functional J in order to hang over the target solution y target Z T Φ v k ,λ k ( θ ) := 1 ( θ ) − y target � 2 + α 2 � λ k + 1 � v k + 1 ( θ ) � 2 dt N 2 0 Use parallel information in order to correct predictor propagator in the sequential part of the algorithm Optimize relaxation of the coupled parareal-control algorithm MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 9 / 21

PITPOC algorithm In this algorithm, our aim is to Reduce complexity by applying the coarse operator G instead of the fine operator F Replace y ( T )( v ) by λ N on the functional J in order to hang over the target solution y target Z T Φ v k ,λ k ( θ ) := 1 ( θ ) − y target � 2 + α 2 � λ k + 1 � v k + 1 ( θ ) � 2 dt N 2 0 Use parallel information in order to correct predictor propagator in the sequential part of the algorithm Optimize relaxation of the coupled parareal-control algorithm MK.Riahi, Y.Maday & J.Salomon () Parallel In Time Optimal Control Solver 22 mai 2011 9 / 21