Optimal control of non-smooth partial differential equations Vu Huu - PowerPoint PPT Presentation

Optimal control of non-smooth partial differential equations Vu Huu Nhu 1 Christian Clason Arnd Rösch Stephan Walther 2 Constantin Christof Christian Meyer 1 Faculty of Mathematics, Universität Duisburg-Essen 2 Faculty of Mathematics, TU Dortmund Workshop “New Trends in PDE-constrained Optimization” RICAM, Linz, October 14, 2019 Overview Semilinear PDEs Quasilinear PDEs 1 / 25

Motivation: PDE-constrained optimization min u , y F ( y ) + G ( u ) such that E ( y , u ) = 0 Standard questions: 1 existence of solutions � direct method of calculus of variations 2 characterization of solutions � (necessary) optimality conditions 3 computation of solutions � gradient, Newton-type methods Current research: F or G not differentiable (constraints, sparsity, impulse noise) E not differentiable Overview Semilinear PDEs Quasilinear PDEs 2 / 25

Overview 1 Non-smooth equations Optimality conditions Semilinear PDEs 2 Optimality conditions Numerical solution Quasilinear PDEs 3 Optimality conditions Numerical solution Overview Semilinear PDEs Quasilinear PDEs 3 / 25

Motivation: non-smooth equations Non-smooth equations: describe models with sharp phase transitions dual formulation of variational inequalities examples: free boundary problems (ice–water), contact problems with friction, non-Newtonian fluid flow, ... Two-phase Stefan problem for all ϕ ∈ H 1 ( Q ) with ϕ ( · , T ) = 0 � – y , ϕ t � + �∇ θ ( y ), ∇ ϕ � = � u , ϕ �  y ( x , t ) y ( x , t ) � 0   θ ( y ( x , t )) = 0 y ( x , t ) ∈ [0, 1]   y ( x , t ) – 1 y ( x , t ) � 1 Overview Semilinear PDEs Quasilinear PDEs 4 / 25

Motivation: non-smooth equations Model problem 1: semilinear “Saran wrap equation” – ∆ y + max{0, y } = u y | ∂ Ω = 0 superposition operator: max : L 2 ( Ω ) → L 2 ( Ω ) pointwise a.e. model for membrane partially in water: y deflection, u force can be extended to arbitrary f ( y ) piecewise differentiable well-posed (in suitable spaces) u �→ y nonlinear, Lipschitz (in suitable spaces) u �→ y not Gâteaux differentiable unless |{ x : y ( x ) = 0}| = 0 Overview Semilinear P DEs Quasilinear PDEs 5 / 25

Motivation: non-smooth equations Model problem 2: quasilinear heat conduction – ∇ · [ a ( y ) ∇ y ] = u y | ∂ Ω = 0 superposition operator: a : L 2 ( Ω ) → L 2 ( Ω ) pointwise a.e. a : R → R bounded from below, Lipschitz (or PC 1 ) nonlinear material-dependent conductivity law e.g., a ( y ) = 1 + | y | well-posed (in suitable spaces) u �→ y nonlinear, continuous (in suitable spaces) u �→ y not Gâteaux differentiable in general Overview Semilinear P DEs Quasilinear PDEs 6 / 25

Motivation: optimality conditions F ( ¯ x ) = min x ∈ X F ( x ) s.t. x ∈ C Optimality conditions ( F differentiable): 1 primal: directional derivative, tangent cone F ′ ( ¯ x ; h ) � 0 for all h ∈ T C ⊂ X 2 dual: (suitable) subdifferential, indicator functional x ) ⊂ X ∗ 0 ∈ ∂ [ F + δ C ]( ¯ 3 primal-dual: calculus rules, normal cone ( � Lagrange multiplier) F ′ ( ¯ x ) ⊂ X ∗ x ) + ¯ p = 0, p ∈ N C ( ¯ ¯ Overview Semilinear P DEs Quasilinear PDEs 7 / 25

Motivation: optimality conditions J ( ¯ u , ¯ y ) = u ∈ X , y ∈ Y J ( u , y ) min s.t. E ( u , y ) = 0 Unique solution y = S ( u ) F ( u ) := J ( u , S ( u )) (differentiable) � 1 primal: directional derivative F ′ ( ¯ u ; h ) � 0 for all h ∈ X 2 dual: Fréchet derivative 0 = F ′ ( ¯ u ) ⊂ X ∗ 3 primal-dual: implicit function theorem � adjoint state J ′ p = S ′ ( ¯ u ) ∗ J ′ u ( ¯ u , S ( ¯ u )) + ¯ p = 0, ¯ y ( ¯ u , S ( ¯ u )) Overview Semilinear P DEs Quasilinear PDEs 8 / 25

Motivation: optimality conditions J ( ¯ u , ¯ y ) = u ∈ X , y ∈ Y J ( u , y ) min s.t. E ( u , y ) = 0 Unique solution y = S ( u ), not Gâteaux differentiable 1 primal: directional derivative F ′ ( ¯ u ; h ) � 0 for all h ∈ X 2 dual: (suitable) subdifferential u ) ⊂ X ∗ 0 ∈ ∂ F ( ¯ 3 primal-dual: chain rule or limit process ( � adjoint state) J ′ u ) ∗ J ′ u ( ¯ u , S ( ¯ u )) + ¯ p = 0, ¯ p ∈ ∂ S ( ¯ y ( ¯ u , S ( ¯ u )) Overview Semilinear P DEs Quasilinear PDEs 9 / 25

Motivation: subdifferentials S : X → Y not Gâteaux differentiable: Bouligand subdifferential � � � there exists { u n } with u n → u � ∂ B S ( u ) := G u ∈ L ( X , Y ) � and S ′ ( u n ) → G u � � (set of all limits of Gâteaux derivatives in nearby points) Clarke subdifferential ∂ C S ( u ) := cl co ∂ B S ( u ) (closed convex hull) X , Y infinite-dimensional � topology matters (strong, weak(- ∗ ),...) Overview Semilinear P DEs Quasilinear PDEs 10 / 25

Overview 1 Non-smooth equations Optimality conditions Semilinear PDEs 2 Optimality conditions Numerical solution Quasilinear PDEs 3 Optimality conditions Numerical solution Overview Semilinear P DEs Quasilinear PDEs 11 / 25

Optimal control of semilinear PDE Semilinear “Saran wrap equation” min 0 ( Ω ) J ( y , u ) s.t. – ∆ y + max{0, y } = u u ∈ L 2 ( Ω ), y ∈ H 1 existence of minimizer ( ¯ u , ¯ y ) for J weakly l.s.c., coercive S : u �→ y Lipschitz from L 2 ( Ω ) → H 1 0 ( Ω ) (standard argument: max Lipschitz and monotone) S Gâteaux differentiable at u if and only if S ( u ) � = 0 a.e. reduced functional F ( u ) := J ( S ( u ), u ) Overview Semilinear P DEs Quasilinear PDEs 12 / 25

Semilinear: primal optimality conditions Primal optimality conditions F ′ ( ¯ u ; h ) = J ′ u ) S ′ ( ¯ u ; h ) + J ′ for all h ∈ L 2 ( Ω ) y ( ¯ y , ¯ u ( ¯ y , ¯ u ) h � 0 if J continuously Fréchet differentiable, partial derivatives J ′ y , J ′ u standard proof: pass to the limit in F ( ¯ u ) � F ( u + th ) directional derivative: w := S ′ ( u ; h ) ∈ H 1 0 ( Ω ) satisfies – ∆ w + 1 { S ( u )>0} w + 1 { S ( u )=0} max{0, w } = h � Gâteaux derivative S ′ ( u ) h = w iff S ( u ) � = 0 a.e. Overview Semilinear P DEs Quasilinear PDEs 13 / 25

Semilinear: primal-dual optimality conditions Primal-dual optimality conditions p + J ′ u ( ¯ y , ¯ u ) = 0, y = S ( ¯ u ) ¯ ¯ p = J ′ – ∆ ¯ p + ξ ¯ y ( ¯ y , ¯ u )  {1} ¯ y ( x ) > 0  ξ ( x ) ∈ ∂ C max( ¯ y ( x )) := a.e. {0} y ( x ) < 0 ¯  [0, 1] y ( x ) = 0 ¯ proof: C 1 approximation max ε , localization � standard conditions pass to limit ε → 0, use regularity of adjoint PDE G ξ := (– ∆ + ξ ) –1 ∈ ∂ w B S ( ¯ u ) (weak limit of Gâteaux derivatives) ξ ( x ) ∈ {0, 1} a.e. G ξ ∈ ∂ B S ( ¯ u ) � � implies dual optimality condition 0 ∈ ∂ B F ( ¯ u ) ⊂ ∂ C F ( ¯ u ) Overview Semilinear P DEs Quasilinear PDEs 14 / 25

Semilinear: strong optimality conditions Strong optimality conditions p + J ′ u ( ¯ y , ¯ u ) = 0, y = S ( ¯ u ) ¯ ¯ p = J ′ – ∆ ¯ p + ξ ¯ y ( ¯ y , ¯ u ) ξ ( x ) ∈ ∂ C max( ¯ y ( x )) a.e. p ( x ) � 0 a.e. where y ( x ) = 0 ¯ ¯ proof: test adjoint equation, use density equivalent to primal optimality condition proof: pointwise argument using structure of max overdetermined: not useful for numerical computation Overview Semilinear P DEs Quasilinear PDEs 15 / 25

Semilinear: numerical solution J ( y , u ) = 1 L 2 ( Ω ) + 1 2 � y – y d � 2 2 � u � 2 L 2 ( Ω ) finite element discretization, mass lumping for max-term eliminate control max convex � proximal point reformulation of ξ i ∈ ∂ C max( y i ) A h y + D h max( y ) = – 1 αM h p A h p + D h ξ ◦ p = M h ( y – y d ) y = prox τ ( y + τ ξ ) Overview Semilinear P DEs Quasilinear PDEs 16 / 25

Semilinear: numerical solution A h y + D h max( y ) = – 1 αM h p A h p + D h ξ ◦ p = M h ( y – y d ) y = prox τ ( y + τ ξ ) � semi-smooth Newton method but: Newton matrix singular for p i = y i + τξ = 0 � eliminate corresponding components in iteration test with constructed y d � S not differentiable at solution Overview Semilinear P DEs Quasilinear PDEs 16 / 25

Semilinear: numerical example � y h – ¯ y � L 2 � p h – ¯ p � L 2 h α τ # SSN � ¯ y � L 2 � ¯ p � L 2 3.030 e– 2 1 e– 4 1 e– 12 8.708 e– 1 1.606 e– 2 4 1.538 e– 2 1 e– 4 1 e– 12 2.281 e– 1 4.541 e– 3 5 7.752 e– 3 1 e– 4 1 e– 12 5.821 e– 2 1.209 e– 3 3 3.891 e– 3 1 e– 4 1 e– 12 1.469 e– 2 3.119 e– 4 3 7.752 e– 3 1 e– 4 1 e– 6 – – no conv. 7.752 e– 3 1 e– 4 1 e– 8 – – no conv. 7.752 e– 3 1 e– 4 1 e– 10 5.821 e– 2 1.209 e– 3 3 7.752 e– 3 1 e– 4 1 e– 14 5.821 e– 2 1.209 e– 3 3 7.752 e– 3 1 e– 2 1 e– 12 3.007 e– 3 1.747 e– 3 2 7.752 e– 3 1 e– 3 1 e– 12 1.659 e– 2 1.512 e– 3 2 7.752 e– 3 1 e– 5 1 e– 12 1.692 e– 1 8.659 e– 4 5 7.752 e– 3 1 e– 6 1 e– 12 – – no conv. Overview Semilinear P DEs Quasilinear PDEs 17 / 25

Overview 1 Non-smooth equations Optimality conditions Semilinear PDEs 2 Optimality conditions Numerical solution Quasilinear PDEs 3 Optimality conditions Numerical solution Overview Semilinear P DEs Quasilinear PDEs 18 / 25