A reduced basis method and ROM-based optimization for batch chromatography

Transcription

1 MAX PLANCK INSTITUTE MedRed, 2013 Magdeburg, Dec A reduced basis method and ROM-based optimization for batch chromatography Yongjin Zhang joint work with Peter Benner, Lihong Feng and Suzhou Li Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Germany FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 1/26

2 Outline 1 Motivation 2 Review on RBM POD-Greedy Algorithm Empirical Interpolation (EI) & Collateral Reduced Basis (CRB) 3 Adaptive Snapshot Selection 4 Error Estimation 5 Numerical Experiments 6 Conclusions and Outlook Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 2/26

3 Motivation : Model Description Batch chromatography: b Feed a + b Column a a b Pump Q t 4 concentration t Solvent concentration t in b a t cyc t Products Principle of batch chromatography for binary separation. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 3/26

4 Motivation : Model Description Batch chromatography: b Feed a + b Column a a b Pump Q t 4 concentration t Solvent concentration t in b a t cyc t Products Principle of batch chromatography for binary separation. Eqs : { cz t + 1 ɛ q z ɛ t = cz x q z t = L Q/(ɛA c ) κ z(q Eq c z Pe x, 0 < x < 1, 2 z q z ), 0 x 1, (1) with suitable initial and boundary conditions. qz Eq = f z (c a (µ), c b (µ)) is nonlinear and nonaffine, z = a, b. µ := (Q, t in ) is the vector of operating parameters. Pe = Ref: Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 3/26

5 Motivation : Model Description (cont.) The optimization of batch chromatography: min { Pr(c z(µ), q z (µ); µ)} µ P s.t. Rec(c z (µ), q z (µ); µ) Rec min, c z (µ), q z (µ) are the solutions to the above system (1). Production rate: Pr = p(µ)q t cyc Recovery yield: Rec = p(µ) t in (c f a +cf b ) p(µ) := t 2 t 1 c b,o (t; µ)dt + t 4 t 3 c a,o (t; µ)dt c z,o (t; µ) := c z (t, x = 1; µ), t [0, T ]: concentrations at the outlet Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 4/26

6 Motivation : General Idea RBM & ROM-based optimization: FOM e.g. POD-Greedy (D)EIM ROM FOM-Opt. Approx. ROM-Opt. Certified error estimation Offline-online decomposition Efficiency of the ROM (-based optimization) Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 5/26

7 Review on RBM Consider a spatial discretized parametrized time dependent system Σ : F (u(t; µ)) = 0, µ P, t [0, T ], where u(t; µ) R N and F is a linear or nonlinear operator. The output of interest y := y(u(µ, t), µ). Assume that a reduced basis (RB) is available, V := [V 1,, V N ], and u û := Va with a := a(t; µ) R N, then a reduced order model (ROM) can be obtained by using the Galerkin projection, ˆΣ : V T F (û) = V T F (Va) = 0, N N. The corresponding output can be approximated by ŷ(µ) y(û(µ)). Remark: Empirical Interpolation Method (EIM) or its variants (e.g. DEIM, empirical operator interpolation) can be employed if F is nonlinear/non-affine. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 6/26

8 Review on RBM POD-Greedy Algorithm Algorithm 1 RB generation using POD-Greedy Input: P train, tol RB (< 1) Output: RB: V = [V 1,..., V N ] 1: Initialization: W N = [ ], N = 0, µ max = µ 0, η N (µ max ) = 1 2: while the error η N (µ max ) > tol RB do 3: Compute the trajectory S max := {u n (µ max )} K n=0 4: Enrich the RB: W N+1 := W N V N+1, where V N+1 is the first POD mode of the matrix Ū = [ū 0,..., ū K ] with ū n := u n (µ max ) Π W N [u n (µ max )], n = 0,..., K, Π W N [u] is the projection of u on the current space W N := span{v 1,..., V N } 5: N = N + 1 6: Find µ max := arg max µ Ptrain η N (µ) 7: end while Influence factors of the cost: N, P train, η(µ max ), K,... Ref: B. Haasdonk and M. Ohlberger, M2NA., 42, Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 7/26

9 Review on RBM Empirical Interpolation (EI) & Collateral Reduced Basis (CRB) Idea: g(x, µ) ĝ m := m i=1 σ i(µ)w i, µ P [Barrault et al. 2004] Algorithm 2 Generation of CRB and EI points T M Input: := {g(x, µ) µ Pcrb train }, tol CRB(< 1) Output: CRB: W = [W 1,..., W M ] and T M = {x 1,..., x M } 1: Initialization: m = 1, Wei 0 := {0}, r 0 = 1 2: while ξ m 1 > tol CRB do L crb train 3: g L crb train, define the best approximation ĝ = m 1 current space W m 1 ei := span{w 1,..., W m 1 } i=1 σ iw i in the 4: Define g := arg max g L crb m := g ˆ g train 5: if ξ m tol CRB then 6: Stop and set M = m 1 7: else 8: Determine: x m := arg sup x Ω r m (x), W m := ξ m /ξ m (x m ) 9: end if 10: m := m : end while Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 8/26

10 Review on RBM Runtime comparison: FOM-Opt. vs. ROM-Opt h vs. 0.58( = 82.83) h Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 9/26

11 Adaptive Snapshot Selection The idea of ASS is to discard the redundant linear information in the trajectory earlier, before performing SVD for the generation of the basis. Given non-zero vectors v 1, v 2, cos θ = < v 1, v 2 > v 1 v 2. v 1 and v 2 are linearly dependent cos θ = 1 (θ = 0 or π) For a trajectory {u n (µ)} n=k n=0, define Ind(u n (µ), u m (µ)) = 1 < un (µ), u m (µ) > u n (µ) u m, (µ) ASS is implemented as follows. θ v 2 v 1 Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 10/26

12 ASS (cont.) Algorithm 3 Adaptive snapshot selection (ASS) Input: Initial vector u 0 (µ), tol ASS Output: Selected snapshot matrix: S A = [u n1 (µ), u n2 (µ),..., u n l (µ)] 1: Initialization: j = 1, n j = 0, S A = [u n j (µ)] 2: for n = 1,..., K do 3: Compute the vector u n (µ). 4: if Ind(u n (µ), u n j (µ)) > tol ASS then 5: j = j + 1 6: n j = n 7: S A = [S A, u n j (µ)] 8: end if 9: end for Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 11/26

13 ASS (cont.) Remark 1: It can be easily combined with other algorithms, e.g. Alg. 1, for the generation of RB and CRB. Remark 2: For the linear dependency, it is also possible to check the angle between the tested vector u n (µ) and the subspace spanned by the selected snapshots S A. More redundant information can be discarded but at more cost. However, the data will be compressed further, e.g. by using the POD-Greedy algorithm, we simply choose the economical case shown in Algorithm 3. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 12/26

14 ASS-POD-Greedy Algorithm 4 RB generation using ASS-POD-Greedy Input: P train, tol RB Output: RB: V = [V 1,..., V N ] 1: Initialization: W N = {0}, N = 0, µ max = µ 0, η(µ max ) = 2: while the error η N (µ max ) > tol RB do 3: Compute the trajectory S := {u n (µ max )} K n=0, adaptively select snapshots using Alg. 3, and get S A := {u n1 (µ max ),..., u n l (µ max )} 4: Enrich the RB: W N+1 := W N V N+1, where V N+1 is the first POD mode of the matrix Ū A = [ū n1,..., ū n l ] with ū ns := u ns (µ max ) Π W N [u ns (µ max )], s = 1,..., n l (n l K), Π W N [u] is the projection of u on the current space W N := span{v 1,..., V N } 5: N = N + 1 6: Find µ max := arg max µ Ptrain η N (µ) 7: end while Remark: the error η N (µ max ) can be an error estimator or the true error. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 13/26

15 Error Estimation Consider a general evolution scheme, Au n+1 (µ) = Bu n (µ) + g n, where A, B R N N are constant matrices, g n := g(u n (µ), µ) R N is nonlinear or non-affine. Let: û n (µ) = Va n (µ): RB approximation of u n (µ), ĝ n (µ) := I M [g(û n (µ))] = W β n (µ): interpolation of g n, V R N N, W R N M : parameter-independent bases. a n R N, β n R M : parameter-dependent. Define the residual: We have the following propositions. r n+1 := Bû n + I M [g(û n )] Aû n+1. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 14/26

16 Error Estimation(cont.) Proposition 1: Assume that the operator g is Lipschitz continuous, i.e., L g R +, s.t. g(x) g(y) L g x y, and the interpolation of g is exact with a certain dimension of W, i.e., I M+M [g(û n )] := M+M m=1 W m β n m = g(û n ). Assume the initial projection error e 0 = 0, then the field variable error e n := u n û n satisfies: n 1 e n (µ) ( A 1 ) n k C n 1 k (ɛ k EI (µ) + r k+1 (µ) ), k=0 where C = B + L g, and ɛ n EI (µ) is the error due to the interpolation. A sharper error bound is given as: n 1 e n (µ) ( A 1 B + L g A 1 ) n 1 k ( A 1 ɛ k EI (µ) + A 1 r k+1 (µ) ) k=0 := η n N,M(µ). Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 15/26

17 Error Estimation(cont.) Proposition 2: Under the assumptions of the Proposition 1, assume the output of interest y(u n (µ)) can be expressed as y(u n (µ)) = Pu n, where P R No N is a constant matrix. Then the output error e n O (µ) := Pun Pû n satisfies: e n+1 O (µ) ηn+1 N,M :=( PA 1 B + L g PA 1 )η n N,M + PA 1 ɛ n EI (µ) + P A 1 r n+1 (µ). Remark: It is easy to show that the output error bound η n+1 N,M than the trivial bound is sharper e n+1 O (µ) = P(un+1 û n+1 ) P e n+1 (µ) P e n+1 (µ) η n+1 N,M. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 16/26

18 RBM & ROM-based Optimization FOM ASS-POD-Greedy ASS-EI ROM Σ : Ac n+1 z q n+1 z = Bcz n + dz n 1 ɛ ɛ thn z = qz n + thz n c n z, q n z R N Approx. ˆΣ : Â cz a n+1 c z a n+1 q z = ˆB cz ac n z + d0 nˆd cz 1 ɛ ɛ tĥ cz βz n = aq n z + tĥ qz βz n a n c z, a n q z R N, a n q z R M N N, M Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 17/26

19 RBM & ROM-based Optimization FOM ASS-POD-Greedy ASS-EI ROM Σ : Ac n+1 z q n+1 z = Bcz n + dz n 1 ɛ ɛ thn z = qz n + thz n c n z, q n z R N Approx. ˆΣ : Â cz a n+1 c z a n+1 q z = ˆB cz ac n z + d0 nˆd cz 1 ɛ ɛ tĥ cz βz n = aq n z + tĥ qz βz n a n c z, a n q z R N, a n q z R M min { Pr(c z(µ), q z (µ); µ)} s.t. µ P Rec(c z (µ), q z (µ); µ) Rec min, c z (µ), q z (µ) : solutions to Σ. N N, M Approx. min { ˆPr(ĉ z (µ), ˆq z (µ); µ)} s.t. µ P Rec(ĉ ˆ z (µ), ˆq z (µ); µ) Rec min, ĉ z (µ), ˆq z (µ) : solutions to ˆΣ. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 17/26

20 Num. Ex.: Performance of ASS Generation of CRBs (W a, W b ) with different tol ASS at the same tolerance tol CRB = tol ASS M (W a W b ) Runtime [h] no ASS (-) ASS ( 90.3%) ASS ( 93.5%) ASS ( 94.4%) Runtime comparison of using POD-Greedy algorithm with early-stop (Alg. 5) with and without ASS. Simulations Num. of RB (N) Runtime [h] POD-Greedy ASS-POD-Greedy ( 50.1%) 1 Done on a Workstation with 4 Intel Xeon E CPUs (8 core per CPU) 2.67 GHz RAM 1TB, others were done on the PC with Quad CPU 2.83GHz RAM 4GB. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 18/26

21 Num. Ex.: Performance of Error Estimation 4 2 Field variable error bound Output error bound True output error max µ Ptrain error Size of RB: N Comparison of the error bound decay during the RB extension and the corresponding true output error. Output error bound: η N := max z {a,b} {max µ Ptrain η N,M,c K z (µ)} Field variable error bound: η N := max z {a,b} {max µ Ptrain η N,M,c K z (µ)} True output error: en max := max µ Ptrain ē N (µ), where ē N (µ) := max z {a,b} ē N,cz (µ), ē N,cz (µ) := 1 K K n=1 cn z,o (µ) ĉn z,o (µ) Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 19/26

22 Num. Ex.: ASS-POD-Greedy with Early-stop Algorithm 5 RB generation using ASS-POD-Greedy with early-stop Input: P train, tol RB, tol decay Output: RB: V = [V 1,..., V N ] 1: Implement Step 1 in Alg. 4 2: while the error η N (µ max ) > tol RB do 3: Implement Steps 3-6 in Alg. 4 4: Compute the decay of the error bound dη = η N 1(µ old 5: if dη < tol deacy then max ) η N (µ max) η N 1 (µ old max ) 6: Compute the true output error at the selected parameter µ max, ē N (µ max ) 7: if ē N (µ max ) < tol RB then 8: Stop 9: end if 10: end if 11: end while Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 20/26

23 Num. Ex.: Error Decay 4 2 Output error bound True output error max µ Ptrain error Size of RB: N Error bound decay during the RB extension using Alg. 5 and the corresponding maximal true output error. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 21/26

24 Num. Ex.: Parameter Location Injection period: t in Feed flow rate: Q Parameter location during the RB extension using Alg. 5. Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 22/26

25 Num. Ex.: ROM Performance Dimensionless Concentration c a FOM c b FOM c a ROM c ROM b Dimensionless Time Concentrations at the outlet of the column with the FOM (N = 1500) and the ROM (N = 47) at the parameter µ = (Q, t in ) = (0.1018, ). Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 23/26

26 Num. Ex.: ROM Validation Runtime comparison of the detailed and reduced simulation over a validation set P val with 600 random sample points. Tolerance for the generation of ROM: tol CRB = , tol RB = , tol ASS = Simulations Max. error Average runtime [s]/fos FOM (N = 1500) (-) ROM, no ASS for POD-Greedy / 53 ROM, ASS-POD-Greedy / 48 Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 24/26

27 Num. Ex.: ROM-based Optimization The optimization for batch chromatography: FOM-based Opt.: min { Pr(c z(µ), q z (µ); µ)} s.t. µ P Rec(c z (µ), q z (µ); µ) Rec min, c z (µ), q z (µ) : solutions to Σ. Approx. ROM-based Opt.: min { ˆPr(ĉ z (µ), ˆq z (µ); µ)} s.t. µ P Rec(ĉ ˆ z (µ), ˆq z (µ); µ) Rec min, ĉ z (µ), ˆq z (µ) : solutions to ˆΣ. P = [0.0667, ] [0.5, 2.0] Pu a = Pu b = 95.0%, Rec min = 80.0% N = 1500, N = 47 Comparison of the optimization based on the ROM and FOM. Simulations Obj. (Pr) Opt. solution (µ) It. Runtime [h]/fos FOM-Opt ( , ) / - ROM-Opt ( , ) / 53 The optimizer: NLOPT GN DIRECT L Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 25/26

28 Conclusions and Outlook FOM ASS-POD-Greedy ASS-EI ROM FOM-Opt. Approx. ROM-Opt. Adaptive Snapshots Selection Output-oriented error estimation Early-stop for (ASS-)POD-Greedy algorithm Outlooks: Error estimation with dual system RBM to Simulated Moving Bed (SMB) chromatography RBM for SMB with uncertainty quantification (UQ) Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 26/26

29 Conclusions and Outlook FOM ASS-POD-Greedy ASS-EI ROM FOM-Opt. Approx. ROM-Opt. Adaptive Snapshots Selection Output-oriented error estimation Early-stop for (ASS-)POD-Greedy algorithm Outlooks: Error estimation with dual system RBM to Simulated Moving Bed (SMB) chromatography RBM for SMB with uncertainty quantification (UQ) Thank you for your attention! Max Planck Institute Magdeburg Y. Zhang, RBM and ROM-based optimization 26/26