Error Control and Adaptivity for Reduced Basis Approximations of Parametrized Evolution Equations. Mario Ohlberger

Transcription

1 Error Control and Adaptivity for Reduced Basis Approximations of Parametrized Evolution Equations Mario Ohlberger In cooperation with: M. Dihlmann, M. Drohmann, B. Haasdonk, G. Rozza Workshop on A posteriori error estimates and mesh adaptivity for evolutionary nonlinear problems, Paris, July 7, 2010

2 Outline Background: Multi-scale and multi-physics problems Mathematical challenges Model reduction for parameterized evolution equations Reduced basis methods for linear problems Adaptive basis enrichment Generalization to non-linear problems Current work and outlook

3 Institut für Numerische Multi-Scale- and Multi-Physics Problems Example: PEM fuel cells Pore Cell Stack System [BMBF-Project PEMDesign: Fraunhofer ITWM and Fraunhofer ISE]

4 Multi-Scale- and Multi-Physics Problems Example: PEM fuel cells Porous layers: Two phase flow with phase transition Species transport with reaction for O 2, H 2, H 2 O Potential flow for electrons Energy balance Membrane: Two phase water transport Potential flow for protons Energy balance Gas channels: flow, species transport and energy balance Bipolar plates: electron flow and energy balance Coupling through interface conditions

5 Multi-Scale- and Multi-Physics Problems Example: Hydrological Modeling [BMBF-Project AdaptHydroMod: Bronstert et al., Potsdam]

6 Multi-Scale- and Multi-Physics Problems [BMBF-Project AdaptHydroMod: Wald & Corbe, Hügelsheim ]

7 Mathematical Challenges and Possible Solutions

8 Mathematical Challenges and Possible Solutions

9 Reduced Basis Method for Linear Parameterized Evolution Equations with B. Haasdonk and G. Rozza Goal: Fast Online -Simulation of Complex Evolution Systems for Parameter Optimization Design Optimization Optimal Control Integration into System Simulation

10 Reduced Basis Method for Linear Parameterized Evolution Equations with B. Haasdonk and G. Rozza Goal: Fast Online -Simulation of Complex Evolution Systems for Parameter Optimization Design Optimization Optimal Control Integration into System Simulation Ansatz: Reduced Basis Method (RB) dim(w N ) << dim(w H )! Some references: notation RB [Noor, Peters 80], initial value problems [Porsching, Lee 87], method [Nguyen et al. 05], book [Patera, Rozza 07],

11 Example: Convection-Diffusion Problem t c(µ) + (v(µ)c(µ) d(µ) u(µ)) = 0 in Ω [0, T max ], c(, 0; µ)=u 0 (µ) in Ω, c(µ)=b dir (µ) in Γ dir [0, T max ], (v(µ)c(µ) d(µ) c(µ)) n=b neu (u; µ) in Γ neu [0, T max ]. Discretization by Finite Volumes = c H (µ) W H.

12 Example: Convection-Diffusion Problem Parameter: Initial Data Boundary Values Diffusion Parameter Possible Variations of the Solution:

13 Model Reduction: Reduced Basis Method Goal: Find c(, t; µ) L 2 (Ω) for t [0, T], µ P IR p with t c(µ) + Lµ(c(µ)) = 0 in Ω [0, T], plus suitable Initial and Boundary Conditions.

14 Model Reduction: Reduced Basis Method Goal: Find c(, t; µ) L 2 (Ω) for t [0, T], µ P IR p with t c(µ) + Lµ(c(µ)) = 0 in Ω [0, T], plus suitable Initial and Boundary Conditions. Assumption: FV/DG Approximation c H (µ) W H for given Parameter µ

15 Model Reduction: Reduced Basis Method Goal: Find c(, t; µ) L 2 (Ω) for t [0, T], µ P IR p with t c(µ) + Lµ(c(µ)) = 0 in Ω [0, T], plus suitable Initial and Boundary Conditions. Assumption: FV/DG Approximation c H (µ) W H for given Parameter µ Ansatz (RB): Define low dimensional Subspace W N W H and project FV/DG Scheme onto the Subspace = RB Approximation c N (µ) W N.

16 Model Reduction: Reduced Basis Method Goal: Find c(, t; µ) L 2 (Ω) for t [0, T], µ P IR p with t c(µ) + Lµ(c(µ)) = 0 in Ω [0, T], plus suitable Initial and Boundary Conditions. Assumption: FV/DG Approximation c H (µ) W H for given Parameter µ Ansatz (RB): Define low dimensional Subspace W N W H and project FV/DG Scheme onto the Subspace = RB Approximation c N (µ) W N. Requirement: Efficient Choice of W N (Exponential Convergence in N) Offline Online Decomposition for fast Calculation of c N (µ) Error Control for c H (µ) c N (µ)

17 Model Reduction: Reduced Basis Method Assumption: FV/DG Scheme for Evolution Equations of the Form c 0 H = P [c 0 (µ)], L k I(µ)[c k+1 H (µ)] = Lk E(µ)[c k H(µ)] + b k (µ). with time step counter k and c k H (µ) W H.

18 Model Reduction: Reduced Basis Method Assumption: FV/DG Scheme for Evolution Equations of the Form c 0 H = P [c 0 (µ)], L k I(µ)[c k+1 H (µ)] = Lk E(µ)[c k H(µ)] + b k (µ). with time step counter k and c k H (µ) W H. RB Method: Let W N W H be given, {ϕ 1,..., ϕ N } a ONB of W N. Sought: c k N (µ) = N n=1 ak n(µ)ϕ n with L k I(µ)a k+1 = L k E(µ)a k + b k (µ). We then have (L k I(µ)) nm := ϕ n L k I(µ)[ϕ m ], (L k E(µ)) nm := ϕ n L k E(µ)[ϕ m ], Ω Ω (a 0 (µ)) n = P [c 0 (µ)]ϕ n, (b k (µ)) n := ϕ n b k (µ). Ω Ω

19 A Posteriori Error Estimates Definition: Residual of the FV/DG Method at Time t k R k+1 (µ)[c N ] := 1 t ( L k I (µ)[c k+1 N (µ)] Lk E(µ)[c k N(µ)] b k (µ) )

20 A Posteriori Error Estimates Definition: Residual of the FV/DG Method at Time t k R k+1 (µ)[c N ] := 1 t ( L k I (µ)[c k+1 N (µ)] Lk E(µ)[c k N(µ)] b k (µ) ) Theorem: A Posteriori Error Estimate in L L 2 c k N (µ) c k H(µ) k 1 L 2 (Ω) t (C E ) k 1 l R l+1 (µ)[c N (µ)] L 2 (Ω) l=0

21 A Posteriori Error Estimates Theorem: A Posteriori Error Estimate in a Weighted Energy Norm c k N(µ) c k H(µ) 2 γ with weighted energy norm 1 4αC(1 γc) ( k 1 l=0 v 2 γ := v k 2 L 2 (Ω) + γ ( k l=1 t v l, L l I[v l ] t R l+1 [c N (µ)] 2 L 2 (Ω) ) )

22 Offline Online Decomposition Goal: All Steps for the Calculation of c N (µ) and for the Calculation of the Error Estimator are split into Two Parts: Offline Step: Complexity depending on dim(w H ) Online Step: Complexity independent of dim(w H )

23 Offline Online Decomposition Goal: Constrained: All Steps for the Calculation of c N (µ) and for the Calculation of the Error Estimator are split into Two Parts: Offline Step: Complexity depending on dim(w H ) Online Step: Complexity independent of dim(w H ) Affine Parameter Dependency of the Evolution Scheme L k I (µ)[ ] = Q q=1 L k,q I [ ] σ q L I (µ) depending on x depending on µ = Precompute offline: (L k,q I ) nm := ϕ n L k,q I [ϕ m ] Ω = Assemble online: (L k I (µ)) nm := Q (L k,q I ) nm σ q L I (µ) q=1

24 Offline Online Decomposition Goal: Constrained: All Steps for the Calculation of c N (µ) and for the Calculation of the Error Estimator are split into Two Parts: Offline Step: Complexity depending on dim(w H ) Online Step: Complexity independent of dim(w H ) Affine Parameter Dependency of the Evolution Scheme L k I (µ)[ ] = Q q=1 L k,q I [ ] σ q L I (µ) depending on x depending on µ = Precompute offline: (L k,q I ) nm := ϕ n L k,q I [ϕ m ] Ω = Assemble online: (L k I (µ)) nm := Q (L k,q I ) nm σ q L I (µ) q=1

25 Numerical Results CPU-Time Comparison for the Convection-Diffusion Problem: Discretization: Elements, K = 200 time steps time dependent data constant data Reference RB online RB offline Reference RB online RB offline implicit s 16.67s s 45.67s 1.02s 2.41s Factor explicit s 16.53s s 1.51s 0.79s 2.31s Factor Discretization: Elements, K = 1000 time steps time dependent data constant data Reference RB online RB offline Reference RB online RB offline implicit s s s s 6.18s 9.22s Factor explicit s s s 17.41s 3.64s 8.83s Factor

26 Efficient Choice of W N Idea: Construct WÑ as the span of Snapshots c H (µ), µ D P. Use Error Estimator for an efficient Choice of the Snapshots with Guaranteed Error Control on a Training Set. Reduce WÑ to W N with Principal Component Analysis (PCA). Goal: Exponential Convergence in N!

27 Efficient Choice of W N Idea: Construct WÑ as the span of Snapshots c H (µ), µ D P. Use Error Estimator for an efficient Choice of the Snapshots with Guaranteed Error Control on a Training Set. Reduce WÑ to W N with Principal Component Analysis (PCA). Goal: Exponential Convergence in N! Preliminary Result: Convergence in N for Training and Test Sets

28 Improvement through Adaptive Basis Enrichment Algorithms for Basis Enrichment: Fixed / Adaptive Training Sets ESGREEDY(Φ 0, M train, ε tol, M val, ρ tol ) 1 Φ := Φ 0 2 repeat 3 µ := arg maxµ M train (µ, Φ) 4 if (µ ) > ε tol 5 then 6 ϕ := ONBASISEXT(u H(µ ), Φ) 7 Φ := Φ {ϕ} 8 ε := maxµ M train (µ, Φ) 9 ρ := maxµ M val (µ, Φ)/ε 10 until ε ε tol or ρ ρ tol 11 return Φ, ε 12 Here, M denotes a Partition of the Parameter Space, and V (M) are the Vertices of M.

29 Improvement through Adaptive Basis Enrichment Algorithms for Basis Enrichment: Fixed / Adaptive Training Sets ESGREEDY(Φ 0, M train, ε tol, M val, ρ tol ) 1 Φ := Φ 0 2 repeat 3 µ := arg maxµ M train (µ, Φ) 4 if (µ ) > ε tol 5 then 6 ϕ := ONBASISEXT(u H(µ ), Φ) 7 Φ := Φ {ϕ} 8 ε := maxµ M train (µ, Φ) 9 ρ := maxµ M val (µ, Φ)/ε 10 until ε ε tol or ρ ρ tol 11 return Φ, ε 12 RBADAPTIVE(Φ 0, M 0, ε tol, M val, ρ tol ) 1 Φ := Φ 0, M := M 0 2 repeat 3 M train := V (M) 4 [Φ, ε] := ESGREEDY(Φ, M train, ε tol, 5 M val, ρ tol ) 6 if ε > ε tol 7 then 8 η = ELEMENTINDICATORS(M, Φ, ε) 9 M := MARK(M, η) 10 M := REFINE(M) 11 until ε ε tol 12 return Φ Here, M denotes a Partition of the Parameter Space, and V (M) are the Vertices of M.

30 Improvement through Adaptive Basis Enrichment Error Distribution for Uniform / Adaptive Training Sets maximum test estimator values Delta N Exponential Convergence and CPU-Efficiency 10 1 test estimator values decrease uniform fixed 4 3 uniform fixed 5 3 uniform refined 2 3 uniform refined 3 3 adaptive refined 2 3 adaptive refined num basis functions N max test estimator value max test estimator over training time uniform fixed 4 3 uniform fixed 5 3 uniform refined 2 3 uniform refined 3 3 adaptive refined 2 3 adaptive refined training time

31 Adaptive Parameter Domain Partition [Dihlmann, Haasdonk, Ohlberger 10] Idea: Ansatz: Construct a reduced model with prescribed error tolerance and an upper bound for the dimension of the reduced space. Parameter domain partition and construction of independent reduced spaces in the parameter sub-domains.

32 Adaptive Parameter Domain Partition [Dihlmann, Haasdonk, Ohlberger 10] Idea: Ansatz: Construct a reduced model with prescribed error tolerance and an upper bound for the dimension of the reduced space. Parameter domain partition and construction of independent reduced spaces in the parameter sub-domains.

33 Adaptive Parameter Domain Partition ADAPTIVEPARAMPARTITION(M 0, ε tol, N max ) 1 M := M 0, Φ(e) := for e E(M) 2 repeat 3 for e E(M) with Φ(e) = 4 do Φ 0 := INITBASIS(e) 5 M train := MTRAIN(e) 6 η(e) := 0 7 [Φ(e), ε(e)] := EARLYSTOPPINGGREEDY(Φ, M train, ε tol,,, N max ) 8 if ε(e) > ε tol 9 then η(e) := 1, Φ(e) := 10 η max := max e E(M) η(e) 11 if η max > 0 12 then M := MARK(M, η) 13 M := REFINE(M) 14 until η max = 0 15 return M, {Φ(e), ε(e)} e E(M)

34 Evaluation of the Online Efficiency (max) test error over online simulation time no P partition fixed P partition adaptive P partition adapt. P partition + adapt. M train (maximal) test error (average) online simulation time [s]

35 Extension to Nonlinear Explicit Operators Reduced Basis Method for Explicit Finite Volume Approximations of Nonlinear Conservation Laws [Haasdonk, Ohlberger 08] A Simple Model Problem t c(µ) + (vc(µ) µ ) = 0 in Ω [0, T], µ [1, 2] plus suitable Initial and Boundary Conditions. µ = 1 = Linear Transport µ = 2 = Burgers Equation

36 Numerical Results Initial values: c 0 (x) = 1/2(1 + sin(2πx 1 ) sin(2πx 2 )) Linear Transport Solution at t = 0.3 Burgers Equation

37 General Framework Nonlinear Equation: t c(µ) + Lµ[c(µ)] = 0 in Ω [0, T ], Explicit Discretization: c k+1 H (µ) = ck H(µ) tl k H(µ)[c k H(µ)]. Problem: Non-Affine Parameter Dependency Non-Linear Evolution Operator

38 General Framework Nonlinear Equation: t c(µ) + Lµ[c(µ)] = 0 in Ω [0, T ], Explicit Discretization: c k+1 H (µ) = ck H(µ) tl k H(µ)[c k H(µ)]. Problem: Idea: Non-Affine Parameter Dependency Non-Linear Evolution Operator Linear Affine Approximation through Empirical Interpolation L k H (µ)[ck H (µ)](x) M y m (c, µ, t k )ξ m (x) m=1

39 General Framework Nonlinear Equation: t c(µ) + Lµ[c(µ)] = 0 in Ω [0, T ], Explicit Discretization: c k+1 H (µ) = ck H(µ) tl k H(µ)[c k H(µ)]. Problem: Idea: Non-Affine Parameter Dependency Non-Linear Evolution Operator Linear Affine Approximation through Empirical Interpolation L k H (µ)[ck H (µ)](x) M y m (c, µ, t k )ξ m (x) m=1 y m (c, µ, t k ) := L k H (µ)[ck H (µ)](x m)

40 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)]

41 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)] Ingredients: Collateral Reduced Basis Space: W M := span{l k m H (µ m )[c k m H (µ m )] m = 1,..., M}

42 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)] Ingredients: Collateral Reduced Basis Space: W M := span{l k m H (µ m )[c k m H (µ m )] m = 1,..., M} Nodal Collateral Reduced Basis: {ξ m } M m=1 = W M = span{ξ m m = 1,..., M} Interpolation Points: {x k } M k=1 with ξ m(x k ) = δ mk

43 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)] Ingredients: Collateral Reduced Basis Space: W M := span{l k m H (µ m )[c k m H (µ m )] m = 1,..., M} Nodal Collateral Reduced Basis: {ξ m } M m=1 = W M = span{ξ m m = 1,..., M} Interpolation Points: {x k } M k=1 with ξ m(x k ) = δ mk Empirical Interpolation: I M [L k H (µ)[ck H (µ)]] := M y m (c, µ, t k )ξ m (x) m=1

44 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)] Ingredients: Collateral Reduced Basis Space: W M := span{l k m H (µ m )[c k m H (µ m )] m = 1,..., M} Nodal Collateral Reduced Basis: {ξ m } M m=1 = W M = span{ξ m m = 1,..., M} Interpolation Points: {x k } M k=1 with ξ m(x k ) = δ mk Empirical Interpolation: I M [L k H (µ)[ck H (µ)]] := M y m (c, µ, t k )ξ m (x) Offline: Collateral Basis {ξ m } M m=1 and Interpolation Points {x m } M m=1 Online: Calculate Coefficients y m = L k H (µ)[ck H (µ)](x m) m=1

45 Empirical Interpolation of Localized Operators Idea: Construct a Collateral Reduced Basis Space W M that approximates the space spanned by L k H (µ)[ck H (µ)] Ingredients: Collateral Reduced Basis Space: W M := span{l k m H (µ m )[c k m H (µ m )] m = 1,..., M} Nodal Collateral Reduced Basis: {ξ m } M m=1 = W M = span{ξ m m = 1,..., M} Interpolation Points: {x k } M k=1 with ξ m(x k ) = δ mk Empirical Interpolation: I M [L k H (µ)[ck H (µ)]] := M y m (c, µ, t k )ξ m (x) Offline: Collateral Basis {ξ m } M m=1 and Interpolation Points {x m } M m=1 Online: Calculate Coefficients y m = L k H (µ)[ck H (µ)](x m) = Localized operators for H-independent point evaluations m=1

46 Local Operator Evaluations and Reduced Basis Scheme Local Operator Evaluations in the Online-Phase require: 1.) Local reconstruction of c k N from coefficients ak 2.) Local operator evaluation: y m = L k H (µ)[ck H (µ)](x m) Requires Offline: Numerical subgrids, local representation of ϕ n W N

47 Local Operator Evaluations and Reduced Basis Scheme Local Operator Evaluations in the Online-Phase require: 1.) Local reconstruction of c k N from coefficients ak 2.) Local operator evaluation: y m = L k H (µ)[ck H (µ)](x m) Requires Offline: Numerical subgrids, local representation of ϕ n W N RB Method: Galerkin projection of interpolated evolution scheme Ω ( c k+1 N (µ) = ck N(µ) ti M [L k H(µ)[c k N(µ)]] ) ϕ, ϕ W N. Offline-Online decomposition analog to the linear and affine case!!

48 Empirical Interpolation: M max = 150 interpolation points Translation symmetry detected Numerical Experiment

49 Empirical Interpolation: M max = 150 interpolation points Translation symmetry detected Numerical Experiment Test error convergence: Exponential convergence for simultaneous increase of N and M

50 Numerical Experiment Comparison of Online-Runtimes Simulation Dimension Runtime [s] Gain Factor detailed H = reduced N=20, M= reduced N=40, M= reduced N=60, M= reduced N=80, M= reduced N=100, M=

51 Numerical Experiment Comparison of Online-Runtimes Simulation Dimension Runtime [s] Gain Factor detailed H = reduced N=20, M= reduced N=40, M= reduced N=60, M= reduced N=80, M= reduced N=100, M= < Demonstration >

52 Extension to Nonlinear Implicit Operators with M. Drohmann and B. Haasdonk t c(µ) + Lµ[c(µ)] = 0 in Ω [0, T ], Mixed Implicit - Explicit Discretization: L k I(µ)[c k+1 H (µ)] = Lk E(µ)[c k H(µ)]. Problem: Non-Affine Parameter Dependency Non-Linear Evolution Operators L k I involves the solution of a non-linear System

53 Extension to Nonlinear Implicit Operators with M. Drohmann and B. Haasdonk t c(µ) + Lµ[c(µ)] = 0 in Ω [0, T ], Mixed Implicit - Explicit Discretization: L k I(µ)[c k+1 H (µ)] = Lk E(µ)[c k H(µ)]. Problem: Ansatz: Non-Affine Parameter Dependency Non-Linear Evolution Operators L k I involves the solution of a non-linear System Newton s Method and Empirical interpolation for the linearized defect equation

54 Newton s Method and Empirical Interpolation Define the defect d k+1,ν+1 H := c k+1,ν+1 H c k+1,ν H. Solve in each Newton step ν for the defect FI k (µ)[c k+1,ν H ]d k+1,ν+1 H = L k I(µ)[c k+1,ν H ] + L k E(µ)[c k H], and update c k+1,ν+1 H = c k+1,ν H + d k+1,ν+1 H. Here FI k is the Frechet derivative of Lk I. Problem: FI k has Non-Affine Parameter Dependency L k I and Lk E can be treated as before!

55 Empirical Interpolation for the Frechet Derivative Starting point: Empirical interpolation for L k I M I M [L k I(µ)[c H ]] = ym(c I k H, µ) ξ m. m=1 Empirical Interpolation for F k I I M [F k I (µ)[c H ]v H ] := H M i ym(c I k H, µ)v i ξ m i=1 m=1! M = i ym(c I k H, µ)v i ξ m. i τ m=1 Properties: Here τ {1,..., H} is the smallest subset, such that the equality holds = card(τ) = O(M), since L k I is supposed to be localized! (v i ) i τ can be evaluated efficiently in case of a nodal basis of W H.

56 Resulting Reduced Basis Formulation of one Newton Step Ansatz: c k,ν N (x) = N n=1 ak,ν n φ n (x), ( a k,ν denotes the coefficient vector) G A[c k+1,ν N ] (a k+1,ν+1 a k+1,ν ) = RHS(a } {{ } k+1,ν, a k ). =:d k+1,ν+1 Thereby the matrices A[c N ], G are given as (A[c N ]) m,n := M i ym(c I N, µ)ϕ n (x i ), G n,m := i=1 Ω ξ m ϕ n with a corresponding offline-online splitting.

57 A Posteriori Error Estimate Definition: Residual of the approximated FV/DG Method R k+1 (µ) [c N ] = I M [ LI (µ) [ c N k+1 ]] I M [ LE (µ) [ c N k ]] Theorem: A Posteriori Error Estimate in L L 2 c k N (µ) c k H(µ) L 2 (Ω) k 1 C k l I C k 1 l E l=0 M+M m=m ( ( y I l+1 m cn, µ ) ( ym E l cn, µ )) ξ m +2ε Newton + R l+1 (µ) [c N ] L 2 (Ω) ) L 2 (Ω)

58 Reduced Basis Approximation for the Heat Equation on Parameterized Geometries with M. Drohmann and B. Haasdonk Transformed heat equation on reference domain ˆΩ t û + κ(µ) (GG û) + κ(µ) (vû) + κ(µ) vû = f

59 Preliminary experimental results Results for µ = (0, 0) and t = 0, t = 0.75, and t = 1.5. Results for µ = (0.2, 0.2) and t = 0, t = 0.75, and t = 1.5.

60 Preliminary experimental results Dimension CPU time H = N = 7, M= N = 7, M= N = 14, M= N = 14, M= N = 20, M= N = 20, M= Fig. L 2 convergence Tab. Computing times Gain factor: 10

61 Further work on reduced basis techniques Coupling with DUNE [ Geometry IF Visualization IF Grape DX UG Albert Grid IF User Code SPGrid Functions & Operators IF Sparse Matrix Vector IF Solvers IF CRS N-Adaptivity [HO 09] Block CRS Sparse BCRS µ=(0,0.01) µ=(0.2,1.3)µ=(0.1,0.01) Non-linear evolution equations [DHO 10] t=0.000 t=0.013 t=0.025 t=0.038 t= max error uh ured L (0,Tmax;L 2 (Ω)) RB size N Reduced basis medthods for dynamical systems [H0 09] M = 80 M = 140 M = 250 RB with output estimation

62 Thank you for your attention! Software: DUNE ALUGrid GRAPE RBmatlab