Time parallelisation for optimal control and data assimilation - PowerPoint PPT Presentation

Time parallelisation for optimal control and data assimilation Julien Salomon Joint work with M. Gander, F. Kwok, S. Reyes-Riffo JLL & INRIA, ANGE project-team (Inria, Cerema, UPMC, CNRS) PinT 2018, Roscoff

Problem 1 : control on a fixed, bounded interval [0 , T ] Given T > 0, consider the optimal control problem associated with the cost functional � T J ( c ) = 1 2 � x ( T ) − x target � 2 + α c 2 ( t ) dt, 2 0 where the state function x evolution is described by an equation : x ( t ) = f ( x ( t ) , c ( t )) , ˙ with initial condition x (0) = x init . Given an optimal control solver, Objective : combine it with a time-parallelization.

Problem 2 : assimilation on an unbounded interval [ t 0 , + ∞ ) Given a (linear) dynamic x ( t ) = Ax ( t ) + Bu ( t ) ˙ whose initial condition is NOT known , and an output y ( t ) = Cx ( t ) , which is known . Combine observer approaches Objective : with a time-parallelization.

Previous works : Hackbusch, 1984 : Multgrid approach Borzì , 2003 : Multigrid for parabolic distributed Heinkenschloss, 2005 : Block symmetric Gauss-Seidel preconditioning Maday, Turinici, J.S. 2007 : intermediate states approach Mathew, Sarkis, 2010 : combination of a shooting method and parareal preconditionning

Outline 1 Non-linear Control 2 Linear Control Time sub-intervals decomposition Use of a coarse solver Numerical examples 3 Unbounded time domains and assimilation Algorithm Analysis Numerical example

"Non-linear control" or "Bilinear control" Linear eq. Non-linear eq. "Linear" control y = Ay + Bc ˙ y = f ( y ) + Bc ˙ Non-linear control y = A ( c ) y ˙ y = f ( y, c ) ˙ y = y ( t, x ) state c = c ( t ) or c ( t, x ) control

Non-linear Control The Intermediate States Method Schematic description • U cible dU dt = f ( U, c ) • Initial points / Λ = � � Intermediate targets λ j • j =1 , ··· , 5 • U initial • • Time of control CPU CPU CPU CPU CPU 1 2 3 4 5 Disclaimer : not a parareal algorithm. Y. Maday, J. Salomon, G. Turinici, SIAM J. Num. Anal., 45 (6), 2007. K. M. Riahi, J. Salomon, S. J. Glaser, D. Sugny, Phys. Rev. A, 93 (4), 2016.

Outline 1 Non-linear Control 2 Linear Control Time sub-intervals decomposition Use of a coarse solver Numerical examples 3 Unbounded time domains and assimilation Algorithm Analysis Numerical example

"Linear control" Linear eq. Non-linear eq. "Linear" control y = Ay + Bc ˙ y = f ( y ) + Bc ˙ Non-linear control y = A ( c ) y ˙ y = f ( y, c ) ˙ y = y ( t, x ) state c = c ( t ) or c ( t, x ) control

Linear Control The optimality condition then reads   y ( t ) ˙ = f ( y ( t )) + c ( t ) ,  − ( f ( y ( t )) ′ ) T λ ( t ) , ˙ λ ( t ) =   αc ( t ) = − λ ( t ) . → Elimination of c :  y = f ( y ) − λ   ˙ α,     λ = − ( f ( y ) ′ ) T λ, ˙ and final condition λ ( T ) = y ( T ) − y target . � � Y Time discretization ⇒ M δt = b Λ

Linear Control Time parallelization Our approach is based on two ideas : 1 Partition the time interval [0 , T ] : T 0 = 0 < T 1 < . . . < T L = T . 2 Coarse approximation of the inverse : M δt → M ∆ t .

Outline 1 Non-linear Control 2 Linear Control Time sub-intervals decomposition Use of a coarse solver Numerical examples 3 Unbounded time domains and assimilation Algorithm Analysis Numerical example

Linear Control Time sub-intervals decomposition Boundary value problems notations : on the subinterval [ T l , T l +1 ] with initial condition y ( T l ) = y l and final condition λ ( T l +1 ) = λ l +1 , we denote � � � � y ( T l +1 ) P ( y l , λ l +1 ) = . λ ( T l ) Q ( y l , λ l +1 )

Linear Control Time sub-intervals decomposition The optimality system is enriched : y 0 − y init = 0 y 1 − P ( y 0 , λ 1 ) = 0 λ 1 − Q ( y 1 , λ 2 ) = 0 y 2 − P ( y 1 , λ 2 ) = 0 λ 2 − Q ( y 2 , λ 3 ) = 0 (1) . . . . . . y L − P ( y L − 1 , λ L ) = 0 λ L − y L + y target = 0 That is : a system of boundary value subproblems, satisfying matching conditions .

Linear Control Time sub-intervals decomposition Collecting the unknowns in the vector ( Y T , Λ T ) := ( y 0 , y 1 , y 2 , . . . , y L , λ 1 , λ 2 , . . . , λ L ) , we obtain the nonlinear system   y 0 − y init   y 1 − P ( y 0 , λ 1 )     y 2 − P ( y 1 , λ 2 )     .   .  .    F ( Y T , Λ T ) :=   y L − P ( y L − 1 , λ L ) = 0 .     λ 1 − Q ( y 1 , λ 2 )     λ 2 − Q ( y 2 , λ 3 )     .   .  .  λ L − y L + y target

Linear Control Time sub-intervals decomposition Newton’s method : F ′ � �� � � � Y n +1 − Y n Y n Y n = −F , Λ n Λ n +1 − Λ n Λ n where the Jacobian matrix of F is given by F ′ � � Y = Λ   1 − P Y ( Y 0 , Λ 1 ) 1 − P Λ ( Y 0 , Λ 1 )   ... ... ...      − P Y ( Y N − 1 , Λ N ) 1 − P Λ ( Y N − 1 , Λ N )    − Q Y ( Y 1 , Λ 2 ) 1 − Q Λ ( Y 1 , Λ 2 )    ... ... ...  − Q Y ( Y N − 1 , Λ N ) 1 − Q Λ ( Y N − 1 , Λ N ) − 1 1

Outline 1 Non-linear Control 2 Linear Control Time sub-intervals decomposition Use of a coarse solver Numerical examples 3 Unbounded time domains and assimilation Algorithm Analysis Numerical example

Linear Control Use of a coarse solver Third idea : coarse approximation of the Jacobian F ′ ≈ finite difference

Linear Control Use of a coarse solver Which concretely corresponds to : ℓ )( y n +1 P G ( y n +1 P y ( y n ℓ − 1 , λ n ℓ − 1 − y n ℓ − 1 , λ n ℓ ) − P G ( y n ℓ − 1 , λ n ℓ − 1 ) ≈ ℓ ) , ℓ )( λ n +1 ℓ − 1 , λ n +1 P λ ( y n ℓ − 1 , λ n − λ n P G ( y n ) − P G ( y n ℓ − 1 , λ n ℓ ) ≈ ℓ ) , ℓ ℓ ℓ )( λ n +1 ℓ − 1 , λ n +1 Q λ ( y n ℓ − 1 , λ n − λ n Q G ( y n ) − Q G ( y n ℓ − 1 , λ n ℓ ) ≈ ℓ ) , ℓ ℓ ℓ )( y n +1 Q G ( y n +1 Q y ( y n ℓ − 1 , λ n ℓ − 1 − y n ℓ − 1 , λ n ℓ ) − Q G ( y n ℓ − 1 , λ n ℓ − 1 ) ≈ ℓ ) . → Inspiration from the Parareal algorithm : J.-L. Lions, Y. Maday, and G. Turinici. A "parareal" in time disretization of pde’s. Comptes Rendus de l’Acad. des Sciences, 2001. → and its interpretation : M. Gander, S. Vandewalle, SISC 2003.

Linear Control Parareal for Control Partial summary : In parallel : all fine propagations on sub-intervals. Sequential part : only coarse solving.

Linear Control Linear dynamics Example : linear dynamics y ( t ) = σy ( t ) + c ( t ) . ˙ Discretizing and setting : � � Y X = , Λ we get : � � X k +1 = X k + M − 1 Id − M − 1 ∆ t M δt ∆ t b.

Linear Control Linear dynamics Example : linear dynamics y ( t ) = σy ( t ) + c ( t ) . ˙ Discretizing and setting : � � Y X = , Λ we get : � � X k +1 = X k + M − 1 Id − M − 1 ∆ t M δt ∆ t b. Analyze the eigenvalues of Id − M − 1 ∆ t M δt !

Linear Control Linear dynamics Results for implicit Euler : Contraction factor : ρ ≤ C (∆ t − δt ) For σ < 0, C can be chosen independent of σ For very large α , C can grow like log( α ) when the number of subdomains becomes large F. Kwok, M. Gander, J. Salomon, to appear ...

Outline 1 Non-linear Control 2 Linear Control Time sub-intervals decomposition Use of a coarse solver Numerical examples 3 Unbounded time domains and assimilation Algorithm Analysis Numerical example

Linear Control Numerical example : Linear dynamics y ( t ) = σy ( t ) + c ( t ) . ˙ Convergence for various values of r = δt/ ∆ t for fixed δt = δt 0 .

Linear Control Numerical example : Linear dynamics Convergence for various with respect to the number of iteration for various number of subintervals.

Linear Control Numerical example : Linear dynamics y ( t ) = σy ( t ) + c ( t ) . ˙ Convergence for various values of r = δt/ ∆ t for fixed δt = δt 0 .

Linear Control Numerical example : Linear dynamics Convergence for various with respect to the number of iteration for various number of subintervals.

Linear Control Numerical example : Non-linear vectorial dynamics Minimize � 1 J ( c ) = 1 2 | y (1) − y target | 2 + 1 | c ( t ) | 2 dt 2 0 with y target = (100 , 20) T , subject to the Lotka-Volterra equation y 1 = a 1 y 1 − b 1 y 1 y 2 + c 1 , ˙ y 2 = a 2 y 1 y 2 − b 2 y 2 + c 2 ˙ with initial conditions y (0) = (20 , 10) T Backward Euler, δt = 10 − 5

Linear Control Numerical example : Non-linear vectorial dynamics Vector example - N = 10, r = δt/ ∆ t = 0 . 01 Iteration #1

Linear Control Numerical example : Non-linear vectorial dynamics Vector example - N = 10, r = δt/ ∆ t = 0 . 01 Iteration #2

Linear Control Numerical example : Non-linear vectorial dynamics Vector example - N = 10, r = δt/ ∆ t = 0 . 01 Iteration #3

Linear Control Numerical example : Non-linear vectorial dynamics Vector example - N = 10, r = δt/ ∆ t = 0 . 01 Iteration #4