A multilevel stochastic gradient algorithm for PDE-constrained - PowerPoint PPT Presentation

A multilevel stochastic gradient algorithm for PDE-constrained optimal control problems under uncertainty Fabio Nobile CSQI – Institute of Mathematics, EPFL, Switzerland Joint work with : M. Martin (Criteo, Grenoble), S. Krumscheid (RWTH Aachen), P. Tsilifis (EPFL) RICAM Special Semestre on Optimization Workshop 3 “Optimization and Inversion under Uncertainty” November 11-15, 2019, Linz, Austria F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 1

Outline Problem setting – quadratic optimal control problem 1 Discretization by finite elements + Monte Carlo 2 Deterministic (CG) iterative solvers versus Stochastic Gradient 3 Multilevel stochastic gradient algorithms 4 Conclusions 5 F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 2

Problem setting – quadratic optimal control problem Outline Problem setting – quadratic optimal control problem 1 Discretization by finite elements + Monte Carlo 2 Deterministic (CG) iterative solvers versus Stochastic Gradient 3 Multilevel stochastic gradient algorithms 4 Conclusions 5 F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 3

Problem setting – quadratic optimal control problem Problem setting (Ω , F , P ): complete probability space F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 4

Problem setting – quadratic optimal control problem Problem setting (Ω , F , P ): complete probability space D ⊂ R d : physical domain. Throughout � · � = � · � L 2 ( D ) . F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 4

Problem setting – quadratic optimal control problem Problem setting (Ω , F , P ): complete probability space D ⊂ R d : physical domain. Throughout � · � = � · � L 2 ( D ) . Forward problem � − div( a ( x , ω ) ∇ y ( x , ω )) = g ( x ) + u ( x ) , for a.e. x ∈ D , ω ∈ Ω (*) y ( x , ω ) = 0 , for a.e. x ∈ ∂ D , ω ∈ Ω with a ( · , ω ) a random field s.t. 0 < a min ≤ a ( x , ω ) ≤ a max , ∀ ( x , ω ) ∈ D × Ω. ⇒ random solution ω �→ y ( · , ω ) ∈ H 1 0 ( D ). In particular y ∈ L 2 P (Ω; H 1 = 0 ( D )). F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 4

Problem setting – quadratic optimal control problem Problem setting (Ω , F , P ): complete probability space D ⊂ R d : physical domain. Throughout � · � = � · � L 2 ( D ) . Forward problem � − div( a ( x , ω ) ∇ y ( x , ω )) = g ( x ) + u ( x ) , for a.e. x ∈ D , ω ∈ Ω (*) y ( x , ω ) = 0 , for a.e. x ∈ ∂ D , ω ∈ Ω with a ( · , ω ) a random field s.t. 0 < a min ≤ a ( x , ω ) ≤ a max , ∀ ( x , ω ) ∈ D × Ω. ⇒ random solution ω �→ y ( · , ω ) ∈ H 1 0 ( D ). In particular y ∈ L 2 P (Ω; H 1 = 0 ( D )). u ∈ L 2 ( D ): control function F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 4

Problem setting – quadratic optimal control problem Problem setting (Ω , F , P ): complete probability space D ⊂ R d : physical domain. Throughout � · � = � · � L 2 ( D ) . Forward problem � − div( a ( x , ω ) ∇ y ( x , ω )) = g ( x ) + u ( x ) , for a.e. x ∈ D , ω ∈ Ω (*) y ( x , ω ) = 0 , for a.e. x ∈ ∂ D , ω ∈ Ω with a ( · , ω ) a random field s.t. 0 < a min ≤ a ( x , ω ) ≤ a max , ∀ ( x , ω ) ∈ D × Ω. ⇒ random solution ω �→ y ( · , ω ) ∈ H 1 0 ( D ). In particular y ∈ L 2 P (Ω; H 1 = 0 ( D )). u ∈ L 2 ( D ): control function Optimal control problem J ( u , y ) := 1 2 E ω [ � y ( · , ω ) − y target � 2 ] + β ˜ 2 � u � 2 , min subject to (*) u ∈ L 2( D ) y ∈ L 2 P (Ω; H 1 0 ( D )) F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 4

Problem setting – quadratic optimal control problem Reduced functional (Stochastic) Affine solution operator: y ω : L 2 ( D ) → H 1 0 ( D ) � − div( a ( · , ω ) ∇ y ω ( u )) = g + u , in D ∀ ω ∈ Ω u �→ y ω ( u ) , solution of y ω ( u ) = 0 , on ∂ D . F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 5

Problem setting – quadratic optimal control problem Reduced functional (Stochastic) Affine solution operator: y ω : L 2 ( D ) → H 1 0 ( D ) � − div( a ( · , ω ) ∇ y ω ( u )) = g + u , in D ∀ ω ∈ Ω u �→ y ω ( u ) , solution of y ω ( u ) = 0 , on ∂ D . Reduced functional: min u ∈ L 2 ( D ) J ( u ) f ( u , ω ) = 1 2 � y ω ( u ) − y target � 2 + β 2 � u � 2 J ( u ) = E ω [ f ( u , ω )] , F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 5

Problem setting – quadratic optimal control problem Reduced functional (Stochastic) Affine solution operator: y ω : L 2 ( D ) → H 1 0 ( D ) � − div( a ( · , ω ) ∇ y ω ( u )) = g + u , in D ∀ ω ∈ Ω u �→ y ω ( u ) , solution of y ω ( u ) = 0 , on ∂ D . Reduced functional: min u ∈ L 2 ( D ) J ( u ) f ( u , ω ) = 1 2 � y ω ( u ) − y target � 2 + β 2 � u � 2 J ( u ) = E ω [ f ( u , ω )] , Adjoint based gradient computation: ∇ u f ( u , ω ) = β u + p ω ( u ) , ∇ u J ( u ) = β u + E ω [ p ω ( u )] where p ω ( u ) solves the adjoint problem ∀ ω ∈ Ω − div( a ( · , ω ) ∇ p ω ( u )) = y ω ( u ) − y target in D , p ω ( u ) = 0 on ∂ D . F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 5

Problem setting – quadratic optimal control problem Reduced functional (Stochastic) Affine solution operator: y ω : L 2 ( D ) → H 1 0 ( D ) � − div( a ( · , ω ) ∇ y ω ( u )) = g + u , in D ∀ ω ∈ Ω u �→ y ω ( u ) , solution of y ω ( u ) = 0 , on ∂ D . Reduced functional: min u ∈ L 2 ( D ) J ( u ) f ( u , ω ) = 1 2 � y ω ( u ) − y target � 2 + β 2 � u � 2 J ( u ) = E ω [ f ( u , ω )] , Adjoint based gradient computation: ∇ u f ( u , ω ) = β u + p ω ( u ) , ∇ u J ( u ) = β u + E ω [ p ω ( u )] where p ω ( u ) solves the adjoint problem ∀ ω ∈ Ω − div( a ( · , ω ) ∇ p ω ( u )) = y ω ( u ) − y target in D , p ω ( u ) = 0 on ∂ D . Lipschitz and strong convexity properties of ∇ u f : ∀ u 1 , u 1 ∈ L 2 ( D ) , ω ∈ Ω L = β + C 4 P �∇ u f ( u 1 , ω ) − ∇ u f ( u 2 , ω ) � ≤ L � u 1 − u 2 � , a 2 min �∇ u f ( u 1 , ω ) − ∇ u f ( u 2 , ω ) , u 1 − u 2 � L 2 ( D ) ≥ ℓ 2 � u 1 − u 2 � 2 , ℓ = 2 β F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 5

Discretization by finite elements + Monte Carlo Outline Problem setting – quadratic optimal control problem 1 Discretization by finite elements + Monte Carlo 2 Deterministic (CG) iterative solvers versus Stochastic Gradient 3 Multilevel stochastic gradient algorithms 4 Conclusions 5 F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 6

Discretization by finite elements + Monte Carlo Finite dimensional approximation Finite Element approximation of the PDE: ∀ u ∈ L 2 ( D ) , ω ∈ Ω � � u �→ y h a ( · , ω ) ∇ y h ∀ v h ∈ Y r ω ( u ) solves ω ( u ) · ∇ v h = ( g + u ) v h , h D D Y r h : space of continuous P r finite element functions vanishing on the boundary F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 7

Discretization by finite elements + Monte Carlo Finite dimensional approximation Finite Element approximation of the PDE: ∀ u ∈ L 2 ( D ) , ω ∈ Ω � � u �→ y h a ( · , ω ) ∇ y h ∀ v h ∈ Y r ω ( u ) solves ω ( u ) · ∇ v h = ( g + u ) v h , h D D Y r h : space of continuous P r finite element functions vanishing on the boundary iid Monte Carlo approximation of expectation: ω i ∼ P , i = 1 , . . . N N J ( u ) = E ω [ f ( u , ω )] ≈ 1 � f ( u , ω i ) N i =1 F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 7

Discretization by finite elements + Monte Carlo Finite dimensional approximation Finite Element approximation of the PDE: ∀ u ∈ L 2 ( D ) , ω ∈ Ω � � u �→ y h a ( · , ω ) ∇ y h ∀ v h ∈ Y r ω ( u ) solves ω ( u ) · ∇ v h = ( g + u ) v h , h D D Y r h : space of continuous P r finite element functions vanishing on the boundary iid Monte Carlo approximation of expectation: ω i ∼ P , i = 1 , . . . N N J ( u ) = E ω [ f ( u , ω )] ≈ 1 � f ( u , ω i ) N i =1 Discrete optimal control problem: N N � 1 � u ∈ L 2 ( D ) J h , N ( u ) := 1 f h ( u , ω i ) = 1 ω i ( u ) − y target � 2 + β � � 2 � y h 2 � u � 2 min N N i =1 i =1 F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 7

Discretization by finite elements + Monte Carlo Finite dimensional approximation Finite Element approximation of the PDE: ∀ u ∈ L 2 ( D ) , ω ∈ Ω � � u �→ y h a ( · , ω ) ∇ y h ∀ v h ∈ Y r ω ( u ) solves ω ( u ) · ∇ v h = ( g + u ) v h , h D D Y r h : space of continuous P r finite element functions vanishing on the boundary iid Monte Carlo approximation of expectation: ω i ∼ P , i = 1 , . . . N N J ( u ) = E ω [ f ( u , ω )] ≈ 1 � f ( u , ω i ) N i =1 Discrete optimal control problem: N N � 1 � J h , N ( u ) := 1 f h ( u , ω i ) = 1 ω i ( u ) − y target � 2 + β � � 2 � y h 2 � u � 2 min N N u ∈ Y r h i =1 i =1 Remark: The unique minimizer u h , N ∈ Y r ⋆ h F. Nobile (EPFL) MLSG for PDE-constrained optimization RICAM 2019, Linz 7