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
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
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
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
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
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:6.04050 Jan Heiland Space-ime Galerkin POD 6/9
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:6.04050 Jan Heiland Space-ime Galerkin POD 7/9
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
Secion 4 Space-Time Galerkin-POD for Opimal Conrol Jan Heiland Space-ime Galerkin POD 9/9
Targe : Sep funcion 0.5 0 0 0.3 0.7 0.5 0 0 0.3 0.7 0.5 0 0 0.3 0.7 0.5 0 0 0.3 0.7 0.5 0 0 0.3 0.7 0.5 0 0 0.3 0.7 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
Targe 2: Hear shape 0.0 0.3 0.6.0 0 0.3 0.6 0.0 0.3 0.6.0 0 0.3 0.6 0.0 0.3 0.6.0 0 0.3 0.6 0.0 0.3 0.6.0 0 0.3 0.6 0.0 0.3 0.6.0 0 0.3 0.6 0.0 0.3 0.6.0 0 0.3 0.6 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
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
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
Numerical Seup The PDE D Burger s equaion I = (0, ], Ω = (0, ) Viscosiy: ν = 5 0 3 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... 20 linear ha funcions Y = Λ... 220 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
Targe : Sep funcion ˆK 24 36 48 72 96 2 ˆ 0 2 L 0.0330 0.0280 0.092 0.02 0.004 2 J (ˆ, û) 0.035 0.0309 0.0234 0.077 0.052 wallime [s] 0. 0.48.8 8.7 55 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 2 0.043 0.006 0.05 0.0303 0.038 0.0357 J (ˆ, û) 0.089 0.059 0.092 0.0340 0.0353 0.0382 wallime.43 2.7.58.42 2.54. Table : Performance of he subopimal conrol versus varying disribuions of space and ime resoluions. Jan Heiland Space-ime Galerkin POD 5/9
Targe : Sep funcion (ˆq, ŝ)/(ˆp, ˆr) (6, 7) (5,0) (2,0) (0,2) (0,5) ( 7,6) 2 ˆ 0 2 L 2 0.043 0.006 0.05 0.0303 0.038 0.0357 J (ˆ, û) 0.089 0.059 0.092 0.0340 0.0353 0.0382 wallime.43 2.7.58.42 2.54. 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 2 0.044 0.09 0.00 0.0090 0.009 0.0087 J (ˆ, û) 0.046 0.022 0.03 0.0093 0.0095 0.009 wallime 2.98 35.04 40.34 4.93 46.54 50.62 Table : Benchmark of an gradien based approach (SQP-POD-BFGS wih α = 6.25 0 5 ) Jan Heiland Space-ime Galerkin POD 6/9
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
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:6.04050, Nov. 206. 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 585 608. 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.66339. Jan Heiland Space-ime Galerkin POD 8/9
Thank you! Thank you for your aenion! I am always open for discussion heiland@mpi-magdeburg.mpg.de www.janheiland.de gihub.com/highlando Jan Heiland Space-ime Galerkin POD 9/9