Model order reduction & PDE constrained optimization

Transcription

1 1 Model order reduction & PDE constrained optimization Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg Leuven, January 8, 01

2 Names Afanasiev, Attwell, Burns, Cordier,Fahl, Farrell, Glavaski, Graham, Gunzburger, Hinze, Iannou, Ito, King, Kunisch, Lall, Leibfritz, Le Letty, Marsden, Noack, Patera, Peraire, Prudhomme, Ravindran, Sachs, Schroeter, Serban, Sorensen, Tang, Volkwein, Willcox,...

3 3 Outline Problem formulation Model reduction Validation of reduced model Optimization using the reduced model Basic adaptive approach Numerical experiments Recent developments

4 4 Motivation We have a validated mathematical model for physical process (here a pde system) We intend to use this model to tailor and/or optimize the physical process.

5 5 Motivation: -dimensional optimization problem with pde constraints min (y,u) W U ad J(y, u) s.t. y + Ay + G(y) t = Bu in Z y(0) = y 0 in H. Central tasks: Develop solution strategies which obey the rule Effort of optimization = K Effort of simulation with K small, Propose surrogate models for the pde and quantify their errors, Present a complete (numerical) analysis.

6 6 Examples of pde systems 1 Heat equation: A :=. y + Ay + G(y) t = Bu in Z y(0) = y 0 in H. Burgers: A :=, G(y) := yy, 3 Ignition (Bratu): A :=, G(y) := δe y, δ > 0, 4 Navier Stokes: A := P, G(y) := P[(y )y], P Leray projector, 5 Boussinesq [ Approximation: P Gr g A := 0 ], G(y) = G(v, θ) := [ P[(v )v] (v )θ ].

7 7 DOF diagram Spatial Discretization DOF for Moving Horizon Approach DOF for full optimization DOF for Moving Horizon combined with Model Reduction DOF for Model Reduction Approach

8 8 System reduction for model dynamics Navier-Stokes equations y + (y )y ν y + p = f t in Q = (0, T ) Ω, div y = 0 in Q, y(t, ) = g y(0, ) = y 0 in Ω. on Σ = (0, T ) Ω, Aim: Reduced description of the Navier-Stokes equations α + Aα + n(α) = r in (0, T ) α(0) = a 0

9 1. Construction and validation of the reduced model MOR & PDE constrained optimization 9

10 10 System reduction: Expansions w.r.t. base flows Let ȳ denote a base flow and Φ i, i = 1,..., n Modes. Ansatz for the flow: n y = ȳ + α i Φ i i=1 Possibilities: ȳ stationary solution of Navier-Stokes system, Φ i eigenfunctions of the Navier-Stokes system linearized at ȳ. ȳ mean value of instationary Navier-Stokes solution, Φ i eigenfunctions of the Navier-Stokes system linearized at ȳ. ȳ mean value of instationary Navier-Stokes solution, Φ i normalized Modes obtained from snapshot form of Proper Orthogonal Decomposition.

11 11 Snapshot form of POD Let s take snapshots:

12 1 POD with Snapshots Let y 1,..., y n denote an ensemble of snapshots (of the flow or the dynamical system). Build mean ȳ and modes Φ i as follows: n 1 Compute mean ȳ = 1 y n i i=1 Build correlation matrix K = k ij, k ij = y i ȳ, y j ȳ 3 Compute eigenvalues λ 1,..., λ n and eigenvectors v 1,..., v n of K 4 Define modes Φ i := n j=1 5 Normalize modes Φ i = Φi Φ i Properties: v i j (y j ȳ) The modes are pairwise orthogonal w.r.t. inner product, No other basis can contain more information in fewer elements (Information w.r.t. the norm induced by, ).

13 MOR & PDE constrained optimization 13 First 10 Modes containing % of the information Vorticity Y Vorticity Y X Y Vorticity Vorticity Vorticity X Vorticity Y Y X Vorticity X - 6 Y Y X Vorticity X - 6 Y Y X Vorticity X - 6 Y Vorticity X X 6 7 8

14 14 Galerkin projection Ansatz for the flow n y = ȳ + α i Φ i i=1 Galerkin method with basis Φ 1,..., Φ n yields reduced system α + Aα + n(α) = r α(0) = a 0. Here,, denotes the L inner product. n A = ( ) n a i,j i=1, a i,j = ν Φ i Φ j dx, n(α) = (y y)φ i dx r = ν Ω Ω n ȳ Φ i + f Φ i dx Note that Φ 1,..., Φ n are solenoidal. i=1 Ω c n and a 0 = y 0 Φ i dx Ω i=1. i=1,

15 MOR & PDE constrained optimization 15 Long time behaviour of the POD model Amplitudes of the first mode in [34,44] when using N modes in the POD model N=4 N=8 N=10 N=16 N=0 amplitude time

16 MOR & PDE constrained optimization 16 Cylinder flow at Re = 100, reduced versus full model, 50 snapshots 4 Amplitudes of the first four modes (DS1 with 50 equidistant snapshots on [0,6,8]) projections predictions amplitude time 6 8

17 17 Error estimate (Kunisch and Volkwein) Let y(t 1 ),..., y(t n) denote snapshots taken on an equidistant time grid of [0, T ] with gridsize δt. Let λ 1 > > λ d > 0 denote the strictly positive eigenvalues of the correlation matrix K. For l d let V l = Φ 1,..., Φ l. Further set Then l Y k := α i (t k )Φ i. i=1 n d δt Y i y(t i ) H C y 0, Φ i V + 1 d δt λ i + δt. i=1 i=l+1 i=l+1 This result also extends to the case of distinguish time and snapshot grids. Improvements of reduced models and error estimate by different weighting of snapshots.

18 . Optimization with reduced model MOR & PDE constrained optimization 18

19 19 System reduction & control Model optimization problem: min J(y, u) := 1 y z γ dxdt + (y,u) W U Q o Q c u dxdt s.t. y + (y )y ν y + p t = Bu in Q = (0, T ) Ω, div y = 0 in Q, y(t, ) = g on Σ = (0, T ) Ω, y(0, ) = y 0 in Ω. Q o := (0, T ) Ω o Observation cylinder, Q c := (0, T ) Ω c control cylinder, B control operator

20 0 POD model as pde surrogate in the optimization problem Ansatz for state, control and desired state n n n y = ȳ + α i Φ i, u = β i Φ i, z = ȳ + i=1 i=1 i=1 α z i Φ i. Optimization problem with surrogate model (reduced order model) min (y,u) J(y, u) = J(α, β) = 1 T (α α z ) t M 1 (α α z ) dt + γ T 0 0 β t Mβ dt s.t. α + Aα + n(α) = r + Mβ, α(0) = a 0.

21 1 Validity of surrogate model Fact: Control changes system dynamics. Consequence: Mean and modes should be suitably modified during the optimization process. Idea: Adaptively modify the surrogate model and thus, the reduced optimization problems.

22 Adaptive POD control Afanasiev, Hinze Snapshots y 0 i, i = 1,..., N 0 given, u 0 given control, δ [0, 1] required relative information content, j=0. Compute M = argmin { I(M) := } M λ k / N λ k ; I(M) δ. k=1 k=1 3 Compute POD modes and solve min J(α, β) (ROM) s.t. α + Aα = n(α) + Mβ, α(0) = a 0. for u j = u j (β j ). 4 Compute y j corresponding to u j and new snapshots y j+1 i, i = N j + 1,..., N j+1 to the snapshot set y j i, i = 1,..., N j. 5 While u j+1 u j is large, j = j+1 and goto.

23 3 Numerical comparison Flow around a circular cylinder at Re=100. Control gain: Tracking of Stokes flow (or mean flow) ȳ in an observation volume Ω obs behind the cylinder by applying a volume force in the control volume Ω c. Cost functional: J(y, u) = γ T u 1 T dxdt + 0 Ω c 0 Ω obs y ȳ dxdt CPU time needed to compute the suboptimal controls 40 times smaller than that needed to compute the optimal open loop control. But the quality of the controls is very similar. Runtime(Optimization Problem) = 6 8 Runtime(PDE)

24 4 Uncontrolled flow, target flow = mean flow, controlled flow at t = 3.4.

25 MOR & PDE constrained optimization 5 Numerical results cont. Cost, Ω o = Ω = Ω c, control in space of fluctuations, tracking of Stokes flow 10 8 Evolution of cost 1 iteration iteration 3 iteration 4 iteration 5 iteration uncontrolled 1 grad. iteration (optimal) optimal control Time 3

26 MOR & PDE constrained optimization 6 Numerical results cont. Cost, Ω o = Ω = Ω c, control in the space of fluctuations, tracking of mean flow J 10 unkontrolliert 1 Iteration (ISO) Iteration 3 Iteration Iteration 5 Iteration 6 Iteration 7 Iteration (ISO) optimale Kontrolle (NM) optimale Kontrolle (GM) Zeit

27 MOR & PDE constrained optimization 7 Numerical results cont. Control cost, Ω o = Ω = Ω c, control in the space of fluctuations, tracking of mean flow Iteration (ISO) Iteration (ISO) 3 Iteration (ISO) 4 Iteration (ISO) 5 Iteration (ISO) 6 Iteration (ISO) 7 Iteration (ISO) optimale Kontrolle (NM) 10 0 J f Phase 1 0 Phase 4 6

28 8 Recent developments TRPOD by Arian, Fahl and Sachs 000 Idea: Use a POD surrogate model as model function in the Trust Region process. Let J(u) = J(y(u), u), Ĵ(u) = J(ŷ(u), u), with ŷ(u) the response of the POD surrogate model. Pseudo Algorithm: 1 Given u, compute POD model Compute s = argmin u s Ĵ(u + s) 3 ρ := J(u + s ) J(u) Ĵ(u + s ) Ĵ(u) large: u = u + s, increase moderate: u = u + s, decrease small: keep u, decrease Global convergence under standard TR assumptions plus J (u) Ĵ (u) Ĵ (u) sufficiently small POD of the adjoint equation required.

29 9 Recent developments OSPOD by Kunisch and Volkwein 006 Idea: Include choice of trajectory dependent POD modes as subsidiary condition into the optimization problem. This reads min α,φ,u Ĵ(α, Φ, u) s.t. M(Φ) α + A(Φ)α + n(φ)(α) = B(Φ)u, M(Φ)α(0) = α 0 (Φ), (POSPOD l ) y t + Ay + G(y) = Bu, y(0) = y 0, R(y)Φ i = λ i Φ i for i = 1,..., l, Φ i X = 1 for i = 1,..., l. Here Φ = [Φ 1,..., Φ l ], y l = l α i (t)φ, and i=1 T R(y)(z) := y(t), z X y(t)dt for z X. 0

30 30 Upcoming Numerical analysis of POD Galerkin approximations Error estimates for POD approximations to pde constrained optimal control problems Goal oriented concepts in POD Future perspectives Break

31 31 Outline Motivation: PDE constrained optimization Mathematical setting Construction of the POD spaces Tailored discrete concept for optimal control problem and numerical analysis Adaptive concepts - why?

32 3 Motivation: optimization problem with pde constraints min (y,u) W U ad J(y, u) s.t. y + Ay + G(y) t = Bu in Z y(0) = y 0 in H. Approach: Solve this problem by using a POD surrogate model; min J l (y l, u l ) s.t. (y l,u l ) W l U ad Tasks: Error estimation, y l t + Al y l + G l (y l ) = Bu l in (Z l ) y l (0) = y l 0 in Hl. adaption of the POD surrogate model during the optimization loop.

33 33 Mathematical setting, state equation V, H separable Hilbert spaces, (V, H = H, V ) Gelfand triple. a : V V R bounded, coercive and symmetric. Set, V := a(, ). U Hilbert space, B : U L (U, L (V )) linear control operator, y 0 H. State equation d dt (y(t), v) H + a(y(t), v) = (Bu)(t), v V,V, t [0, T ], v V, (y(0), v) H = (y 0, v) H, v V. For every u U the solution y = y(u) W := {w L (V ), w t L (V )} is unique.

34 34 Optimization problem Cost functional J(y, u) := 1 y z L (H) + α u U. Admissibility: u U ad U closed, convex, y y(u) unique solution of state equation associated to u, i.e. d dt (y(t), v) H + a(y(t), v) = (Bu)(t), v V,V, t [0, T ], v V, (y(0), v) H = (y 0, v) H, v V. Minimization problem: (P) min J(y, u) s.t. Admissibility. (y,u) W (0,T ) U ad (P) admits a unique solution (y, u) W U ad.

35 35 Optimality conditions With the reduced cost functional Ĵ(u) := J(y(u), u) there holds (Ĵ (u), v u) 0 for all v U ad. Here Ĵ (u) = αu + B p(y(u)). The function p solves the adjoint equation d dt (p(t), v) H + a(v, p(t)) = (y z, v) H, t [0, T ], v V, (p(t ), v) H = 0, v V. Variational inequality equivalent to nonsmooth operator equation u = P Uad ( 1 ) α B p(y(u)) with P Uad denoting the orthogonal projection onto U ad.

36 36 Construction of the POD Galerkin spaces Let (X = H or X = V ) T Y : L (0, T ) X, Yϕ = ϕ(t)y(t) dt for ϕ L (0, T ). 0 Then Y : X L (0, T ), ( Y z ) (t) = z, y(t) X for z X and t [0, T ], and T R = YY : X X, Rz = z, y(t) X y(t) dt for z X, 0 K = Y Y : L (0, T ) L (0, T ), ( Kϕ ) T (t) = y(s), y(t) X ϕ(s) ds for ϕ L (0, T ) 0 K continuous version of the correlation matrix, and K is compact, R is compact.

37 37 Construction of the POD spaces cont. Right-eigenfunctions: Kv i = λ i v i, λ 1 λ..., and λ i 0 as i Left eigenfunctions: Rψ i = λ i ψ i, i = 1,,... with {ψ i } i N a complete orthonormal basis for X and non negative eigenvalues {λ i } i N. Eigenvalues contain information: T y(t) X dt = λ i and y 0 X = y0, ψ i X. 0 i=1 i=1

38 38 Discrete concept for the state equation For l N choose a POD subspace V l := χ 1,..., χ l of V with the property l y(t) (y(t), χ k ) V χ k L (0,T ;V ) λ k. k=1 k=l+1 Galerkin semi-discretization y l of state y using subspace V l : ( d dt y l (t), v ) H + a(y l (t), v) = (Bu)(t), v V,V, t [0, T ], v V l, (y(0), v) H = (y 0, v) H, v V l. If needed, define similarly a Galerkin semi-discretization p l of p; d ( dt p l (t), v ) H + a(v, pl (t)) = ( y l z, v ) H, t [0, T ], v V l, ( p l (T ), v ) = 0, v V l. H

39 39 Optimization problem with POD surrogate model Discrete minimization problem: ( ˆP l ) min u U ad Ĵ l (u) := J(y l (u), u). ( ˆP l ) admits a unique solution u l U ad. Optimality condition: (Ĵl (u), v u l ) 0 for all v U ad. Here Ĵ l (u) = αu + B p l (y l (u)). The function p l solves the adjoint equation d ( dt p l (t), v ) H + a(v, pl (t)) = ( y l z, v ) H, t [0, T ], v V l, ( p l (T ), v ) = 0, v V l. H Variational inequality equivalent to nonsmooth operator equation u l = P Uad ( 1 ) α B p l (y l (u)).

40 40 Error estimate Theorem: Let u, u l denote the unique solutions of (P) and ( ˆP l ), respectively. Then u u l U 1 {( ) B (p(y(u)) p l (y(u))), u l u α + U T + 0 ( ) y l (u l ) y l (u), y(u) y l (u) dt H Using the analysis of Kunisch and Volkwein for POD approximations one gets u u l U y 0 P l y 0 H + λ k + k=l+1 + y t P l y t L (0,T ;V ) + p(y(u)) Pl (p(y(u))) W (0,T )

41 41 Conclusions from the analysis Get rid of (y P l y) t L (0,T ;V ) include derivative information into your snapshot set. Get rid of p P l p W (0,T ) include adjoint information into your snapshot set. Recipe: For l N choose a POD subspace V l := χ 1,..., χ l of V with the property l y(t) (y(t), χ k ) V χ k W (0,T ) λ k, k=1 k=l+1 and if one intends to solve optimization problems, also ensure l p(t) (p(t), χ k ) V χ k W (0,T ) λ k, k=1 k=l+1

42 41 Conclusions from the analysis Get rid of (y P l y) t L (0,T ;V ) include derivative information into your snapshot set. Get rid of p P l p W (0,T ) include adjoint information into your snapshot set. Recipe: For l N choose a POD subspace V l := χ 1,..., χ l of V with the property l y(t) (y(t), χ k ) V χ k W (0,T ) λ k, k=1 k=l+1 and if one intends to solve optimization problems, also ensure l p(t) (p(t), χ k ) V χ k W (0,T ) λ k, k=1 k=l+1

43 4 Numerical confirmation of approximation order of the states Linear heat equation with y 0 0 and inhomogeneous boundary data. FE-solution {y h (t j )} m j=0 computed on equi distant time grid. Snapshots: y h (t j 1 ) for 1 j m + 1, y j = y h (t j m 1 ) y h (t j m ) for m + j m + 1. t Correlation matrix (k ij ) m+1 i,j=1, k ij = y i, y j V Expected decay of its eigenvalues: Experimental order of decay: λ i = λ 1 e α(i 1) for i 1. Q(l) = ln y l y L (0,T ;X ) y l+1 y EOD := 1 l max Q(k) α. l L (0,T ;X ) max k=1

44 43 Decay of eigenvalues and of norms 10 0 Decay of the eigenvalues and estimated rates Decay of the norms L (H 1 (Ω)) error L (L (Ω)) error λ i λ 1 e α (i 1) mit L (0,T;H 1 (Ω)) λ 1 e α (i 1) mit L (0,T;L (Ω)) i axis Number of POD basis functions

45 44 Optimization problem with POD surrogate model Discrete minimization problem: ( ˆP l ) min u U ad Ĵ l (u) := J(y l (u), u). ( ˆP l ) admits a unique solution u l U ad. Optimality condition: (Ĵl (u), v u l ) 0 for all v U ad. Here Ĵ l (u) = αu + B p l (y l (u)). The function p l solves the adjoint equation d ( dt p l (t), v ) H + a(v, pl (t)) = ( y l z, v ) H, t [0, T ], v V l, ( p l (T ), v ) = 0, v V l. H Variational inequality equivalent to nonsmooth operator equation u l = P Uad ( 1 ) α B p l (y l (u)).

46 45 Error estimate Theorem: Let u, u l denote the unique solutions of (P) and ( ˆP l ), respectively. Then u u l U 1 {( ) B (p(y(u)) p l (y(u))), u l u α + U T + 0 ( ) y l (u l ) y l (u), y(u) y l (u) dt H Consequently, if u y l (u) is Lipschitz continuous w.r.t. the L (H)-norm, there holds u u l U y 0 P l y 0 H + λ k + k=l+1 + y t P l y t L (0,T ;V ) + p(y(u)) Pl (p(y(u))) W (0,T )

47 46 Proof of error estimate Add necessary optimality conditions and use u, u l as test functions: (αu l + B p l (y l (u)), v u l ) 0 for all v U ad. Set v = u (αu + B p(y(u)), v u) 0 for all v U ad. Set v = u l α u u l U (B (p p l (y(u)) + (p l (y(u)) p l (y l (u)))+ ) + (p l (y l (u)) p l )), u l u =: U 1. gives first term in estimate.. gives second term in estimate 3. 0.

48 47 Error between state and and its orthogonal projection 10 0 y P l y L (0,T;H 1 (Ω)) 10 0 y P l y L (0,T;L (Ω)) {ψ 1 } {ψ } {ψ 3 } Number l of POD basis functions {ψ 1 } {ψ } {ψ 3 } Number l of POD basis functions

49 48 Error between co-state and its orthogonal projection 10 0 p P l p L (0,T;H 1 (Ω)) 10 p P l p L (0,T;L (Ω)) {ψ 1 } {ψ } {ψ 3 } Number l of POD basis functions {ψ 1 } {ψ } {ψ 3 } Number l of POD basis functions

50 49 Neumann boundary control of the heat equation Comparison of the optimal controls Comparison of u h u u h u 15 for ψ i 1 u 15 for ψ i u 15 for ψ i t axis 0. u h u 15 for ψ i 1 u h u 15 for ψ i u h u 15 for ψ i t axis

51 50 Recent developments TRPOD by Arian, Fahl and Sachs 000 Idea: Use a POD surrogate model as model function in the Trust Region process. Let J(u) = J(y(u), u), Ĵ(u) = J(ŷ(u), u), with ŷ(u) the response of the POD surrogate model. Pseudo Algorithm: 1 Given u, compute POD model Compute s = argmin u s Ĵ(u + s) 3 ρ := J(u + s ) J(u) Ĵ(u + s ) Ĵ(u) large: u = u + s, increase moderate: u = u + s, decrease small: keep u, decrease Global convergence under standard TR assumptions plus J (u) Ĵ (u) Ĵ (u) sufficiently small.

52 51 Recent developments OSPOD by Kunisch and Volkwein 006 Idea: Include choice of trajectory dependent POD modes as subsidiary condition into the optimization problem. This reads min α,φ,u Ĵ(α, Φ, u) s.t. M(Φ) α + A(Φ)α + n(φ)(α) = B(Φ)u, M(Φ)α(0) = α 0 (Φ), (POSPOD l ) y t + Ay + G(y) = Bu, y(0) = y 0, R(y)Φ i = λ i Φ i for i = 1,..., l, Φ i X = 1 for i = 1,..., l. Here Φ = [Φ 1,..., Φ l ], y l = l α i (t)φ, and i=1 T R(y)(z) := y(t), z X y(t)dt for z X. 0 A very similar approach is proposed by Ghattas, van Bloemen Waanders and Willcox 005.

53 5 How many snapshots? Meyer, Matthies, Heuveline, H. How many snapshots? iterative goal oriented procedure. Goal: Resolve J(y) Start on coarse equi-distant time grid and compute snapshots Build POD model and compute y h and adjoint z h of reduced dynamics Becker and Rannacher: J(y) J(y h ) η(y h, z h ) η(y h, z h ) > tol: double number of snapshots (re-computation)

54 53 Where to take snapshots? Where to take snapshots? time-step adaption via sensitivity of POD model. Goal: Optimal time-grid for system dynamics Start on coarse (equi-distant) time grid and compute snapshots Build POD model and compute y h and adjoint z h of reduced dynamics Becker, Johnson, Rannacher: η(y h, z h ) = I j ρ loc j (y h )ωj loc (z h ) New time-grid: equi-distribute ρ loc j (y h )ωj loc (z h )

55 54 Further developments and improvements Efficient treatment of nonlinearities Chaturantabut, Sorensen (010) MOR for the input output map Heiland, Mehrmann A posteriori POD concept Tröltzsch and Volkwein (010) Which modes? DWR concepts (Matthies, Meyer 003) How many snapshots? iterative goal oriented DWR procedure Were take snapshots? time step adaption via sensitivity of the POD model POD in the context of space mapping Sampling of parameter ( control) space Greedy sampling by Patera and Rozza (007) Use of linear MOR techniques for nonlinear problems SQP context, semi linear time integration, domain decomposition Thank you for your attention

56 54 Further developments and improvements Efficient treatment of nonlinearities Chaturantabut, Sorensen (010) MOR for the input output map Heiland, Mehrmann A posteriori POD concept Tröltzsch and Volkwein (010) Which modes? DWR concepts (Matthies, Meyer 003) How many snapshots? iterative goal oriented DWR procedure Were take snapshots? time step adaption via sensitivity of the POD model POD in the context of space mapping Sampling of parameter ( control) space Greedy sampling by Patera and Rozza (007) Use of linear MOR techniques for nonlinear problems SQP context, semi linear time integration, domain decomposition Thank you for your attention