Proper Orthogonal Decomposition (POD) for Nonlinear Dynamical Systems. Stefan Volkwein

Transcription

1 Proper Orthogonal Decomposition (POD) for Nonlinear Dynamical Systems Institute for Mathematics and Scientific Computing, Austria DISC Summerschool 5

2 Outline of the talk POD and singular value decomposition (SVD) Snapshot POD for nonlinear dynamical systems Reduced-order modeling (ROM) Numerical examples: heat flow, Burgers equation, Navier-Stokes equations Continuous POD for nonlinear dynamical systems Numerical examples

3 POD as a minimizing problem Given: y,..., y n R m ; set V = span {y,..., y n } R m Goal: Find l dim V orthonormal vectors {ψ i } l i= in Rm minimizing J(ψ,..., ψ l ) = n y j j= l i= with the Euclidean norm y = y T y Constrained optimization: min J(ψ,..., ψ l ) subject to ψ T i ψ j = ( ) y T j ψ i ψi min! { if i = j otherwise

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

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

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

7 Computation of the POD basis (Part ) Data ensemble: V = span {y,..., y n } R m and d = dim V POD basis of rank l: ψ i = u i and λ i = σi > for i =,..., 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 = σ i Yv i (if m n) Error formula for the POD basis of rank l: J(ψ,..., ψ l ) = n y j j= l i= ( y T j ψ i ) ψi = d i=l+ λ i

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

9 Snapshot POD for dynamical systems Nonlinear dynamical system in a Hilbert space X : ẏ(t) = f (t, y(t)) for t (, T ) and y() = y with continuous f : [, T ] X X and given y X Time grid: t < t <... t n T, δt j = t j t j for j n Available or known snapshots: y j = y(t j ) X, j n Snapshot ensemble: V = span {y,..., y n } X, d = dim V n POD basis of rank l < d: with weights α j n yj min α j j= i= l y j, ψ i X ψ i X s.t. ψ i, ψ j X = δ ij Application: X = R m and α j = for j n

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

11 ROM (Part ) Heat equation (for instance): y t y = f y in Q = (, T ) Ω n = g on Σ = (, T ) Γ y() = y in Ω Variational formulation: V = H (Ω), y(t) V y t (t)ϕ + y(t) ϕ dx = f (t)ϕ dx + g(t)ϕ ds Ω Ω Γ ϕ V FE discretization: y m (t) V m = span {ϕ,..., ϕ m } V yt m (t)ϕ + y m (t) ϕ dx = f (t)ϕ dx + g(t)ϕ ds ϕ V m Ω Ω Γ

12 ROM (Part ) Time grid: t < t <... t n T, δt j = t j t j for j n FE snapshots: y j = y m (t j ) V = H (Ω), j n Topology: V H = L (Ω), X = H or X = V Sizes: # FE s # time instances, i.e., m n Computation of the correlation K n : α j = n n y j m, yi m X = n Y ik Y jl ϕ l, ϕ k X = ( n Y T MY ) ij k,l= with M ij = ϕ j, ϕ i X (mass [X = H] or stiffness matrix [X = V ]) ROM for heat equation: y l (t) V l = span {ψ,..., ψ l } V m yt l (t)ψ + y l (t) ψ dx = f (t)ψ dx + g(t)ψ ds ψ V l Ω Ω Γ

13 Heat flow in a block (Part ) y t y = in Q = (, 5) Ω y = on Γ = {(.5, y) :.8 y.8} y n =. on Γ = {(.5, y) :.8 y.8} y n = on Ω \ (Γ Γ ) y(, ) = in Ω R.8 Domain Ω and triangular mesh FE solution at the terminal time Triangulation of Ω and FE solution at T = 5 computed with 844 degrees of freedom, backward Euler and y.9 n = 6 equidistant time instances = t <... < t n = x.8 y axis.5 x axis.5.85

14 Heat flow in a block (Part ) FE space V m = span {ϕ,..., ϕ m } H (Ω) L (Ω), m = 844 Time grid: T = 5, n = 6, δt = Snapshots: y m j = m Y ij ϕ i, j n i= Topology for POD: X = L (Ω) or X = H (Ω) T n, t j = (j )δt, j n Computation of the correlation matrix (m n): K n = n Y T MY with M = (( ϕ j, ϕ i X )) EVD for K n : ( n Y T MY ) n v i = λ i v i and ψ i = n (v λ i ) j y m i j V m j=

15 Heat flow in a block (Part 3) Decay of the first eigenvalues for X = L (Ω) and X = H (Ω) Dacay rate of the first eigenvalues for X=H First eigenvalues λ k for X=V 4 λ k k k Rapid decay of the eigenvalues, e.g., for l = 5 n n y m j= j 5 i= y m j, ψ i X ψ i X 6 = λ i < 6 i=6

16 Heat flow in a block (Part 4) ROM: l = 5 POD basis functions, X = L (Ω) and X = H (Ω) FE-/POD-solution and error: FE solution at the terminal time POD solution at the terminal time.4 t > u(t) FE POD (X=V) POD (X=H).85.8 y axis.5 x axis y axis.5 x axis t axis

17 Outline POD and SVD Snapshot POD ROM Numerical examples Continuous POD POD error Burgers equation Eigenvalues of the correlation matrix K= u(t ),u(t ) yt νyxx + yyx = f y (, = y (, ) = y (, ) = y in Ω = (, π) R j L (,π) i in Q = (, T ) Ω λ % on (, T ) λ+λ % λ+λ+λ % y (x ) = sin(x ) and ν = finite elements 6 Time integration with Matlab s ode5s Snapshots V = span {y (t ),..., y (t )} ψ 8 ψ 3 4 i axis Solution of the Burgers equation x axis 4 6 ψ x axis 4 6 ψ 3 t axis x axis x axis x axis 4 5 6

18 Outline POD and SVD Snapshot POD ROM Numerical examples Continuous POD POD error Navier-Stokes equation ut + uux + vuy + px = ν u vt + uvx + vvy + py = ν v ux + v y = in Q = (, T ) Ω in Q in Q ν = finite elements (Femlab) Time integration with Matlab s ode5s Snapshots V(u) = span {u(t ),..., u(t )} and V(v ) = span {v (t ),..., v (t )} Eigenvalues of K= u(ti),u(tj) L(Ω) and K= v(ti),v(tj) L(Ω) ψ.4 u: 99.74% v: 96.8% ψ i axis

19 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 = θ t + uθ x + vθ y = α θ in Q in Q in Q in Q α = 5, β =, ν = finite elements (Femlab) Time integration with Matlab s ode5s Snapshots at t,..., t for u, v and θ 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 99.7% λ +λ 99.84% 3 λ +λ +λ % i axis i axis

20 Discussions Problems: Choice of the grid t <... < t n T, i.e., of the snapshots Dependence of {(λ i, ψ i )} l i= on the time grid {t j} n j= Choice for the weights α j Procedere: continuous version of POD

21 Continuous POD for dynamical system Nonlinear dynamical system in a Hilbert space X : ẏ(t) = f (t, y(t)) for t (, T ) and y() = y with continuous f : [, T ] X X and given y X Available or known snapshots: y(t) X for all t [, T ] Snapshot ensemble: V = {y(t) t [, T ]} X, d = dim V POD basis of rank l < d: T min y(t) l y(t), ψ i X ψ i i= X dt s.t. ψ i, ψ j X = δ ij

22 Computation of the POD basis EVD for linear and symmetric R in X : Ru i = T u i, y(t) X y(t) dt = σ i u i (YY T u i = σ i u i) and set λ i = σ i, ψ i = u i EVD for linear and symmetric K in L (, T ): (Kv i )(t) = T and set λ i = σi, ψ i = λ i y(s), y(t) X v i (s) dt = σ i v i(t) (Y T Yv i = σ i v i) n α j (v i ) j y j j= Error formula for the POD basis of rank l: T l y(t) y(t), ψi X ψi dt = X i= i=l+ λ i

23 Relation to Snapshot POD Operators R n and R: n R n ψ = α j ψ, y j X y j Rψ = j= T ψ, y(t) X y(t) dt for ψ X for ψ X Convergence of R R n : y j = y(t j ) and appropriate α j s Perturbation theory [Kato]: n λ i λ i n ψ i ψi d(n) n λ i λ i i= i= for i l for i l

24 Relation to Snapshot POD Choice of times: capture the dynamics y([, T ]) as good as possible Choice of weights: ensure operator convergence R n R

25 Burgers equation: y t νy xx + yy x = f in Q = (, T ) Ω y(, = y(, ) = on (, T ) y(, ) = in Ω = (, ) R exact solution y(t, x) = (x x) sin(πt) and f = y t νy xx + yy x FE with error < and fine time grid with n = 5, δt = T n Snapshots at t i,..., ti n i with δt i = i δt, t i j = jδt i, i Error: e(n i ) = δt i n i y FE (tj i) y POD(tj i) L (Ω) j= Implicit Euler: error O(δt) e(n i ) O(δt i ) = O( n i ) Quotient: n i = 4n i e(ni ) e(n i ) 4 i e(n i ) e(n i )

26 References Karhunen-Loéve Decomp., Principal Component Analysis, SVD,... Kunisch & V.: Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, JOTA, :345-37, 999 Kunisch & V.: Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 9:7-48, V.: Optimal and suboptimal control of partial differential equations: augmented Lagrange-SQP methods and reduced-order modeling with proper orthogonal decomposition, Habilitation, Kunisch & V.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SINUM, 4:49-55,