Empirical Interpolation of Nonlinear Parametrized Evolution Operators

Transcription

1 MÜNSTER Empirical Interpolation of Nonlinear Parametrized Evolution Operators Martin Drohmann, Bernard Haasdonk, Mario Ohlberger 03/12/2010

2 MÜNSTER 2/20 > Motivation: Reduced Basis Method RB Scenario: Parametrized partial differential equations for (non-stationary) problems Applications relying on time-critical or many repeated simulations Goals: Offline-/Online decomposition Efficient reduced simulations A posteriori error control References: [Patera&Rozza, 2006], [Haasdonk et al., 2008] manifold of "interesting" snapshots U K H (µ) W h

3 MÜNSTER 2/20 > Motivation: Reduced Basis Method RB Scenario: Parametrized partial differential equations for (non-stationary) problems Applications relying on time-critical or many repeated simulations Goals: Offline-/Online decomposition Efficient reduced simulations A posteriori error control References: [Patera&Rozza, 2006], [Haasdonk et al., 2008] manifold of "interesting" snapshots U K H (µ) W h

4 MÜNSTER 2/20 > Motivation: Reduced Basis Method RB Scenario: Parametrized partial differential equations for (non-stationary) problems Applications relying on time-critical or many repeated simulations Goals: Offline-/Online decomposition Efficient reduced simulations A posteriori error control W h References: [Patera&Rozza, 2006], [Haasdonk et al., 2008] W red U K H (µ) U K N (µ)

5 MÜNSTER 3/20 > Motivation: Parametrized Evolution Equation Analytical Formulation For µ P R p, find u : [0, T max] W h L 2 (Ω), s.t. u(0) = u 0(µ), t u(t) L(µ) [u(t)] = 0 plus (parameter dependent) boundary conditions.

6 MÜNSTER 3/20 > Motivation: Parametrized Evolution Equation Analytical Formulation For µ P R p, find u : [0, T max] W h L 2 (Ω), s.t. u(0) = u 0(µ), t u(t) L(µ) [u(t)] = 0 plus (parameter dependent) boundary conditions. Discretization (implicit/explicit with Newton scheme) For µ P find {u h } K k=0 W h L 2 (Ω), s.t. with Newton iteration Id + tdl I h u k+1,ν h u 0 h := P h [u 0(µ)], u k+1 h u k+1,0 := u k h h, uk+1,ν+1 h «h δ k+1,ν+1 h := u k+1,ν h i = u k h uk+1,ν t h := u k+1,νmax(k) h + δ k+1,ν+1, h h L I h u k+1,ν h i h i + L E h u k h.,

7 MÜNSTER 4/20 > Empirical interpolation: Idea General operator approximation Approximate operator evaluations i i L h (µ) hu k h (µ) I M [L h (µ)] hu k h (µ) MX = m=1 l m(µ) ˆu k h (µ) ξ m, where l m(µ) : W h R are efficiently computable functionals and ξ m collateral reduced basis functions.

8 MÜNSTER 4/20 > Empirical interpolation: Idea Approximate operator evaluations i i L h (µ) hu k h (µ) I M [L h (µ)] hu k h (µ) MX = m=1 l m(µ) ˆu k h (µ) ξ m. Empirical interpolation[barrault et al, 2004] collateral reduced basis W M made out of operator evaluations: n h io M L h (µ m ) u km h (µ m ) ( Greedy search in parameter space ) m=1 coefficients are exact operator evaluations at interpolation points {x m} M m=1 : l m(µ) := L h (µ) [ ] (x m).

9 MÜNSTER 4/20 > Empirical interpolation: Idea Approximate operator evaluations i i L h (µ) hu k h (µ) I M [L h (µ)] hu k h (µ) MX = m=1 l m(µ) ˆu k h (µ) ξ m. Empirical interpolation[barrault et al, 2004] collateral reduced basis W M made out of operator evaluations: n h io M L h (µ m ) u km h (µ m ) ( Greedy search in parameter space ) m=1 coefficients are exact operator evaluations at interpolation points {x m} M m=1 : l m(µ) := L h (µ) [ ] (x m). CRB functions ξ m are nodal in interpolation points: ξ m(x n) = δ nm.

10 MÜNSTER 5/20 > Empirical interpolation: Online evaluations Efficient evaluations of l m (µ) The coefficient functionals l m(µ) = L h (µ)[ ](x m) can be computed efficiently during online phase, if operator has localized structure (small stencil) and local geometry information is precomputed in offline phase. Restrict u h to subgrid Evaluate l m(µ) [u h ] local subgrid x m x m

11 MÜNSTER 6/20 > Empirical interpolation: CRB generation CRB extension algorithm m 0, q 0 0 Initialization. repeat for u h L train do u h arg inf v h span{q i } m u h v i=1 h. Find the best approximation. end for u m arg sup uh L train u h u. Find approximation with worst error. h r m u m I m [u m] Compute the residual between u m and its interpolant. x m arg sup x XH r m(x i ) Find new interpolation point. q m rm Find new CRB function. r m(x m) m m + 1 until r m ε tol or m = M max

12 MÜNSTER 6/20 > Empirical interpolation: CRB generation CRB extension algorithm m 0, q 0 0 Initialization. repeat for u h L train do u h arg inf v h span{q i } m u h v i=1 h. Find the best approximation. end for u m arg sup uh L train u h u. Find approximation with worst error. h r m u m I m [u m] Compute the residual between u m and its interpolant. x m arg sup x XH r m(x i ) Find new interpolation point. q m rm Find new CRB function. r m(x m) m m + 1 until r m ε tol or m = M max Use L or L 2

13 MÜNSTER 6/20 > Empirical interpolation: CRB generation CRB extension algorithm m 0, q 0 0 Initialization. repeat for u h L train do u h arg inf v h span{q i } m u h v i=1 h. Find the best approximation. end for u m arg sup uh L train u h u. Find approximation with worst error. h r m u m I m [u m] Compute the residual between u m and its interpolant. x m arg sup x XH r m(x i ) Find new interpolation point. q m rm Find new CRB function. r m(x m) m m + 1 until r m ε tol or m = M max Use L or L 2 For notation: nodal basis ξ := {ξ m} M m=1 made out of {qm}m m=1

14 MÜNSTER 7/20 > Emprical interpolation: Fréchet derivative Observation DI M [L h (µ)] uh [v h ] = P M m=1 Dlm(µ) u h [v h] ξ m parameter dependent parameter independent

15 MÜNSTER 7/20 > Emprical interpolation: Fréchet derivative DI M [L h (µ)] uh [v h ] = P M m=1 Dlm(µ) u h [v h] ξ m Coefficient functionals parameter dependent parameter independent W h is finite dimensional Dl m(µ) uh [v h ] = MX HX m=1 i=1 ψ i l m(µ) [u h ] (v h,i )

16 MÜNSTER 7/20 > Emprical interpolation: Fréchet derivative DI M [L h (µ)] uh [v h ] = P M m=1 Dlm(µ) u h [v h] ξ m Coefficient functionals parameter dependent W h is finite dimensional MX HX Dl m(µ) uh [v h ] = m=1 i=1 MX X = m=1 i I xm L h has local stencil parameter independent ψ i l m(µ) [u h ] (v h,i ) ψ i l m(µ) [u h ] (v h,i ).

17 MÜNSTER 7/20 > Emprical interpolation: Fréchet derivative DI M [L h (µ)] uh [v h ] = P M m=1 Dlm(µ) u h [v h] ξ m Coefficient functionals parameter dependent W h is finite dimensional MX HX Dl m(µ) uh [v h ] = m=1 i=1 MX X = m=1 i I xm L h has local stencil ψ i l m(µ) [u h ] (v h,i ) ψ i l m(µ) [u h ] (v h,i ). Note: card(i xm ) < C for all m = 1,..., M. Complexity still O(M). parameter independent

18 MÜNSTER 8/20 > Reduced basis method for nonlinear schemes Generate reduced basis space with POD-Greedy algorithm: W red := span {ϕ i } N i=1 W h Reduced model order by Galerkin projection: P red : W h W red Offline-/online decomposition of operators: `Pred [u h,0 (µ)] n = P Q u0 q=1 σq u0(µ) R X Ω uq 0ϕ n assuming: u 0(µ) = σu0(µ)u q q 0 q=1 online offline i L I red hu (µ) k red (µ) = P M R m=1 n li m(µ) [u red )] Ω ξmϕn online DL I red (µ) u [δ red red] = P M m=1 l n ψ I i m (µ) [u red] online offline R Ω ξmϕn offline Q u0

19 MÜNSTER 8/20 > Reduced basis method for nonlinear schemes Projection of Operators: P red : W h W red Galerkin projection onto W red W h L I red := P red I M L I h L E red := P red I M L E h Reduced simulation For µ P find {u red } K k=0 W red, s.t. u k+1 red with Newton iteration Id + tdl I red u k+1,ν red := uk+1,νmax(k), u 0 red red := P red ˆuh,0 (µ) u k+1,0 := u k red red, uk+1,ν+1 := u k+1,ν + δ k+1,ν+1, red red red «h i h δ k+1,ν+1 = u k red red uk+1,ν t L I red red u k+1,ν red i h i + L E red u k red.

20 MÜNSTER 9/20 > Reduced basis generation POD-greedy algorithm INPUT: M train P, ε tol, N max OUTPUT: W red Initialize reduced basis space: Φ N0 {ϕ n} N0 n=1, N N0 repeat 1. Find worst approximated reduced solution: µ max arg max µ Mtrain η(µ) 2. Compute trajectory: u k h (µ ) K max k=0. 3. Compute new reduced basis function: ϕ N+1 POD uk h (µ max ) P red ˆuk h (µ max ) K k=0 N N + 1 until error indicator η(µ max ) falls beneath tolerance ε tol or N max is reached. W red span {ϕ n} N n=1

21 MÜNSTER 10/20 > A posteriori error estimator Theorem (A posteriori error estimator ) Assumptions: Then: Operators and DL I are Lipschitz-continuous. h DL I has bounded inverse h Empirical interpolations exact for larger CRB space W M+M and P h [u 0(µ)] W red u k red (µ) uk h (µ) k N,M (µ) with N,M (µ) := recursively defined through the Newton step error estimator Xk 1 i=0 ν max(i) X k+1,j+1 := t C 1 k 1,ν + C k+1,ν 2 + δ k+1,ν+1 red + R k+1,ν+1 D,M + R k+1,ν I,M + R k+1,0 E,M + R k+1,ν. j=1 i,j

22 MÜNSTER 10/20 > A posteriori error estimator Theorem (A posteriori error estimator cont.) Then: u k red (µ) uk h (µ) k N,M (µ) with N,M (µ) := recursively defined through the Newton step error estimator Xk 1 i=0 ν max(i) X k+1,j+1 := t C 1 k1,ν + C 2 k+1,ν + δ k+1,ν+1 red + R k+1,ν+1 D,M + R k+1,ν I,M + R k+1,0 E,M + R k+1,ν. The residuals R,M measure the empirical interpolation error, e.g. M+M R k+1,ν X I,M := l I m m=m h u k+1,ν red i ξ m j=1 i,j

23 MÜNSTER 11/20 > Numerical results: Porous Medium Equation Problem definition: t u m u p = 0 in Ω [0, 1], u(, 0) = c 0 + u 0 on Ω {0} with homogeneous boundary conditions, µ = (p, m, c 0) [1, 5] [0, 0.01] [0, 0.2] Initial data: Sample trajectories: c0+0.5 c0+0.4 c0+0.3 c0+0.2 c0+0.1 c0 untransformed a) t=0.1 untransformed b) t=0.1 untransformed t=1.0 untransformed t=1.0 untransformed c) t=0.1 untransformed d) t=0.1 a) µ = (1, 0.01, 0.2), b) µ = (4, 0.01, 0.2), c) µ = (1, 0.01, 0.0), d) µ = (4, 0.01, 0.0) untransformed t=1.0 untransformed t=1.0

24 MÜNSTER 12/20 > Numerical results: Porous Medium Equation First 6 reduced basis functions

25 MÜNSTER 13/20 > Numerical results: Porous Medium Equation Slices of first 6 reduced basis functions at y = Φ i (x,0.6) x Φ i (x,0.6) x

26 MÜNSTER 14/20 > Numerical results: Empirical interpolation of L I h Subgrid at interpolation points y x 1.0 max. Error in L train Error decrease during EI-Offline algorithm CRB Dim: M

27 MÜNSTER 15/20 > Numerical results: Reduced basis scheme max µ Mtrain η(µ) Dimension Runtime[s] Error H= N=15, M= N=30, M= N=40, M= N=50, M= RB Dim: N Figure: RB error convergence on 100 test samples Table: average time measurements on 100 test samples Number of base functions in W h : Number of Newton steps ν max: 1 20 Time gain factor for online phase:

28 MÜNSTER 16/20 > Numerical results: A posteriori error estimator POD-Greedy with efficient error control (explicit Burgers problem) maxµ P train u h (µ) u red (µ) "True" error for POD-greedy(M ) Basis size: N M = M = 1 M = 20 M = 50 k N,M,M Estimates for POD-Greedy(M ) Basis size: N M = 1 M = 20 M = 50

29 MÜNSTER 17/20 > Numerical results: A posteriori error estimator Efficiency of error estimator η: λ(µ) := η(µ) u h (µ) u red (µ) 11 9 efficiency λ Figure: Error bar plot showing mean and standard deviation of error estimator efficiency over a sample of 20 random parameters for different values of M. The dots indicate the minimum ( ) and maximum ( ) efficiency. M

30 MÜNSTER 18/20 > Outlook Conclusion Model order reduction of general (scalar) parametrized evolution schemes Main ingredient is the empirical interpolation for discrete operators Rigorous error control via a posteriori error estimator is possible

31 MÜNSTER 18/20 > Outlook Conclusion Model order reduction of general (scalar) parametrized evolution schemes Main ingredient is the empirical interpolation for discrete operators Rigorous error control via a posteriori error estimator is possible Some Problems Control and measurement of complexity of parametrized solution manifold ( Kolmogorov n-width ) Big reduced basis space are inefficient and instable through cancellation Variable time step width

32 > Outlook Some Problems MÜNSTER 18/20 Control and measurement of complexity of parametrized solution manifold ( Kolmogorov n-width ) Big reduced basis space are inefficient and instable through cancellation Variable time step width Some solutions/ideas Do intensive tests with real-world discretizations (including systems) Improve reduced basis generation generate many small reduced bases spaces for parameter and time ranges improve RB approximation by adaptive search algorithms change reduced basis spaces over the time Stability analysis of reduced matrices for time-stepping

33 MÜNSTER 19/20 > Software concepts (DUNE-RB/RBMATLAB interface) Goals: Use C++ software DUNE-RB for high-dimensional computations and RBMATLAB for reduced basis algorithms Access to open source implementations of real world problems. (Several available in DUNE) Testing the reduced basis methods on these implementations. For further information see:

34 MÜNSTER 19/20 > Software concepts (DUNE-RB/RBMATLAB interface) Goals: Use C++ software DUNE-RB for high-dimensional computations and RBMATLAB for reduced basis algorithms Access to open source implementations of real world problems. (Several available in DUNE) Testing the reduced basis methods on these implementations. Illustration of interface concept: abstraction TCP/IP communication of low-dimensional data generic procedure of RB generation und reduced simulation linear evolution problems For further information see: non-linear evolution problems...

35 MÜNSTER 19/20 > Software concepts (DUNE-RB/RBMATLAB interface) Goals: Status: Use C++ software DUNE-RB for high-dimensional computations and RBMATLAB for reduced basis algorithms Access to open source implementations of real world problems. (Several available in DUNE) Testing the reduced basis methods on these implementations. Communication interface between RBMATLAB and DUNE-RB exists Example implementation: linear heat equation in Dune affinely parameter dependent structure. (No empirical interpolation necessary) Empirical interpolation of simple operators. For further information see:

36 MÜNSTER 19/20 > Software concepts (DUNE-RB/RBMATLAB interface) dune-rb Control structures parameter model Visualization Reconstruction Offline data reduced basis space grid high dim operators Solvers dune comsol others high dim computation Network RBmatlab Reduced simulation Error estimators Server Client Visualization control For further information see: Reduced basis generation greedy direct mexfunction usage communication of low dim data low dim computation,

37 MÜNSTER 20/20 > References [Patera&Rozza, 2006] A. Patera and G. Rozza. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering [Haasdonk&Ohlberger, 2008] B. Haasdonk und M. Ohlberger Reduced Basis Method for Finite Volume Approximations of Parametrized Linear Evolution Equations M2AN, Math. Model. Numer. Anal. 42, , 2008 [Haasdonk et al., 2008] B. Haasdonk, M. Ohlberger und G. Rozza. A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators ETNA, Electronic Transactions on Numerical Analysis, vol. 32, 2008 [Barrault et al., 2004] M. Barrault, M., Y. Maday, N. Nguyen and A. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations C. R. Math. Acad. Sci. Paris Series I, 2004, 339, [Drohmann et al., 2009] M. Drohmann, B. Haasdonk and M. Ohlberger, Reduced Basis Method for Finite Volume Approximation of Evolution Equations on Parametrized Geometries Proceedings of ALGORITMY 2009, [Drohmann et al.,2010] M. Drohmann, B. Haasdonk and M. Ohlberger, Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation FB 10, Universität Münster num. 10/02 - N Preprint - october 2010,