Proper Orthogonal Decomposition. POD for PDE Constrained Optimization. Stefan Volkwein

Transcription

1 Proper Orthogonal Decomposition for PDE Constrained Optimization Institute of Mathematics and Statistics, University of Constance Joined work with F. Diwoky, M. Hinze, D. Hömberg, M. Kahlbacher, E. Kammann, K. Kunisch, O. Lass, F. Leibfritz, H. Müller, T. Tonn, F. Tröltzsch, K. Urban AICES, 3 November 011

2 Motivation 1: Parameter identification Model equations (linear, for simplicity): div ( c u) + β u + au = f in Ω R d c u n + qu = g N on Γ N Γ = Ω ( ) u = g D on Γ D = Γ \ Γ N Problem: estimate parameters (e.g., c, β, a or q) in ( ) from given (perturbed) measurements u d for the solution u on (parts of) Γ Mathematical formulation: ( -dimensional) optimization problem min α u u d ds + κ µ s.t. (u,µ) solves ( ) and µ M ad Γ s.t. subject to Numerical strategy: combine optimization methods with fast (local) rate of convergence and POD model reduction for the PDEs

3 Motivation : Optimal control of time-dependent problems Model problem: min 1 y(t) y T dx + κ T u dxdt Ω 0 Γ y t y + f (y) = 0 in Q = (0,T) Ω s.t. y Γ = u on Σ = (0,T) Γ y(0) = y on Ω R d Adjoint system (for gradient computation): p t p + f (y) p = 0, p Γ = 0, p(t) = y T y(t) Optimizer: second-order methods like SQP or (semismooth) Newton Challenge: large-scale fast/real-time optimization

4 Motivation 3: Closed-loop control for time-dependent PDEs Open-loop control: input u(t) ẋ(t)=f (t,x(t),u(t)) x(0)=x R l (after spatial discretization) output y(t) = Cx(t) + Du(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 eq. (v value function) v t (t,y ) + H(v y (t,y ),y ) = 0 in (0,T) R l Strategy: l-dim. spatial approximation by, e.g., POD basis

5 Model reduction: POD basis containing characteristic information 0,5% der Matrixbasis > 45% Information

6 Model reduction: POD basis containing characteristic information 0,5% der Matrixbasis > 45% Information 1% der Matrixbasis > 56% Information

7 Model reduction: POD basis containing characteristic information 0,5% der Matrixbasis > 45% Information 1% der Matrixbasis > 56% Information 5% der Matrixbasis > 76% Information

8 Model reduction: POD basis containing characteristic information 0,5% der Matrixbasis > 45% Information 1% der Matrixbasis > 56% Information 5% der Matrixbasis > 76% Information 10% der Matrixbasis > 85% Information

9 Model reduction: POD basis containing characteristic information 0,5% der Matrixbasis > 45% Information 1% der Matrixbasis > 56% Information 5% der Matrixbasis > 76% Information 10% der Matrixbasis > 85% Information 0% der Matrixbasis > 9% Information Originalbild

10 General outline of the lecture 1.) The Proper Orthogonal Decomposition (POD) method: What is a POD basis? How can we compute the basis numerically? Which theory is behind?.) Reduced-order modelling (ROM) with POD: What is a POD reduced-order model? How can we derive a-priori error estimates? Can we determine an improved ROM in an adaptive way? 3.) POD suboptimal control: How do we apply the POD in PDE constrained optimization? Can we control the error? What can be done for feedback control?

11 Part 1: The POD method What is a POD basis? How can we compute the basis numerically? Which theory is behind?

12 Outline of the first part: The POD method POD and singular value decomposition (SVD) POD method for ordinary differential equations (ODEs) Continuous POD method for ODEs POD method for partial differential equations (PDEs) References

13 POD method & SVD [Kunisch/V. 99, V. 01, V. 09] Given: y 1,...,y n R m ; set V = span {y 1,...,y n } R m Goal: Find l dim V orthonormal vectors {ψ i } l i=1 in Rm minimizing J(ψ 1,...,ψ l ) = n y j j=1 with the Euclidean norm y = y T y Constrained optimization: l i=1 min J(ψ 1,...,ψ l ) subject to ψ T i ψ j = ( y T j ψ i ) ψi { 1 if i = j 0 otherwise Equivalent problem: Find orthonormal ψ 1,...,ψ l R m maximizing J (ψ 1,...,ψ l ) = n j=1 i=1 l y T j ψ i since y j = m i=1 ( y T j ψ i ) ψi

14 Necessary optimality conditions (Part 1) Lagrange functional: l ( ) L(ψ 1,...,ψ l,λ 11,...,λ ll ) = J(ψ 1,...,ψ l ) + λ ij ψ T i ψ j δ ij i,j=1 with the Kronecker symbol δ ij = 1 for i = j and δ ij = 0 otherwise Optimality conditions: L (ψ 1,...,ψ l,λ 11,...,λ ll ) = 0 R m ψ i L (ψ 1,...,ψ l,λ 11,...,λ ll ) = 0 R λ ij for i = 1,...,l for i,j = 1,...,l

15 Necessary optimality conditions (Part ) L(ψ 1,...,ψ l,λ 11,...,λ ll ) = J(ψ 1,...,ψ l ) + l L ψ i = 0 n j=1 L λ ij = 0 ψ T i ψ j = δ ij i,j=1 λ ij ( ψ T i ψ j δ ij ) y j (y T j ψ i ) = λ ii ψ i and λ ij = 0 for i j Setting λ i = λ ii and Y = [y 1,...,y n ] R m n we have YY T ψ i = λ i ψ i for i = 1,...,l i.e., necessary optimality conditions are given by a symmetric m m eigenvalue problem Here: necessary optimality conditions are already sufficient

16 Computation of the POD basis (Part 1) Optimality conditions: YY T ψ i = λ i ψ i for i = 1,...,l Solution by SVD for Y R m n : d = rank Y, σ 1... σ d > 0, U = [u 1,...,u m ] R m m und V = [v 1,...,v n ] R n n orthogonal with ( ) U T D 0 YV = = Σ R m n 0 0 where D = diag (σ 1,...,σ d ) R d d. Moreover, for 1 i d Yv i = σ i u i, Y T u i = σ i v i, YY T u i = σ i u i, Y T Yv i = σ i v i POD basis: ψ i = u i and λ i = σ i > 0 for i = 1,...,l d = dim V with V = span {y 1,...,y n }

17 Computation of the POD basis (Part ) Data ensemble: V = span {y 1,...,y n } R m and d = dim V POD basis of rank l: ψ i = u i and λ i = σ i > 0 for i = 1,...,l d Three choices to compute the ψ i s SVD for Y R m n : Yv i = σ i u i EVD for YY T R m m : YY T u i = σi u i (if m n) EVD for Y T Y R n n : Y T Yv i = σi v i and u i = 1 σ i Yv i (if m n) Essential error formula for the POD basis of rank l: J(ψ 1,...,ψ l ) = n y j j=1 l i=1 ( y T j ψ i ) ψi = d i=l+1 λ i

18 Computation of the POD basis (Part 3) Essential error formula for the POD basis of rank l: n l ( ) J(ψ 1,...,ψ l ) = y j y T j ψ i ψi = j=1 i=1 YY T ψ i = λ i ψ i, 1 i l, and YY T ψ i = n j=1 j=1 d i=l+1 λ i ( y T j ψ i ) yj give λ i = λ i ψi T ψ i = ( ( YY T ) n ) Tψi ( ) T ψ i = y T j ψ i yj ψ i = y j = d i=1 ( y T j ψ i ) ψi, j = 1,...,m, and ψ T i ψ j = δ ij imply n y j j=1 l i=1 ( y T j ψ i ) ψi = n d j=1 i=l+1 y T j ψ i = n y T j ψ i j=1 d i=l+1 λ i

19 Outline of the first part: The POD method POD and SVD POD method for ODEs Continuous POD method for ODEs POD method for PDEs References

20 POD method for ODEs Nonlinear dynamical system in R m : ẏ(t) = f (t,y(t)) for t (0,T) and y(0) = y with given y 0 R m and f : [0,T] R m R m Time grid: 0 t 1 < t <...t n T, δt j = t j t j 1 for j n Available or known snapshots: y j = y(t j ), 1 j n Snapshot ensemble: V = span {y 1,...,y n }, d = dim V n POD basis of rank l < d: with weights α j 0 min n yj α j j=1 l i=1 ( y T j ψ i ) ψi s.t. ψ T i ψ j = δ ij

21 Computation of the POD basis EVD for linear and symmetric R n in ODE space R m : n R n ( ) u i = α j y j y T j u i = σ i u i (YY T u i = σi u i) j=1 and set λ i = σ i, ψ i = u i EVD for linear and symmetric K n = (( αi α j y T j y i )) in R n : and set λ i = σ i, ψ i = 1 λi K n v i = σ i v i (Y T Yv i = σ i v i) n αj (v i ) j y j j=1 Error formula for the POD basis of rank l: n yj l ( ) α j y T j ψ i ψi = j=1 i=1 d i=l+1 λ i

22 Outline of the first part: The POD method POD and SVD POD method for ODEs Continuous POD method for ODEs POD method for PDEs References

23 Continuous POD method [Kunisch/V. 0, Henri 04, V. 09, Chapelle et al. 11] Snapshots: y(t) for all t [0,T] Snapshot ensemble: V = {y(t) t [0,T]}, d = dim V POD basis of rank l < d: T l ( min y(t) y(t) T ) ψ i ψi dt s.t. ψ T i ψ j = δ ij 0 i=1 Optimality conditions: EVP for linear, symmetric, compact R Rψ i = T 0 ( ψ T i y(t) ) y(t)dt = λ i ψ i for i N Error for the POD basis of rank l: T l ( y(t) ψ T i y(t) ) d ψ i dt = 0 i=1 i=l+1 λ i

24 Relationship between discrete and continuous POD Operators R n and R: R n ψ = Rψ = n ( α j ψ T y(t j ) ) y(t j ) j=1 T 0 ( ψ T y(t) ) y(t)dt Operator convergence of R n R: y smooth and appropriate α j s Perturbation theory [Kato 80]: (λ n i,ψn i ) n (λ i,ψ i ) for 1 i l Choice of the weights α j?: ensure convergence R n n R

25 Outline of the first part: The POD method POD and SVD POD method for ODEs Continuous POD method for ODEs POD method for PDEs References

26 POD method for PDEs Heat equation (for instance): y t y = f y in Q = (0,T) Ω n = g on Σ = (0,T) Γ y(0) = y in Ω R l Variational formulation: for V = H 1 (Ω), f.a.a. t [0,T] y t (t)ϕ + y(t) ϕdx = f (t)ϕdx + g(t)ϕds Ω Ω Γ ϕ V FE Galerkin: y m (t) V m = span {ϕ 1,...,ϕ m }, f.a.a. t [0,T] yt m (t)ϕ + y m (t) ϕdx = f (t)ϕdx + g(t)ϕds ϕ V m Ω Ω Γ

27 POD basis computation Time grid: 0 t 1 < t <...t n T, δt = t j t j 1 for j n FE snapshots: y j = y m (t j ) V m, 1 j n Inner product: u,v = Ω uv dx or u,v = uv + u v dx Ω Sizes: # FE s # time instances, i.e., m n Computation of the correlation K n : α j = O(δt) (1 < j < n) αi α j y m j,y m i = α i α j n Y ik Y jl ϕ l,ϕ k = ( DY T MYD ) ij =: Kn ij k,l=1 with M ij = ϕ j,ϕ i (mass or stiffness matrix) and D = diag ( α i ) POD basis: K n v i = λ i v i and ψ i = 1 λi YDv i

28 Motivation Outline POD and SVD POD for ODEs Continuous POD POD for PDEs References Numerical example: Burgers equation Eigenvalues of the correlation matrix K= u(t ),u(t ) yt νyxx + yyx = f y (, 0 = y (, 1) = 0 y (0, ) = y j L (0,π) i 0 10 in Q = (0, T ) Ω λ % on (0, T ) λ1+λ % in Ω = (0, π) R 10 λ1+λ+λ % y (x) = sin(x) and ν = finite elements 6 10 Time integration with Matlab s ode15s Snapshots V = span {y (t1 ),..., y (t100 )} i axis ψ1 ψ Solution of the Burgers equation x axis x axis ψ ψ 3 6 t axis x axis x axis x axis 4 5 6

29 Numerical example: Energy transport (Boussinesq) u t + uu x + vu y + p x = ν u v t + uv x + vv y + p y = ν v + βθ u x + v y = 0 θ t + uθ x + vθ y = α θ in Q in Q in Q in Q α = 10 5, β = 10, ν = finite elements (Femlab) Time integration with Matlab s ode15s Snapshots at t 1,..., t 1 for u, v and θ 10 0 Eigenvalues of K= u(t i ),u(t j ) L (Ω) und K= v(t i ),v(t j ) L (Ω) u: 97.83% v: 95.98% Eigenvalues of the correlation matrix K= T(t i ),T(t j ) L (Ω) λ % 10 λ 1 +λ 99.84% λ 1 +λ +λ % i axis i axis

30 POD for λ-ω systems [Müller/V. 06] PDEs: s = u + v, λ(s) = 1 s, ω(s) = βs ( ) ( )( ut λ(s) ω(s) u = ω(s) λ(s) v v t ) + ( σ u σ v ) Homogeneous boundary conditions: u = v = 0 or u n = v n = 0 Initial conditions: u (x 1,x ) = x 0.5, v (x 1,x ) = (x 1 0.5)/ u for β=1 and t=100 u for β= and t=100 u for β=3 and t=100 u for β=1 and t=100

31 POD basis for λ-ω systems Offsets: ū(x) = 1 n n u(t j,x) or ū 0 j=1 Snapshots: û j (x) = u(t j,x) ū(x) for 1 j n POD eigenvalue problem: u,v = uv dx Ω K n v i = λv i, 1 i l, with Kij n = α i α j POD basis computation: ψ i = 1 λi 10 0 Decay of the first eigenvalues n αj (v i ) j û j j=1 Ω û j (x)û i (x)dx 10 0 Decay of the first eigenvalues β=1 β=1.5 β= β= β=1 β=1.5 β= β=

32 Elliptic-parabolic systems [Lass/V. 11] Elliptic-parabolic systems: T = 1, Ω = (a, b) y t (c 1 y) N(y,p,q;µ) = 0 (c p) N(y,p,q;µ) = 0 (c 3 q) + N(y,p,q;µ) = 0 in Q = (0,T) Ω in Q in Q

33 Elliptic-parabolic systems [Lass/V. 11] Elliptic-parabolic systems: T = 1, Ω = (a, b) y t (c 1 y) N(y,p,q;µ) = 0 (c p) N(y,p,q;µ) = 0 (c 3 q) + N(y,p,q;µ) = 0 in Q = (0,T) Ω in Q in Q Parameter-dependent nonlinearity: µ = (µ 1,µ ) 0 N(y,p,q;µ) = µ y sinh(µ1 (q p ln y)) Boundary conditions: y x (t,a) = y x (t,b) = p(t,a) = p x (t,b) = 0, q x (t,a) = q(t,b) = 0 Discretization: FE (nd order) and implicit Euler method Numerical solution method: (damped) Newton algorithm

34 POD basis computation POD criterium: l dim(span {y(t) t [0,T]}) T min y(t) 0 l y(t),ψ i ψ i dt s.t. ψi,ψ j = δ ij i=1 Inner product: L (Ω) or H 1 (Ω) (+b.c.) Solution to optimization problem: Rψ i = T 0 y(t),ψ i y(t)dt = λ i ψ i, i = 1,...,l (Kv i )(t) = T 0 y(t),y( ) v i ds = λ i v i (t), i = 1,...,l Relation via SVD: ψ i = T 0 v i(t)y(t)dt/ λ i Discrete variant: α j = O(N 1 t ) N t y(tj l min α j ) y(t j ),ψ i ψ i j=1 i=1 s.t. ψ i,ψ j = δ ij

35 Compare: POD and SVD Singular Values for POD σ Y σ P σ Q Singular and Eigen Values for Y σ λ σ σ and λ # #

36 References Antoulas, Chapelle, Heinkenschloss, Hinze, Petzold, Sachs... Karhunen-Loéve Decomp., Principal Component Analysis, SVD,... Kunisch/V. 99: Control of Burgers equation by a reduced order approach using POD Kunisch/V. 0: Galerkin POD methods for a general equation in fluid dynamics Lass/V. 11: POD Galerkin schemes for nonlinear elliptic-parabolic systems Müller/V. 06: Model reduction by POD for lambda-omega systems V. 01 : Optimal control of a phase-field model using POD V. 09: Model Reduction using POD. Lecture notes