Reduced-Order Methods in PDE Constrained Optimization

Transcription

1 Reduced-Order Methods in PDE Constrained Optimization M. Hinze (Hamburg), K. Kunisch (Graz), F. Tro ltzsch (Berlin) and E. Grimm, M. Gubisch, S. Rogg, A. Studinger, S. Trenz, S. Volkwein University of Konstanz, Department of Mathematics and Statistics Recent Trends and Future Developments in Computational Science & Engineering Koppelsberg, Plo n, March -, 5 Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

2 Introduction Motivation Motivation for our Research Areas Problem: time-variant, nonlinear, parametrized PDE systems Efficient and reliable numerical simulation in multi-query cases finite element or finite volume discretizations too complex Multi-query examples fast simulation for different parameters on small computers parameter estimation, optimal design and feedback control usage of a reduced-order SURROGATE MODEL Time-variant, nonlinear coupled PDEs methods from linear system theory not directly applicable Nonlinear model-order reduction proper orthogonal decomposition and reduced-basis method Error control for reduced-order model new a-priori and a-posteriori error analysis [Martin Gubisch, Stefan Trenz, Andrea Wesche] reliable PDE constrained multi-objective optimization [Laura Iapichino, Stefan Trenz] economic model predicitive control [Lars Grüne, Thomas Meurer, Sabrina Rogg] certified closed-loop strategies [Alessandro Alla, Maurizio Falcone] PDE Partial Differential Equation Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

3 Introduction Introductory Example Minimization with Inequality Constraints 5 Minimiere f(x)=x Problem : min {f (x) = x : < x < } 5 f(x)=x Optimality condition: f (x )! = with f (x) = x Solution: x = Problem: min {f (x) = x : x } Lagrange functional: L(x, µ a, µ b ) = f (x) + µ a ( x) + µ b (x ) Optimality condition: f (x ) µ a +µ b µ a ( x ) µ b (x ) =! f (x )(x x ) x [,] }{{} (x ) 5 * x = * f (x )= x Achse Solution: x =, µ a =, µ b = x Achse Minimiere f(x)=x mit x aus [,] f(x)=x x * = f (x * )=x * Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

4 Introduction Outline of the Talk Outline of the talk Semilinear optimal control problems Proper orthogonal decomposition (POD) A-posteriori error for linear-quadratic optimal control Basis update by optimality-system POD (OS-POD) Extension to semilinear optimal control problems Conclusions and outlook Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

5 Semilinear Optimal Control Problems Semilinear Optimal Control Problems Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 5 / 6

6 Semilinear Optimal Control Problems Problem Formulation Optimal Control of Semilinear Parabolic Problems [Tröltzsch ] PDE constrained optimization problem: Minimize the quadratic cost J(y,u) = α tf Q y(t,x) yq (t,x) dxdt + α Ω y(tf,x) y Ω (x) dx t Ω Ω + κd tf u d (t,x) u d (t,x) dxdt + κb tf u b (t,s) u b (t,s) dsdt t Ω t Ω with respect to (y, u) subject to the semilinear evolution equation c p y t (t,x) y(t,x) + N (y(t,x)) = f (t,x) + u d (t,x), y n (t,s) + qy(t,s) = ub (t,s), and the bilateral control constraints (t,x) Q = (t,t f ) Ω (t,s) Σ = (t,t f ) Ω y(t,x) = y (x), x Ω R n, n {,,} U ad = { u = (u d,u b ) ua d u d ub d in Q and ub a u b ub d on Σ} Assumption: for any control u U ad there exists a unique state y = y(u) satisfying (SE) Reduced problem: Setting Ĵ(u) = J(y(u),u) we consider the reduced problem (SE) minĵ(u) subject to u U ad (ˆP) nonconvex problem possibly many local optimal solutions Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 6 / 6

7 Semilinear Optimal Control Problems Characterization of Solutions by First-Order Optimality Conditions First-Order Necessary Optimality Condition [Hinze/Pinnau/Ulbrich/Ulbrich 9, Ito/Kunisch 8, Tröltzsch ] Reduced problem: Setting Ĵ(u) = J(y(u),u) we consider the reduced problem minĵ(u) subject to u U ad (ˆP) Existence of local optimal controls ū: properties of N, regularity hypotheses Optimality conditions: based on constraint qualifications () State equation [ ȳ = y(ū) ] c p ȳ t ȳ + N (ȳ) = f + ū d in Q = (t,t f ) Ω ȳ + qȳ = ūb n on Σ = (t,t f ) Ω ȳ(t ) = y in Ω () Adjoint equation [ p = p(ū) ] c p p t p + N (ȳ) p = α Q ( yq ȳ ), p n + q p = on Σ ( p(t f ) = α Ω yω ȳ(t f ) ) in Ω () Variational inequality [ ū = (ū d,ū b ] ) U ad tf ( κ d( ū d u d ) ) (u p ū d ) tf ( dxdt + κ b( ū b u b ) ) (u p b ū b) dsdt t Ω t Ω for all (admissible) u = (u d,u b ) U ad in Q Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 7 / 6

8 Semilinear Optimal Control Problems Projected Newton-CG Method with Armijo Linesearch Globalization Computation of the Reduced Hessian [Hinze/Pinnau/Ulbrich/Ulbrich 9] Reduced problem: Setting Ĵ(u) = J(y(u),u) we consider the reduced problem minĵ(u) subject to u U ad (ˆP) ( [κ Reduced gradient: Ĵ (u) = d ( u d u d ) ] p Q [ κ b ( u b u b ) ] ) p Σ Evaluation of the reduced hessian Ĵ at u U ad in a direction u δ = ( uδ d ),ub δ : () solve the linearized state equation [ y δ = y δ (u δ ;u) with y = y(u) ] c p (y δ ) t y δ + N (y)y δ = u d δ in Q y δ n + qy δ = uδ b on Σ y δ (t ) = in Ω () solve the linearized adjoint equation [ p δ = p δ (u δ ;u) with y = y(u), p = p(u) ] c p (p δ ) t p δ + N (y)p δ = α Q y δ N (y)y δ p, in Q p δ n + qp δ = p δ (t f ) = α Ω y δ (t f ) ( [κ () Set Ĵ (u)u δ = d uδ d p ] Q [ δ κ b uδ b p ] ) Σ δ on Σ in Ω Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 8 / 6

9 Semilinear Optimal Control Problems Locally Second-Order Optimization Method Numerical Solution Algorithm Reduced problem: Setting Ĵ(u) = J(y(u),u) we consider the reduced problem minĵ(u) subject to u U ad (ˆP) Newton s method: solve at u k with y k = y(u k ) and p k = p(u k ) minĵ(uk ) + Ĵ (u k )u δ + Ĵ (u k ) ( u δ,u δ ) subject to u δ with u k + u δ U ad with a (truncated) conjugate gradient method [Nocedal/Wright 6] ( [κ Reduced gradient: Ĵ (u k ) = d ( u k,d u d ) p k ] [ Q κ b ( u k,b u b ) p k ] ) Σ Evaluation of the reduced hessian at u U ad in direction u δ = ( uδ d ),ub δ : () solve the linearized state equation for y δ = y δ (u δ ;u k ) () solve the linearized ( adjoint equation for p δ = p δ (u δ ;u k ) [κ () Set Ĵ (u k )u δ = d uδ d p ] [ Q δ κ b uδ b p ] ) Σ δ Globalization: negative curvature test and Armijo line search Control constraints: projected BFGS/Newton method [Kelley 99] Numerical realization: finite elements (FE) and implicit Euler method Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 9 / 6

10 Semilinear Optimal Control Problems Numerical Test Experiment Numerical Example: Distributed Optimal Control [Kammann/Tröltzsch/V., Trenz 5] Control input: u d (t,x) = i= u i(t)w i (x) and u b, start with u () (t,x) = i=.w i(x) Bilateral control bounds: u d a = and u d b = Nonlinearity: N (y) = y with N (y) = y Discretization: N = 79 FE unknown and N t = time instances Exact optimal state: ȳ(t,x) = cos(x )cos(x ) = y (x) for x = (x,x ) Ω = (,π) Exact optimal control: ū = ū and ū = ū in [t,t f ] = [,].5 Initial state y(x,x) Weighting functions w, w, w, w Analytical optimal control u i (t) ua,ub i = i = i = i =.5 x x wi(x,x) 6 x x t Newton-CG BFGS BFGS-Inv # Iterations 6 Time 59 s s 8 s J(ȳ FE,ū FE ).e-.e-.e- ū ū FE 6.69e-.5e- 5.7e- Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

11 Proper Orthogonal Decomposition (POD) Proper Orthogonal Decomposition (POD) Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

12 Proper Orthogonal Decomposition (POD) Motivation of the Reduced-Order Modeling Galerkin-Based Reduced-Order Modeling Model problem (weak form of our PDE): y(t ) = y and ( c p y t (t),ϕ + y(t) ϕ + N (y(t))ϕ dx = f (t) + u d (t) ) ϕ dx + u b (t)ϕ ds Ω Ω Ω for all ϕ V in (t,t f ], where we write y(t) = y(t, ) etc. Typical choices: H (Ω) V H (Ω) and H = L (Ω) Finite element approximation: y N (t) V N = span{ϕ,...,ϕ N } V with y N (t ) = P N y and c p yt N (t),ϕ + y N (t) ϕ + N (y N ( (t))ϕ dx = f (t) + u d (t) ) ϕ dx + u b (t)ϕ ds Ω Ω Ω for all ϕ V N in (t,t f ] Alternatives: finite volume or finite difference schemes Reduced-order model: y l (t) V l = span{ψ N,...,ψN l } V N and l N with y l (t ) = P l y and c p yt l (t),ψ + y l (t) ψ + N (y l ( (t))ψ dx = f (t) + u d (t) ) ψ dx + u b (t)ψ ds Ω Ω Ω for all ψ V l in (t,t f ] Reduced-order subspace V l : Proper Orthogonal Decomposition or Reduced-Basis Nonlinear problems: (Discrete) Empirical Interpolation Method (D)EIM Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

13 Proper Orthogonal Decomposition (POD) A Continuous Variant of Proper Orthogonal Decomposition (POD) Proper Orthogonal Decomposition (POD) Dynamical system in separable Hilbert space X (e.g., X = H or V ): ẏ(t) = F(t,y(t); µ(t)) in (t,t f ], y(t ) = y X with given parameter or control µ(t), and data y, f Given multiple snapshots: solutions y k (t) X, e.g., for parameters {µ k } k= Snapshot subspace: V = span { y k (t) t [t,t f ] and k } X Continuous variant of POD: for every l solve tf y min k l (t) k= t y k (t),ψ i X ψ i dt s.t. {ψ i} l i= X, ψ i,ψ j X X = δ ij, i,j l i= Integral operator: define R : X X as Rψ = k= tf t ψ,y k (t) X y k (t)dt for ψ X Theorem [Hilbert-Schmidt, Riesz-Schauder; Perturbation theory for the spectrum] ( ) ) R is linear, compact, selfadjoint and nonnegative. ) There are eigenfunctions { ψ i } i= and eigenvalues { λ i } i= with ) { ψ i } l i= solves ( ) and k= R ψ i = λ i ψ i, λ λ..., lim i λi = t f t y k (t) l i= y k (t), ψ i X ψ i dt = λi X i=l+ Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

14 Proper Orthogonal Decomposition (POD) Numerical Test Experiment Numerical Example: POD-Basis Computation [Trenz 5] Initial state y (x, x ) Eigenvalues Control input: u d (t, x) = i=.wi (x).8.6 Nonlinearity: N (y) = y..5. λi 6.5 Discretization: N = 79 FEs, Nt = POD topology: X = L (Ω).8 x Number i x Snapshots of the state y at t =.5,.5,.75, and : Snapshot: state y(x, t) at t =.5 Snapshot: state y(x, t) at t =.5.5 Snapshot: state y(x, t) at t = Snapshot: state y(x, t) at t = x x x x x x x x First four POD basis functions: POD basis function ψ POD basis function ψ. POD basis function ψ.. POD basis function ψ x Stefan Volkwein x. x x x x Reduced-Order Methods in PDE Constrained Optimization x x / 6

15 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control A-Posteriori Error for Linear-Quadratic Optimal Control Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 5 / 6

16 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Problem Formulation Linear-Quadratic, Time-Variant Optimal Control Problems Quadratic programming (QP) problem: min x=(y,u) J(x) = y(t f) y tf H + κ subject to the linear evolution problem tf t u(t) U dt y t (t),ϕ + a(t;y(t),ϕ) = (f + Bu)(t),ϕ ϕ V in (t,t f ] with y(t ) = y and to bilateral control constraints u U ad = { v U u a (t) v(t) u b (t) in [t,t f ] } State: y(t) V H with Hilbert spaces V, H Control (Hilbert) space: U = L (t,t f ;U) with U = R Nu, U = L (Ω) or U = L (Γ) Input/control: u U ad (boundary or distributed control) Bilinear form: a(t;, ) continuous and a(t;ϕ,ϕ) γ ϕ V γ ϕ H, e.g. a(t;ϕ,ψ) = ϕ ψ + N (y(t))ϕψ dx for ϕ,ψ V Ω with N (y) = y or N (y) = sinhy Control operator: B : U L (t,t f ;V ) linear, bounded Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 6 / 6

17 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Characterization of Solutions by First-Order Optimality Conditions First-Order Necessary and Sufficient Optimality Conditions Quadratic programming (QP) problem: min x=(y,u) J(x) = y(t f ) y tf H + κ tf t u(t) U dt s.t. y t (t),ϕ + a(t;y(t),ϕ) = (f + Bu)(t),ϕ ϕ V in (t,t f ] y(t ) = y and u a (t) u(t) u b (t), t [t,t f ] Optimal control ū U ad = {u u a u u b in [t,t f ]}, associated state ȳ = y(ū) Adjoint/dual equation: p t (t),ϕ + a(t;ϕ, p(t)) = ϕ V in [t,t f ), p(t f ) = ȳ(t f ) y tf Variational inequality: tf t κū(t) (B p)(t),u(t) ū(t) dt u U ad (VI) Reduced cost: Ĵ(u) = J(y(u),u) with Ĵ (ū) = κū B p U, i.e., (VI) reads Ĵ (ū),u(t) ū(t) u U ad Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 7 / 6

18 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control POD Galerkin Discretization POD Galerkin Scheme for the State and Dual Variable State equation: y t (t),ϕ + a(t;y(t),ϕ) = (f + Bu)(t),ϕ ϕ V in (t,t f ], y(t ) = y POD space: V l = span{ψ,...,ψ l } V Orthogonal projection: P l ϕ = l i= ϕ,ψ i X ψ i V l for ϕ V POD state: y l = y l (u) with y l (t) V l in [t,t f ] solves yt l (t),ψ + a(t;yl (t),ψ) = (f + Bu)(t),ψ ψ V l in (t,t f ], y l (t ) = P l y Adjoint/dual equation: p = p(y(u)) solves p t (t),ϕ + a(t;ϕ,p(t)) = ϕ V in [t,t f ), p(t f ) = y(t f ) y tf POD dual: p l = p l (y l (u)) with p l (t) V l in [t,t f ] solves pt l (t),ψ + a(t;ψ,pl (t)) = ψ V l in [t,t f ), p l (t f ) = y l (t f ) y tf same POD basis for state and adjoint variable Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 8 / 6

19 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Perturbation Argument POD A-Posteriori Error Analysis [Malanowski/Büskens/Maurer 97, Tröltzsch/V. 9] Variational inequality: tf t κū(t) (B p)(t),u(t) ū(t) dt u U ad (VI) Optimal POD solution: ū l U ad, associated state ȳ l = y l (ū l ) and dual p l = p l (y l (ū l )) POD variational inequality: tf Misfit in the variational inequality: tf with ỹ l = y(ū l ) and p l = p(y(ū l )) t κū l (t) (B p l )(t),u(t) ū l (t) dt u U ad (VI l ) t κū l (t) (B p l )(t),u(t) ū l (t) dt u U ad Perturbation analysis: there exists a perturbation ζ U satisfying tf t κū l (t) (B p l )(t) + ζ (t),u(t) ū l (t) dt u U ad (ṼIl ) A-posteriori analysis: choose u = ū l in (VI), u = ū in (ṼIl ) and add A-posteriori error estimate for the control: ū ū l ζ /κ Reduced-basis method: related work [Dede, Grepl/Kärcher, Negri/Rozza/Manzoni/ Quarteroni ] Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 9 / 6

20 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Algorithmic Realization Algorithm with POD A-Posteriori Analysis A-posteriori error estimate: ū ū l ζ /κ and p l = p(y(ū l )) ( κū l (t) (B p l )(t) ) if u a (t) < ū l (t) < u b (t) Computation of ζ : ζ (t) = min (,κū l (t) (B p l )(t) ) if ū l (t) = u a (t) max (,κū l (t) (B p l )(t) ) if ū l (t) = u b (t) Algorithmus (Optimal control with a-posteriori error estimation) : Choose POD basis {ψ i } l i= for the Galerkin approximation of the QP problem; : Determine the reduced-order model for the QP problem; : Calculate suboptimal control ū l U ad, e.g., by a semismooth Newton method; : Compute perturbation ζ l = ζ (ū l ); 5: if ζ l /κ > TOL then 6: Enlarge l and go back to Step ; 7: else 8: Stop; 9: end if Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

21 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Numerical Test Experiment Numerical Example: Problem Formulation and Optimal Control [Studinger/V. ] Consider: min J(y, u) = Z Ω y(tf ) yd dx + tf Z Z u dsdt Ω y y + = u on Σ, n.5 u on Σ = (, tf ) Ω y() = y in Ω = (, ) s.t. yt. y = in Q, Method & discretization: semismooth Newton & implizit Euler, finite elements x = Course of the FE optimal control u*fe Course of the control uref used for snapshot generation and thus POD basis determination x = y=.5 y= x axis.5 time axis x axis.5 time axis.5.5 Stefan Volkwein.5 x axis.5 time axis time axis x axis x=.5.8 time axis x= x = x = x axis x axis.5 time axis y axis time axis Reduced-Order Methods in PDE Constrained Optimization.5.5 y axis.5 time axis / 6

22 A-Posteriori Error Estimation for Linear-Quadratic Optimal Control Numerical Test Experiment Numerical Example: POD Error Analysis [Studinger/V. ] Course of the control uref used for snapshot generation and thus POD basis determination y= y= x axis time axis x axis x= y axis.5.5 x axis.5 time axis time axis.5 x = x = time axis.9.8 y axis.5 x axis time axis time axis x=.5 λi / trace x = W=M W=M+A. Course of the FE optimal control u*fe x = decay of normalized eigen values on semilog scale, Variant eigs index i kζ (u ` )k [Tro ltzsch/v. 9] κ FE A-posteriori error within FE: ku FE u ` k kζ FE (u ` )k =: εape κ x axis.5 time axis x axis..5 time axis A-posteriori error: ku u ` k [Gubisch/Neitzel/V. 5] ` FE εape u FE u ` FE εape ku FE u ` k FE εape u FE u ` FE εape ku FE u ` k e- 5.9e-.e-.e-.e- 9.e-.e-.e-.e- 9.7e e- 7.5e- 8.e-5.e-5.7e-6 5.6e- 7.e- 8.e-5.e-5.7e Difficulty: choice of expected control for computation of the POD basis Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

23 Basis Update by Optimality-System POD Basis Update by Optimality-System POD Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

24 Basis Update by Optimality-System POD Idea of the POD Basis Update Optimality-System POD (OS-POD) [Kunisch/V. 8] Optimal control problem: minj(y,u) s.t. (y,u) Y ad U ad, ẏ(t) = F(t,y(t),u(t)) in (t,t f ], y(t ) = y (P) POD-Galerkin approximation: {ψ i } l i= POD basis of rank l minj l (y l,u) s.t. (y l,u) Y l ad U ad, ẏ l (t) = F l (t,y l (t),u(t)) in (t,t f ], y l (t ) = y l (P l ) POD basis: eigenvalue problem with λ λ... Rψ i = tf y = y(u), i.e., ψ i = ψ i (u) and λ i = λ i (u) t y(t),ψ i y(t)dt = λ i ψ i, i =,...,l ( ) A-priori analysis for linear-quadratic, time-variant problems [Hinze/V. 8]: ) ū ū l = O( λ i (ū) for ψ i = ψ i (ū) i=l+ but ū ū l l with no rate otherwise [Tröltzsch/V. 9] OS-POD: augment (P l ) by the additional constraints ( ) improve the quality of the initial basis by applying a few gradient steps proceed with a fixed basis and utilize a-posteriori error [Gubisch/V., Tröltzsch/V. 9] Optimization method: variants of semismooth Newton methods Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

25 Basis Update by Optimality-System POD Numerical Test Experiment OS-POD for Linear-Quadratic, Control Constrained Control [Grimm, Grimm/Gubisch/V. 5] Optimal control problem: minj(y,u) = y(t f ) y Ω κ tf dx + u dsdt Ω Ω s.t. c p y t y = in Q, n y + qy = u on Σ, y() = y in Ω = (,) u a = u = u b on Σ = (,t f ) Ω initial state y desired final state yω optimal final state ȳ(tf) 6.5 y (x,x).5 yω(x,x) ȳ(tf,x,x).5 x.. x k = k = with ū FE req. l CPU 7. s 8. s.5 s εape FE.e-.8e-.9e- ū FE ū l 9.5e-.6e-.9e- ū FE ū l / ū FE 7.7e-.5e-.59e- x.. x x.. x.6.8 k = k = with ū FE different u a () different u b 8 6 (89) Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 5 / 6

26 Basis Update by Optimality-System POD x x x x x x Numerical Test Experiment x x x x x x x x x OS-POD for Linear-Quadratic, Control Constrained Control [Grimm, Grimm/Gubisch/V. 5] First four POD basis functions generated from u = : POD basis function ψ POD basis function ψ POD basis function ψ POD basis function ψ x x x First four POD basis functions generated from the optimal control ū FE : POD basis function ψ POD basis function ψ POD basis function ψ POD basis function ψ x x x First four POD basis functions generated after k = OS-POD gradient steps: POD basis function ψ POD basis function ψ POD basis function ψ POD basis function ψ x x x Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 6 / 6

27 Basis Update by Optimality-System POD State Constraints State and Control Constrained Optimal Control Problem Quadratic programming (QP) problem: min x=(y,u) J(x) = y(t f) y d L (Ω) + κ subject to the linear evolution problem tf t u(t) U dt y t (t),ϕ + a(t;y(t),ϕ) = (f + Bu)(t),ϕ ϕ V in (t,t f ] with y(t ) = y and to bilateral control constraints u U ad = { v U u a (t) v(t) u b (t) in [t,t f ] } Lavrentiev regularization: ε > y Y ad = { z L (Q) y a (t,x) z(t,x) y b (t,x) in Q } tf J(x,w) = y(t f) y d L (Ω) + κ u(t) U dt + σ w(t) t L t (Ω) dt (y,w) { z L (Q) y a (t,x) εw(t,x) + z(t,x) y b (t,x) in Q } Regular Lagrange multipliers [Tröltzsch 5]: formulation as a control constrained problem A-posteriori error [Grimm/Gubisch/V. 5]: extension from the control-constrained case OS-POD and semismooth Newton method also applicable tf Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 7 / 6

28 Basis Update by Optimality-System POD Numerical Test Experiment x x Numerical Example [Grimm/Gubisch/V. 5] optimal state optimal control Two-dimensional heat equation.5..5 Distributed control Semismooth Newton method y(t,x,x)... Bu(T,x,x).5.5 direction x direction x First four POD basis functions generated from the optimal control ū FE : st pod basis element nd pod basis element rd pod basis element th pod basis element ψ(x,x) ψ6(x,x) ψ(x,x) ψ(x,x) direction x direction x direction x direction x direction x direction x direction x direction x First four POD basis functions generated after k = OS-POD gradient steps: st pod basis element nd pod basis element rd pod basis element th pod basis element ψ(x,x) ψ6(x,x) ψ(x,x) ψ(x,x) direction x direction x direction x direction x direction x direction x direction x direction x Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 8 / 6

29 Extension to Semilinear Optimal Control Problems Extension to Semilinear Optimal Control Problems Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 9 / 6

30 Extension to Semilinear Optimal Control Problems POD A-Posteriori Error for Semilinear Optimal Control Problems A-Posteriori Analysis for Semilinear Optimal Control Problems [Kammann/Tröltzsch/V., DFG grant) Reduced problem: min u U ad Ĵ(u) with hessian Ĵ (u) First-order optimality conditions: Ĵ (ū)(u ū) for all u U ad Second-order sufficient optimality conditions: there is a constant η = η(ū) > with Ĵ (ū)(u,u) η u u Ĵ (ũ)(u,u) η u u, ũ U ad provided ũ ū sufficiently small Theorem [Kammann/Tröltzsch/V. ] ū optimal control, ū l suboptimal control. Then, second-order sufficient optimality implies ū ū l ε ape with ε ape = η ζ (ūl ) provided ū ū l is sufficiently small Computation of ζ (ū l ): based on state y = y(ū l ) and p = p(ū l ) Problem: estimate η = η(ū) smallest eigenvalue of Ĵ (ũ) Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

31 Extension to Semilinear Optimal Control Problems Numerical Test Experiment Numerical Example: Distributed Optimal Control [Kammann/Tröltzsch/V., Trenz 5] Control input: u d (t,x) = i= u i(t)w i (x) and u b Bilateral control bounds: u d a = and u d b = Nonlinearity: N (y) = y with N (y) = y Discretization: N = 79 FE unknowns, N t = time instances and l = 7 PODs Exact optimal state: ȳ(t,x) = cos(x )cos(x ) = y (x) for x = (x,x ) Ω = (,π) Exact optimal control: ū = ū and ū = ū in [t,t f ] = [,].8.6. Analytical optimal control u i (t) ua,ub i = i = i = i = Error u i (t) uh, i (t) (NEWTCG) i = 9 x u i (t) ul, i (t) (NEWTCG), l = i = t t t POD (l = 7) Newton-CG BFGS BFGS-Inv # Iterations 9 Time.7 s 6.5 s 7. s J(ȳ l,ū l ).5e-.5e- -5e- ū ū l.6e- 5.7e- 5.9e- εape.7e-.7e- (.6e-).7e- (.9e-) λ min 5.7e- 5.7e- (.7e-) 5.7e- (.e-) FE (N = 79) Newton-CG # Iterations 6 Time 59 s J(ȳ FE,ū FE ).e- ū ū FE 6.69e- Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

32 Extension to Semilinear Optimal Control Problems Trust-Region POD Methods POD Basis Update by Trust-Region Optimization [Arian/Fahl/Sachs, Schu, Rogg ] QP problem in Newton s method: minq k (u δ ) = Ĵ(uk ) + Ĵ (u k ),u δ + Ĵ (u k )(u δ,u δ ) s.t. u a u k + u δ u b Expensive parts: computation of gradient Ĵ (u k ) and hessian Ĵ (u k ) utilze POD approximations Ĵ l (uk ) and Ĵ l (uk ) in a trust region around u k Trust-region POD subproblem: minĵ(uk ) + Ĵ l (uk ),u δ + Ĵ l (uk )(u δ,u δ ) s.t. u a u k + u δ u b and u δ k Convergence criterium: Carter condition Ĵ (u k ) Ĵ l (uk ) γ Ĵ l (uk ) with γ (,) Numerical experiments [Rogg ] - Steihaug CG [Nocedal/Wright 6], global convergence - faster for two-dimensional, semilinear parabolic PDE Combination with a-posteriori error analysis: use a-posteriori error for the dual Ĵ (u k ) Ĵ l (uk ) = B (p(y(u k )) p l (y l (u k )) Cerr dual (u k ) efficient computation by evaluating primal and dual residuals at u k [Rogg/Trenz/V. 5] Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

33 Extension to Semilinear Optimal Control Problems Numerical Test Experiment Numerical Example: Boundary Optimal Control [Rogg ] Boundary control: u b (t,x) = i= u i(t)w i (x) and u d Nonlinearity: N (y) = y with N (y) = y Discretization: N = 77 FE unknowns, N t = time instances and l = 7 PODs ū h 8 6 ū h ū h ū h ū h l Ĵ(u k ) Ĵ (u k ) # CG it. k.7667.e e e e time t Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization / 6

34 Extension to Semilinear Optimal Control Problems Numerical Test Experiment Numerical Example [Rogg 5] First four POD basis functions generated from the u = : Basis x x x x Basis x x ψ (x) ψ (x) ψ (x) x x x x 6 x. Basis.. u FE : Basis... ψ (x) First four POD basis functions generated from the optimal control Basis x ψ (x). Basis Basis ψ (x). ψ (x) ψ (x) Basis x x x. x First four POD basis functions generated after one trust-region step: Basis Basis x ψ (x) Stefan Volkwein.. x. Basis ψ (x) ψ (x) Basis ψ (x).9 x x x x Reduced-Order Methods in PDE Constrained Optimization x. x / 6

35 Conclusions and Outlook Summary and Ongoing Research POD method for constrained optimal control problems A-priori and a-posteriori error analysis for the controls Change of the POD basis by OS-POD and TR-POD POD basis updates by optimality conditions (OS-POD) POD basis updates by trust region constraints (TR-POD) Combination of SQP and OS-POD [Metzdorf/V.] Reduced-order method in PDE constrained multiobjective optimization [Iapichino/Trenz/V.] Model predictive control in pharmaceutical application [Renal Research Institute New York & Rogg/V.] A-priori error analysis for closed-loop control (Hamilton-Jacobi-Bellman) [Alla/Falcone/V. 5] Reduced Basis Methods Model Reduction for Parametrized Systems Balanced Truncation Are YOU interested in... POD Methods Optimization Low Rank Tensor Approximation Reduced Basis Summer School Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 5 / 6

36 References Literature P. Benner, E.W. Sachs, S.V.: Model order reduction for PDE constrained optimization. ISNM, C. Gräßle: POD based inexact SQP methods for optimal control problems governed by a semilinear heat equation. Diploma thesis, University of Konstanz, E. Grimm, M. Gubisch, S.V.: Numerical analysis of optimality-system POD for constrained optimal control. LNCSE, 5 E. Grimm: Optimality system POD and a-posteriori error analysis for linear-quadratic optimal control problems. Master thesis, University of Konstanz, M. Gubisch, S.V.: Proper orthogonal decomposition for linear-quadratic optimal control. Submitted, M. Gubisch, S.V.: POD a-posteriori error analysis for optimal control problems with mixed control-state constraints. COAP, M. Hinze, S. V.: Error estimates for abstract linear-quadratic optimal control problems using POD. COAP, 8 E. Kammann, F. Tröltzsch, S.V.: A method of a-posteriori error estimation with application to POD. ESAIM: MAN, K. Kunisch, S.V.: POD for optimality systems. ESAIM: MAN, 8 S. Rogg: Trust region POD for optimal boundary control of a semilinear heat equation. Diploma thesis, University of Konstanz, A. Studinger, S.V.: Numerical analysis of POD a-posteriori error estimation for optimal control. ISNM, F. Tröltzsch, S.V.: POD a-posteriori error estimates for linear-quadratic optimal control problems. COAP, 9 S.V.: Optimality system POD and a-posteriori error analysis for linear-quadratic problems. Control and Cybernetics, Stefan Volkwein Reduced-Order Methods in PDE Constrained Optimization 6 / 6