Space-time Galerkin POD for optimal control of Burgers equation. April 27, 2017 Absolventen Seminar Numerische Mathematik, TU Berlin

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1 Space-ime Galerkin POD for opimal conrol of Burgers equaion Manuel Baumann Peer Benner Jan Heiland April 27, 207 Absolvenen Seminar Numerische Mahemaik, TU Berlin

2 Ouline. Inroducion 2. Opimal Space Time Produc Bases 3. Relaion o POD 4. Space-Time Galerkin-POD for Opimal Conrol Jan Heiland Space-ime Galerkin POD 2/9

3 Inroducion ẋ = f (, ) Consider he soluion of a PDE: L 2 (I ; L 2 (Ω)) and is numerical approimaion: wih I R... he ime-inerval Ω R n... he spaial domain S Y wih S L 2 (I )... discreized ime Y L 2 (Ω)... a FE space Task: Find Ŝ S and Ŷ Y of much smaller dimension o epress. Jan Heiland Space-ime Galerkin POD 3/9

4 Space-Time Spaces Consider finie dimensional subspaces S = span{ψ,, ψ s } L 2 (I ) Y = span{ν,, ν q } L 2 (Ω) wih he mass marices M S = [ ] (ψ i, ψ j ) L 2 and M i,j=,...,s Y = [ ] (ν i, ν j ) L 2 and he produc space S Y L 2 (I ; L 2 (Ω)). PDE soluion L 2 (I ; L 2 (Ω)) S L 2 (I )... discreized ime Y L 2 (Ω)... a FE space i,j=,...,q Jan Heiland Space-ime Galerkin POD 4/9

5 Space-Time Spaces We represen a funcion = s q i j ν i ψ j S Y j= i= via is mari of coefficiens X = [ i j ] j=,...,s i=,...,q Rq,s and vice versa. Jan Heiland Space-ime Galerkin POD 5/9

6 Opimal Bases Lemma (Opimal low-rank bases in space ) Given S Y and he associaed mari of coefficiens X. The bes-approimaing subspace Ŷ in he sense ha Π S Ŷ S Y is minimal over all subspaces of Y of dimension ˆq is given as span{ˆν i } i=,...,ˆq, where ˆν ν ˆν 2. = V q Ṱ ν 2 M /2 Y., ν q ˆνˆq where Vˆq is he mari of he ˆq leading lef singular vecors of M /2 Y XM/2 S. BM&PB&JH 6: ArXiv: Jan Heiland Space-ime Galerkin POD 6/9

7 Opimal Bases The same argumens apply o he ranspose of X: Lemma (Opimal low-rank bases in ime 2 ) Given S Y and he associaed mari of coefficiens X. The bes-approimaing subspace Ŝ in he sense ha ΠŜ Y S Y is minimal over all subspaces of S of dimension ŝ is given as span{ ˆψ j } j=,...,ŝ, where ˆψ ψ ˆψ 2 ψ 2. ˆψŝ = UṰ s M /2 S., where Uŝ is he mari of he ŝ leading righ singular vecors of M /2 Y XM/2 S. ψ s 2 BM&PB&JH 6: ArXiv: Jan Heiland Space-ime Galerkin POD 7/9

8 The soluion of a spaially discreized PDE Relaion o POD : τ R q is projeced o S R q via (, ψ ) L 2... (, ψ s ) L 2 Π S Y =..... M S. ( q, ψ ) L 2... ( q, ψ s ) L 2 In he (degeneraed) case ha ψ j is a dela disribuion cenered a τ j I, he coefficien mari degeneraes o (τ )... (τ s )..... q (τ )... q (τ s ) he sandard POD snapsho mari. Jan Heiland Space-ime Galerkin POD 8/9

9 Secion 4 Space-Time Galerkin-POD for Opimal Conrol Jan Heiland Space-ime Galerkin POD 9/9

10 Targe : Sep funcion Figure : Illusraion of he sae, he adjoin, and he arge and heir approimaion via POD-reduced space-ime bases. Jan Heiland Space-ime Galerkin POD 0/9

11 Targe 2: Hear shape Figure : Illusraion of he sae, he adjoin, and he arge and heir approimaion via POD-reduced space-ime bases. Jan Heiland Space-ime Galerkin POD /9

12 Space-Time Galerkin-POD for Opimal Conrol For a arge rajecory L 2 (0, T ; L 2 (Ω)) and a penalizaion parameer α > 0, consider J (, u) := 2 2 L 2 + α 2 u 2 L 2 subjec o he generic PDE min u L 2 (0,T ;L 2 (Ω)) ẋ + N() = f + u, (0) = 0. (FWD) If he nonlineariy is smooh, hen necessary opimaliy condiions for (, u) are given hrough u = αλ, where λ solves he adjoin equaion λ λ + D N() T λ + =, λ(t ) = 0. (BWD) Jan Heiland Space-ime Galerkin POD 2/9

13 Space-Time Galerkin-POD for Opimal Conrol Algorihm:. Do sandard forward/backward solves o compue he mari of measuremens for and λ. 2. Compue opimal low-dimensional spaces Ŝ, ˆR, Ŷ, and ˆΛ for he space and ime discreizaion of he sae and he adjoin sae λ. 3. Solve he space-ime Galerkin projeced necessary opimaliy condiions (FWD)-(BWD) 3 for he reduced cosae ˆλ. 4. Define he subopimal conrol via û = α ˆλ and inflae i o he full space. 5. Apply i in he full order simulaion. 3 (FWD)-(BWD) is a wo-poin boundary value problem wih iniial and erminal condiions for which ime sepping schemes like RKM do no apply. Jan Heiland Space-ime Galerkin POD 3/9

14 Numerical Seup The PDE D Burger s equaion I = (0, ], Ω = (0, ) Viscosiy: ν = Sepfuncion as iniial value Zero Dirichle condiions The opimizaion arge : keep he iniial sae arge 2: make a hear parameer: α = 0 3 The full model Equidisan space and ime grids S = R linear ha funcions Y = Λ linear ha funcions The reduced model Ŷ = ˆΛ... of dimension ˆq = ˆp Ŝ ˆR... of dimensions ŝ = ˆr ˆq, ˆp, ŝ, ˆr... varying n... varying 4 4 dimension of ime paramerizaion for an gradien based approach Jan Heiland Space-ime Galerkin POD 4/9

15 Targe : Sep funcion ˆK ˆ 0 2 L J (ˆ, û) wallime [s] Table : Performance of he subopimal conrol versus he cumulaive dimension ˆK = ˆp + ˆq + ˆr + ŝ of he reduced bases wih ˆp = ˆq = ˆr = ŝ. (ˆq, ŝ)/(ˆp, ˆr) (6, 7) (5,0) (2,0) (0,2) (0,5) ( 7,6) 2 ˆ 0 2 L J (ˆ, û) wallime Table : Performance of he subopimal conrol versus varying disribuions of space and ime resoluions. Jan Heiland Space-ime Galerkin POD 5/9

16 Targe : Sep funcion (ˆq, ŝ)/(ˆp, ˆr) (6, 7) (5,0) (2,0) (0,2) (0,5) ( 7,6) 2 ˆ 0 2 L J (ˆ, û) wallime Table : Performance of he subopimal conrol versus varying disribuions of space and ime resoluions. (ˆq, n ) (3, 8) (5, 9) (6, 20) (9, 5) (20, 6) (8, 3) 2 ˆ 0 2 L J (ˆ, û) wallime Table : Benchmark of an gradien based approach (SQP-POD-BFGS wih α = ) Jan Heiland Space-ime Galerkin POD 6/9

17 Conclusion The space-ime Galerkin POD approach allows for consrucion of opimized Galerkin bases in space and ime in a funcional analyical framework The resuling space-ime Galerkin discreizaion approimaes PDEs by a small sysem of algebraic equaions and naurally eends o boundary value problems in ime can be used for efficien compuaions of (sub)opimal conrols Fuure work: Use he funcional analyical framework for error esimaes. Eploi he freedom of he choice of he measuremen funcions in Y, o produce, e.g., opimal measuremens or o compensae for sochasic perurbaions. Jan Heiland Space-ime Galerkin POD 7/9

18 Furher Reading and Coding M. Baumann, P. Benner, and J. Heiland. Space-Time Galerkin POD wih applicaion in opimal conrol of semi-linear parabolic parial differenial equaions. ArXiv: , Nov M. Baumann, J. Heiland, and M. Schmid. Discree inpu/oupu maps and heir relaion o POD. In P. Benner e al., ediors, Numerical Algebra, Mari Theory, Differenial-Algebraic Equaions and Conrol Theory, pages Springer, 205. J. Heiland and M. Baumann. spaceime-galerkin-pod-bfgs-ess Pyhon/Malab implemenaion space-ime POD and BFGS for opimal conrol of Burgers equaion. 206, doi:0.528/zenodo Jan Heiland Space-ime Galerkin POD 8/9

19 Thank you! Thank you for your aenion! I am always open for discussion heiland@mpi-magdeburg.mpg.de gihub.com/highlando Jan Heiland Space-ime Galerkin POD 9/9