POD for Parametric PDEs and for Optimality Systems

Transcription

1 POD for Parametric PDEs and for Optimality Systems M. Kahlbacher, K. Kunisch, H. Müller and S. Volkwein Institute for Mathematics and Scientific Computing University of Graz, Austria DMV-Jahrestagung 26, Bonn

2 Motivation : Parameter identification Model equations: ( ) div ( c u) + β u + au = f c u n + qu = g N u = g D in Ω R d on Γ N Γ on Γ D = Γ \ Γ N Problem: estimate parameters (e.g., β or a) in ( ) from given (perturbed) measurements u d for the solution u on (parts of) Γ Mathematical formulation: ( -dim.) optimization problem min α u u d 2 ds + κ p 2 s.t. (p, u) solves ( ) and p P ad Γ Numerical strategy: combine optimization methods with fast (local) rate of convergence and POD model reduction for the PDEs s.t. subject to

3 Motivation 2: Optimal control of time-dependent problems Model problem: min 2 s.t. Adjoint system: Ω y(t) y T 2 dx + κ 2 y t y + f (y) = y Γ = u y() = y T Γ u 2 dxdt p t p + f (y) p =, p Γ =, p(t) = y T y(t) Optimizer: second-order algorithms like SQP or Newton methods Challenge: large-scale fast/real-time optimizer

4 Motivation 3: Closed-loop control for time-dependent PDEs Open-loop control: input u(t) ẏ(t)=f (t,y(t),u(t)) y()=y R l (after spatial discretization) output y(t) Closed-loop control: determine F with u(t) = F(t, y(t)) (feedback law) Linear case: LQR and LQG design Nonlinear case: Hamilton-Jacobi-Bellman equation v t (t, y ) + H(v y (t, y ), y ) = in (, T) R l Strategy: l-dim. spatial approximation by POD model reduction

5 Outline of the talk POD in Hilbert spaces Parameter estimation in elliptic systems POD for optimality systems (OS-POD) Conclusions

6 POD in Hilbert spaces Topology: Hilbert space X with inner product, Snapshots: y,..., y n X Snapshot ensemble: V = span {y,..., y n } X, d = dim V n POD basis of any rank l {,..., d}: with weights α j n yj l 2 min α j y j, ψ i ψ i j= i= s.t. ψ i, ψ j = δ ij Constrained optimization: min J(ψ,..., ψ l ) s.t. ψ i, ψ j = δ ij = { if i = j otherwise s.t. subject to

7 Optimality conditions and computation of POD basis EVP for linear, symmetric R n in X: n R n u i = α j u i, y j y j = λ i u i j= and set ψ i = u i EVP for linear, symmetric K n = (( y j, y i )) in R n : and set ψ i = λi K n v i = λ i v i n α j (v i ) j y j (methods of snapshots) j= Error for the POD basis of rank l: n yj l 2 d α j y j, ψ i ψ i = j= i= i=l+ λ i

8 Properties of the POD basis Uncorrelated POD coefficients: n α j y j, ψ i y j, ψ k = δ ik λ i j= Optimality of the POD basis: l n α j y j, ψ i l n 2 α j y j, χ i 2 i= j= i= j= where {χ i } l i= orthonormal in X

9 POD for λ-ω systems [Müller/V.] PDEs: s = u 2 + v 2, λ(s) = s, ω(s) = βs ( ) ( ) ( ut λ(s) ω(s) u = v t ω(s) λ(s) v Homogeneous boundary conditions: ) + ( σ u σ v ) u = v = or u n = v n = Initial conditions: u (x, x 2 ) = x 2.5, v (x, x 2 ) = (x.5)/2 u for β= and t= u for β=2 and t= u for β=3 and t= u for β= and t=

10 POD basis for λ-ω systems Offsets: ū(x) = n n u(t j, x) or ū j= Snapshots: û j (x) = u(t j, x) ū(x) for j n POD eigenvalue problem: X = L 2 (Ω) Kv i = λv i, i l, with K ij = POD basis computation: ψ i = λi n j= α j(v i ) j û j Ω û j (x)û i (x)dx Decay of the first eigenvalues Decay of the first eigenvalues β= β=.5 β=2 β= β= β=.5 β=2 β=

11 ROM for λ-ω systems POD Galerkin ansatz: u l (t, x) = ū(x) + l u j l (t)ψ j(x), j= Reduced-order model (ROM): insert ansatz into PDEs v l (t, x) = v(x) + l v j l (t)φ j(x) multiply by POD basis functions ψi respectively φ i integrate over Ω j= Numerical results: relative error of u for β=.5 t u l(t) u(t) 2 L 2 (Ω) u(t) 2 L 2 (Ω) 5 l= l=5 l=25 l= t axis

12 Relative POD errors for λ-ω systems Offsets: u m (x) = n Relative POD errors: n u(t j, x) or ū j= ū = ū = u m l = l = l = E rel (u) = np α j u l (t j) u(t j) 2 L j= 2 (Ω) np j= α j u(t j) 2 L 2 (Ω) ū = ū = u m l = l = l = for β =.5 (left) and β = 2 (right)

13 Continuous POD in Hilbert spaces [Henri/Yvon, Kunisch/V.,...] Snapshots: y(µ) X for all µ I Snapshot ensemble: V = {y(µ) µ I} X, d = dim V POD basis of rank l < d: l 2 min y(µ) y(µ), ψ i ψ i dµ s.t. ψi, ψ j = δ ij I i= EVP for linear, symmetric R in X: Rψi = ψi, y(µ) y(µ)dµ = λ i ψi Error for the POD basis of rank l: l y(µ) y(µ), ψi, ψi I I i= 2 dµ = i=l+ λ i

14 Relationship between discrete and continuous POD Operators R n and R: n R n ψ = α j ψ, y(µ j ) y(µ j ) j= Rψ = ψ, y(µ) y(µ)dµ I for ψ X = L 2 (Ω) for ψ X = L 2 (Ω) Operator convergence of R n R: y smooth and appropriate α j s Perturbation theory [Kato]: (λ i, ψ i ) n (λ i, ψ i ) for i l Choice of the weights α j?: ensure convergence R n n R

15 Parameter estimation [Kahlbacher/V.] Model equations: β(x) = ( x x 2 ) 2 u + β u + au = in Ω = (, ) (, ) 2 u n u = on Γ Snapshots: (FE) solutions {u j } 2 j= for a j = j POD basis of rank l: 2 (P l ) min u j j= l u j, ψ i ψ i s.t. ψi, ψ j = δ ij i= with ϕ, φ = Ω ϕφdx and ϕ = ϕ, ϕ Solution to (P l ): correlation matrix K ij = u i, u j Kv i = λ i v i, λ λ 2... λ l, ψ i = λi 2 (v i ) j u j j=

16 λ i Reduced-order modelling (ROM) Ansatz: u l = i l ul i ψ i and Galerkin projection Error estimate: u l (a) u(a) 2 da i>l λ i Exponential decay of the eigenvalues: λ i = λ e η(i ) Experimental order of decay (EOD) [Hinze/V.]: EOD := l max l max Q(l) with l= Q(l) = ln R u l (a) u(a) 2 da R u l+ (a) u(a) 2 da η Decay of Eigenvalues λ i 2 Eigenvalues exponential decay Relative errors for different numbers of POD functions upod(k) ufe(k) / ufe(k) i axis 5 npod= npod=2 npod=3 npod=4 npod= q axis

17 Parameter identification Model equations: g(x) = x ( ) 3 4 u + ( ) u + au = g in Ω = (, ) (, ) 3 u 4 n u = on Γ Data: choose a id and compute (FE) solution u(a id ) to ( ) Reconstruction: estimate a from u d = ( + εδ)u(a id ) Γ with random ε and factor δ = 5% Constrained optimization: min J(a, u) = α u u d 2 ds+κ a 2 s.t. (a, u) solves ( ) and a Relaxation of the inequality: Γ min J λ (a, u) = J(a, u)+ max{, λ + ( a) } 2 s.t. (a, u) solves ( )

18 Global convergent optimization method Outer loop: augmented Lagrangian method control of k and λ k Inner loop: globalized SQP algorithm with fixed ( k, λ k ) for min J k λ k (a, u) s.t. { 3 4 u + ( ) u + au = g in Ω 3 u 4 n u = on Γ Numerical results: α = 5, κ =.5, a id = 25, l = 7 Decay of the first eigenvalues 2 4 relative errors: u l u(a id ) u(a id ) a l a id a id

19 OS-POD [Kunisch/V.] Minimize: J(y, u) = β 2 T s.t. (e.g., Navier-Stokes) y(t) z(t) 2 H dt + 2 T u(t) T Ru(t)dt d dt y(t) + Ay(t) + N(y(t)) = m u k (t)b k in [, T] and k= y() = y H, V Hilbert spaces, V H = H V (e.g., H = L 2, V = H ) R R m m with R, z L 2 (, T;H), β > a : V V R bounded, symmetric, coercive A : V V with Aφ, ϕ V,V = a(φ, ϕ) for all φ, ϕ V N : V V, u L 2 (, T; R m ), y H, b k H

20 POD modelling Choose snapshots, e.g., V = {y(t) t [, T]} for fixed u Compute POD basis ψ,...,ψ l Galerkin ansatz: x(t) = l x i (t)ψ i i= Model reduction low dimension Problems: Quality of the basis for unknown optimal control? Can we avoid the computation of y for various inputs? Can we avoid the computation of p for various observations? Compare: Balanced truncation for ẋ(t) = Ax(t) + Bu(t), t > and x() = x y(t) = Cx(t), t >

21 Optimality System-POD Minimize for fixed l > J l (x, ψ, u) = β 2 s.t. T T x(t) T( E(ψ)x(t) 2z l (t, ψ) ) dt+ 2 u(t) T Ru(t)dt E(ψ)ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t) t, E(ψ)x() = x D E ij = ψ j, ψ i H, A ij = a(ψ j, ψ i ), B ij = b j, ψ i H, N i = N lp j= x j ψ j, ψ i E, z l i = z(t), ψ i H

22 Optimality System-POD Minimize for fixed l > J l (x, ψ, u) = β 2 s.t. T T x(t) T( E(ψ)x(t) 2z l (t, ψ) ) dt+ 2 u(t) T Ru(t)dt E(ψ)ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t) t, E(ψ)x() = x ẏ(t) + Ay(t) + N(y(t)) = (Bu)(t) = m u k (t)b k t, y() = y R(y)ψ i = T k= y(t), ψ i y(t)dt = λ i ψ i, ψ i, ψ j = δ ij D E ij = ψ j, ψ i H, A ij = a(ψ j, ψ i ), B ij = b j, ψ i H, N i = N lp j= x j ψ j, ψ i E, z l i = z(t), ψ i H

23 OS-POD 2 Constraints: E(ψ)ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t) t, E(ψ)x() = x ẏ(t) + Ay(t) + N(y(t)) = m u k (t)b k t, y() = y k= R(y)ψ i = λ i ψ i, ψ i, ψ j = δ ij State variables: w = (x, ψ,...,ψ l, y, λ,...,λ l ) POD basis: ψ i = ψ i (u) computed from trajectory y = y(u) Existence of optimal solutions under assumptions for N and A Existence of Lagrange multipliers provided λ >... > λ l >

24 Optimality system for OS-POD Optimal solution: w = (x, ψ,..., ψ l, y, λ,..., λ l ) Dual equation for x (POD) dynamics: q : [, T] R l, q(t) = E(ψ) q(t) + ( A(ψ) + N x (x(t), ψ) T) q(t) = β ( z l (t, ψ) E(ψ)x(t) ) Dual equations for y dynamics: p : [, T] V, p(t) = ṗ(t) + (A + N (y(t)) )p(t) = l ) ( y(t), µ i ψ i + y(t), ψ i µ i i= Optimality condition: Ru(t) = B(ψ) T q(t) + B p(t) Right-hand side: (R λ i I)µ i = G i (w) ker (R λ i I), i l

25 Boundary control of the Burgers equation Consider s.t. min J(y, u) = 2 T ( y z 2 dx + β ( u(t) 2 + v(t) 2) ) dt Ω y t νy xx + yy x = f in Q = (, T) Ω νy x (, ) = u in (, T) νy x (, ) + σy(, ) = v in (, T) y(, ) = y in Ω = (, ) Desired state z=z(t,x).5 T =, β =., ν =.5, f (t, x) = e 3t sin(2πx), σ =., y = sin(2πx) x axis.2.4 t axis.6.8

26 Numerical strategy: Update of POD basis Idea: combine fast POD solver and OS-POD information ) Choose u and set n = 2) Determine PODs {ψi n}l i= for yn = y n (u n ) 3) Solve inexactly [ũ n, q n ] = SQP l (m) (POD solver) 4) Solve dual quation for p n (OS-POD information) 5) Compute u n+ = ũ n τ ( Rũ n (t) B(ψ n ) T q n (t) B p n (t) ), τ > 6) Set n = n + and go back to 2) if n n max Stopping criterium: OS-POD gradient small, POD solver exact SQP l (m): m SQP steps for the control problem with fixed POD basis

27 Numerical test (figures) Finite element solution y=y(t,x) POD basis ψ POD basis ψ Initial condition y x axis 2.5 x axis.5 POD basis ψ 3 4 POD basis ψ x axis x axis.2.4 t axis x axis x axis Desired state z=z(t,x) POD optimal state y * =y * (t,x) POD basis ψ POD basis ψ x axis 2.5 x axis.5 2 POD basis ψ 3 4 POD basis ψ x axis t axis x axis.2.4 t axis x axis x axis

28 Numerical test (tables) J(y, u, v) Solve for u = v =.2234 OS-POD.383 FE-SQP.3765 Steps Generate snapshots Determine POD Determine ROM SQP solver Compute µ i s Dual FE solver Backtracking in 5) CPU time.62 s.9 s.3 s 4.6 s 2.7 s.7 s.32 s n = n = 4 with FE controls λ /tr (K h ) λ 2 /tr (K h ) λ 3 /tr (K h ) λ 4 /tr (K h )

29 Conclusions POD in Hilbert spaces: choice of weights and norms convergence estimates [Kunisch/V., Hinze/V., Kahlbacher/V.] Parameter estimation in elliptic systems Helmholtz equation [ACC Graz] OS-POD: update of the PODs within optimization optimal POD basis at optimal solution more complex problems Preprints: POD scriptum: