Reduced Basis Methods for Nonlinear Parametrized Partial Differential Equations

Transcription

1 Reduced Basis Methods for Nonlinear Parametrized Partial Differential Equations M. Grepl Aachen Institute for Advanced Study in Computational Engineering Science AICES Progress since 2009 Overview of the A New Principal Inv New Young Rese New facilities Recent highlights Goals and sched Workshop on Model Order Reduction of Transport-dominated Phenomena Advisory Board Meeting Brandenburg-BerlinRWTH Academy of Sciences and Humanities Aachen University July 12, 2011 May 19-20, 2015 Overview of the AICES Project Overvie

2 Acknowledgments Collaborators: Eduard Bader Mark Kärcher Dirk Klindworth Robert O Connor Mohammad Rasty Karen Veroy Sponsors: Excellence Initiative of the German federal and state governments German Research Foundation through Grant GSC 111 M. Grepl (RWTH Aachen) 1

3 Outline PART I: A Reduced Basis Primer PART II: Nonlinear (and Nonaffine) Problems M. Grepl (RWTH Aachen) 2

4 Outline PART I: A Reduced Basis Primer Introduction Key Ingredients Extensions & References PART II: Nonlinear (and Nonaffine) Problems M. Grepl (RWTH Aachen) 2

5 Outline PART I: A Reduced Basis Primer Introduction Key Ingredients Extensions & References PART II: Nonlinear (and Nonaffine) Problems Empirical Interpolation Method Nonlinear Elliptic Problems Extensions & References Numerical Results Nonlinear Reaction Diffusion Systems M. Grepl (RWTH Aachen) 2

6 Part I A Reduced Basis Primer M. Grepl (RWTH Aachen) 3

7 Introduction The Reduced Basis Method M. Grepl (RWTH Aachen) 4

8 Introduction The Reduced Basis Method is a model order reduction technique that provides M. Grepl (RWTH Aachen) 4

9 Introduction The Reduced Basis Method is a model order reduction technique that provides rapid and reliable approximation of solutions M. Grepl (RWTH Aachen) 4

10 Introduction The Reduced Basis Method is a model order reduction technique that provides rapid and reliable approximation of solutions to parametrized partial differential equations (µpdes) M. Grepl (RWTH Aachen) 4

11 Introduction The Reduced Basis Method is a model order reduction technique that provides rapid and reliable approximation of solutions to parametrized partial differential equations (µpdes) for design and optimization, control, parameter estimation,... M. Grepl (RWTH Aachen) 4

12 Introduction The Reduced Basis Method is a model order reduction technique that provides rapid and reliable approximation of solutions to parametrized partial differential equations (µpdes) for design and optimization, control, parameter estimation,... i.e., for parameter space exploration. M. Grepl (RWTH Aachen) 4

13 The Reduced Basis Method FE SPACE a(u(µ), v; µ) = f(v; µ), for all v X M. Grepl (RWTH Aachen) 5

14 The Reduced Basis Method FE SPACE a(u(µ), v; µ) = f(v; µ), for all v X M. Grepl (RWTH Aachen) 5

15 The Reduced Basis Method FE SPACE a(u(µ), v; µ) = f(v; µ), for all v X M. Grepl (RWTH Aachen) 5

16 The Reduced Basis Method HIGH-DIMENSIONAL FE SPACE a(u(µ), v; µ) = f(v; µ), for all v X M. Grepl (RWTH Aachen) 5

17 The Reduced Basis Method SNAPSHOTS u(µ i ) HIGH-DIMENSIONAL FE SPACE X N = span { u(µ i ), i = 1,..., N } M. Grepl (RWTH Aachen) 5

18 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) HIGH-DIMENSIONAL FE SPACE a(u(µ), v; µ) = f(v; µ), for all v X M. Grepl (RWTH Aachen) 5

19 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) u N (µ) (1) APPROXIMATION HIGH-DIMENSIONAL FE SPACE a(u N (µ), v; µ) = f(v; µ), for all v X N. M. Grepl (RWTH Aachen) 5

20 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) u N (µ) (1) APPROXIMATION (2) ERROR BOUND N (µ) HIGH-DIMENSIONAL FE SPACE u(µ) u N (µ) X u N (µ) := r N( ; µ) X α LB (µ) M. Grepl (RWTH Aachen) 5

21 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) u N (µ) (1) APPROXIMATION (2) ERROR BOUND N (µ) HIGH-DIMENSIONAL FE SPACE (3) OFFLINE-ONLINE COMPUTATIONAL DECOMPOSITION a(w, v; µ) = Q q=1 θ q (µ) }{{} a q (w, v), }{{} w, v X µ-dependent µ-independent COEFFICIENTS BILINEAR FORMS M. Grepl (RWTH Aachen) 5

22 The Reduced Basis Method SNAPSHOTS u(µ i ) EXACT SOLUTION u(µ) u N (µ) (1) APPROXIMATION (2) ERROR BOUND N (µ) HIGH-DIMENSIONAL FE SPACE (3) OFFLINE-ONLINE COMPUTATIONAL DECOMPOSITION (4) GREEDY ALGORITHM µ n+1 = arg max µ D s u n (µ) u N (µ) X M. Grepl (RWTH Aachen) 5

23 Problem Statement We wish to compute, for any parameter µ D, s(µ) = l(u(µ); µ) OUTPUT where u(µ) X satisfies and a(u, v; µ) = f(v; µ), for all v X, FEM N (µ) f(v; µ), l(v; µ) a(, ; µ) : X X R are bounded linear functionals is continuous and coercive for all µ D. M. Grepl (RWTH Aachen) 6

24 (1) Approximation We let X N = span { u(µ i ) }{{}, i = 1,..., N } SNAPSHOTS and compute our approximation as s N (µ) = l(u N (µ); µ) where u N (µ) X N satisfies a(u N, v; µ) = f(v; µ), for all v X N. ROM N (µ) M. Grepl (RWTH Aachen) 7

25 (1) Approximation (Algebraic Formulation) We let X N = [ ζ 1 }{{} ζ 2 }{{} ]... ζ }{{} N X -ORTHONORMAL SNAPSHOTS ζ i = u(µ i ) s.t. u N (µ) = X N u N (µ) M. Grepl (RWTH Aachen) 8

26 (1) Approximation (Algebraic Formulation) We let X N = [ ζ 1 }{{} ζ 2 }{{} ]... ζ }{{} N X -ORTHONORMAL SNAPSHOTS ζ i = u(µ i ) and compute our approximation as where u N (µ) R N satisfies s N (µ) = L(µ) T X N }{{} u N (µ) = L T N (µ) u N(µ) s.t. u N (µ) = X N u N (µ) X T A(µ) X N N }{{} u N (µ) = X T F (µ) N }{{} A N (µ) u N (µ) = F N (µ) M. Grepl (RWTH Aachen) 8

27 (1) Approximation (Algebraic Formulation) We let X N = [ ζ 1 }{{} ζ 2 }{{} ]... ζ }{{} N X -ORTHONORMAL SNAPSHOTS ζ i = u(µ i ) and compute our approximation as where u N (µ) R N satisfies s N (µ) = L(µ) T X N }{{} u N (µ) = L T N (µ) u N(µ) s.t. u N (µ) = X N u N (µ) X T A(µ) X N N }{{} u N (µ) = X T F (µ) N }{{} A N (µ) u N (µ) = F N (µ) How do we quantify the error? M. Grepl (RWTH Aachen) 8

28 (2) Error Estimation Let α(µ) be the coercivity constant of a a(v, v; µ) α(µ) := inf v X v 2 X and define residual r N (v; µ) := f(v; µ) a(u N, v; µ), v X. M. Grepl (RWTH Aachen) 9

29 (2) Error Estimation Let α(µ) be the coercivity constant of a a(v, v; µ) α(µ) := inf v X v 2 X and define residual r N (v; µ) := f(v; µ) a(u N, v; µ), v X. The error, e N (µ) := u(µ) u N (µ), then satisfies a(e N, v; µ) = r N (v; µ), v X, M. Grepl (RWTH Aachen) 9

30 (2) Error Estimation Let α(µ) be the coercivity constant of a a(v, v; µ) α(µ) := inf v X v 2 X and define residual r N (v; µ) := f(v; µ) a(u N, v; µ), v X. The error, e N (µ) := u(µ) u N (µ), then satisfies a(e N, v; µ) = r N (v; µ), v X, and, for any µ D and α LB (µ) α(µ), we have e N (µ) X r N( ; µ) X α LB (µ) }{{} =: u N (µ) and s s N l X u N }{{} =: l N (µ) M. Grepl (RWTH Aachen) 9

31 (2) Error Estimation Let α(µ) be the coercivity constant of a a(v, v; µ) α(µ) := inf v X v 2 X and define residual r N (v; µ) := f(v; µ) a(u N, v; µ), v X. The error, e N (µ) := u(µ) u N (µ), then satisfies a(e N, v; µ) = r N (v; µ), v X, and, for any µ D and α LB (µ) α(µ), we have e N (µ) X r N( ; µ) X α LB (µ) }{{} =: u N (µ) and s s N l X u N }{{} =: l N (µ) How do we compute u N, s N, u N, l N efficiently? M. Grepl (RWTH Aachen) 9

32 (3) Offline/Online Decomposition We assume a(v, w; µ) = Q so that A N (µ) }{{} RB-MATRIX q=1 θ q (µ) }{{} a q (v, w) }{{} µ-dependent µ-independent COEFFICIENTS BILINEAR FORMS = X T A(µ) X N N = Q θ q (µ) X T N Aq X N }{{} q=1 }{{} FE-MATRIX µ-independent M. Grepl (RWTH Aachen) 10

33 (3) Offline/Online Decomposition Example Affine Parameter Dependence: Example 1 Parameters: µ = ( k 0, k 1, k 2, k 3, k 4 ) }{{} THERMAL CONDUCTIVITIES Governing Equation: k i 2 y i = 0 in Ω i, i = 0 to 4 M. Grepl (RWTH Aachen) 11

34 (3) Offline/Online Decomposition Example Affine Parameter Dependence: Example 1 Parameters: µ = ( k 0, k 1, k 2, k 3, k 4 ) }{{} THERMAL CONDUCTIVITIES Governing Equation: k i 2 y i = 0 The matrix A(µ) then represents the bilinear form a(v, w; µ) = 4 k i v w i=0 Ω }{{}} i {{} = A q θ q (µ) in Ω i, i = 0 to 4 M. Grepl (RWTH Aachen) 11

35 (3) Offline/Online Decomposition Example Affine Parameter Dependence: Example 2 1 x 2 Ω(µ) µ x 1 PARAMETRIZED DOMAIN Bilinear Form on Parameter-Dependent Domain ( ṽ w ã(ṽ, w; µ) = + ṽ ) w d Ω(µ), Ω(µ) x 1 x 1 x 2 x }{{ 2 } ṽ w ṽ, w H 1 ( Ω(µ)) M. Grepl (RWTH Aachen) 12

36 (3) Offline/Online Decomposition Example Affine Parameter Dependence: Example x 2 Ω(µ) µ x 2 Ω 1 x 1 x 1 PARAMETRIZED DOMAIN REFERENCE DOMAIN Bilinear Form on Parameter-Dependent Domain ( ṽ w ã(ṽ, w; µ) = + ṽ ) w d Ω(µ), Ω(µ) x 1 x 1 x 2 x }{{ 2 } ṽ w After (Affine) Mapping to a Reference Domain ( v w a(v, w; µ) = + 1 ) v w µdω, x 1 x 1 µ 2 x 2 x 2 Ω ṽ, w H 1 ( Ω(µ)) v, w H 1 (Ω) M. Grepl (RWTH Aachen) 12

37 (3) Offline/Online Decomposition We assume a(v, w; µ) = Q so that A N (µ) }{{} RB-MATRIX q=1 θ q (µ) }{{} a q (v, w) }{{} µ-dependent µ-independent COEFFICIENTS BILINEAR FORMS = X T A(µ) X N N = Q θ q (µ) X T N Aq X N }{{} q=1 }{{} FE-MATRIX µ-independent Since all required quantities can be decomposed in this manner, we can OFFLINE: ONLINE: Form and store µ-independent quantities at cost O(N ) For any µ D, compute approx and error bounds at cost O(QN 2 + N 3 ) and O(Q 2 N 2 ) M. Grepl (RWTH Aachen) 13

38 (3) Offline/Online Decomposition We assume a(v, w; µ) = Q so that A N (µ) }{{} RB-MATRIX q=1 θ q (µ) }{{} a q (v, w) }{{} µ-dependent µ-independent COEFFICIENTS BILINEAR FORMS = X T A(µ) X N N = Q θ q (µ) X T N Aq X N }{{} q=1 }{{} FE-MATRIX µ-independent Since all required quantities can be decomposed in this manner, we can OFFLINE: ONLINE: Form and store µ-independent quantities at cost O(N ) For any µ D, compute approx and error bounds at cost O(QN 2 + N 3 ) and O(Q 2 N 2 ) How do we choose the snapshots? M. Grepl (RWTH Aachen) 13

39 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? M. Grepl (RWTH Aachen) 14

40 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 1 µ 2 M. Grepl (RWTH Aachen) 14

41 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 1 µ 2 M. Grepl (RWTH Aachen) 14

42 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 3 = arg max µ D s 2 (µ) u 2 (µ) X where D s D is a finite train sample µ 1 µ 3 µ 2 M. Grepl (RWTH Aachen) 14

43 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 3 = arg max µ D s 2 (µ) u 2 (µ) X where D s D is a finite train sample µ 1 µ 3 µ 2 X 3 = span{u(µ 1 ) u(µ 2 ) u(µ 3 )} M. Grepl (RWTH Aachen) 14

44 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 3 = arg max µ D s 2 (µ) u 2 (µ) X where D s D is a finite train sample µ 1 µ 3 µ 2 X 3 = span{u(µ 1 ) u(µ 2 ) u(µ 3 )} M. Grepl (RWTH Aachen) 14

45 (4) (Weak) Greedy Algorithm Given X 2 = span{u(µ 1 ), u(µ 2 )}, how do we choose µ 3? n u n X µ 3 = arg max µ D s 2 (µ) u 2 (µ) X where D s D is a finite train sample µ 1 µ 3 µ 2 X 3 = span{u(µ 1 ) u(µ 2 ) u(µ 3 )} Key points: n (µ) is sharp and inexpensive to compute (online) Error bounds enable choice of good approximation spaces M. Grepl (RWTH Aachen) 14

46 Summary The reduced basis method provides accurate u N u APPROX reliable N u u N X ERR EST efficient surrogates cost O(N ) DECOMP N small to solutions of parametrized PDEs for the many-query, real-time, GREEDY and slim-computing contexts. M. Grepl (RWTH Aachen) 15

47 Extensions & References Review Articles: [RHP08], [QRM11],... Book: [HRS15] Reduced Basis Methods for: elliptic coercive [PRV + 02], [PRVP02],... elliptic noncoercive [VPRP03], [MPR02],... saddle point (Stokes) [Rov03], [RV07], [GV12],... parabolic [GP05], [RMM06], [HO08], [UP14],... hyperbolic (transport-dominated) [PR07], [HKP11], [DPW14],... Disclaimer: List contains to the best of my knowledge only the first papers on these topics and is thus not meant to be exhaustive References to nonaffine and nonlinear problems to follow... M. Grepl (RWTH Aachen) 16

48 Extensions & References References by topic for: error bounds (adjoint techniques) [PRV + 02], [GP05],... stability factor lower bounds [VRP02], [HRSP07], [HKC + 10], [CHMR09], a priori convergence [MPT02],... greedy algorithm original papger [VPRP03] convergence analysis [BMP + 12], [BCD + 11], [DPW12], [Haa13],... variations (hprb,...) [EPR10], [BTWGvBW07], [BTWG08],... geometry parametrization [RHP08], [LR10],... multiscale problems [Ngu08] stochastic problems [BBM + 09], [BBL + 10]... Disclaimer: see previous slide... M. Grepl (RWTH Aachen) 17

49 Part II Nonlinear (and Nonaffine) Problems M. Grepl (RWTH Aachen) 18

50 Nonlinear Problems Distinguish between problems involving quadratic nonlinearities higher-order or nonpolynomial nonlinearities M. Grepl (RWTH Aachen) 19

51 Nonlinear Problems Distinguish between problems involving quadratic nonlinearities (Petrov-)Galerkin projection A posteriori error bounds: Brezzi-Rappaz-Raviart [BRR80] Offline/online: dominant online cost O(N 4 ) Ref.: [VPP03], [VP05], [Dep08], [DL12], [RG12], [MPU14], [Yan14],... higher-order or nonpolynomial nonlinearities M. Grepl (RWTH Aachen) 19

52 Nonlinear Problems Distinguish between problems involving quadratic nonlinearities (Petrov-)Galerkin projection A posteriori error bounds: Brezzi-Rappaz-Raviart [BRR80] Offline/online: dominant online cost O(N 4 ) Ref.: [VPP03], [VP05], [Dep08], [DL12], [RG12], [MPU14], [Yan14],... higher-order or nonpolynomial nonlinearities Additional approximation of nonlinearity (EIM) A posteriori error estimation (in general no bounds) Offline-online decomposition possible References: to follow... M. Grepl (RWTH Aachen) 19

53 Problem Statement We wish to compute, for any parameter µ D, s(µ) = l(u(µ); µ) where u(µ) X satisfies a(u, v; µ) + g nl (u; x; µ) v = f(v), v X, and Ω f(v; µ), l(v; µ) are bounded linear functionals a(, ; µ) : X X R is continuous and coercive and the nonlinearity satisfies g nl : R Ω D R continuous; g nl (u 1 ; x; µ) g nl (u 2 ; x; µ), u 1 u 2 ; u g nl (u; x; µ) 0, u R, for any x Ω, µ D. M. Grepl (RWTH Aachen) 20

54 RB-Galerkin Approximation We let X N = span { u(µ i ) }{{}, i = 1,..., n } SNAPSHOTS and compute our approximation as where u N (µ) X N a(u N, v; µ) + s N (µ) = l(u N (µ); µ) satisfies Ω g nl (u N ; x; µ) v = f(v), v X N. M. Grepl (RWTH Aachen) 21

55 RB-Galerkin Approximation We let X N = span { u(µ i ) }{{}, i = 1,..., n } SNAPSHOTS and compute our approximation as where u N (µ) X N a(u N, v; µ) + s N (µ) = l(u N (µ); µ) satisfies Ω g nl (u N ; x; µ) v = f(v), v X N. Can we still compute u N efficiently? M. Grepl (RWTH Aachen) 21

56 RB-Galerkin Approximation Sample Computation: N We expand u N (µ) = u N j (µ) ζ j, j=1 and obtain (v = ζ i, 1 i N ) ( N ) g nl (u N (µ); x; µ) ζ i = g nl u N j (µ) ζ j ; x; µ Ω Ω j=1 ζ i M. Grepl (RWTH Aachen) 22

57 RB-Galerkin Approximation Sample Computation: N We expand u N (µ) = u N j (µ) ζ j, j=1 and obtain (v = ζ i, 1 i N ) ( N ) g nl (u N (µ); x; µ) ζ i = g nl u N j (µ) ζ j ; x; µ Ω Ω j=1 Issues Online assembly requires N -dependent cost ζ i M. Grepl (RWTH Aachen) 22

58 RB-Galerkin Approximation Sample Computation: N We expand u N (µ) = u N j (µ) ζ j, j=1 and obtain (v = ζ i, 1 i N ) ( N ) g nl (u N (µ); x; µ) ζ i = g nl u N j (µ) ζ j ; x; µ Issues Ω Ω j=1 Online assembly requires N -dependent cost Similar to nonaffine problems: assume g na is nonaffine, then b(ζ i ; µ) = g na x µ ( (x; µ) ζ i e.g. e σ )2 Ω ζ i requires N -dependent cost. M. Grepl (RWTH Aachen) 22

59 Empirical Interpolation Method Main Idea: g na (x; µ) g na (x; µ) = M M Then: b(ζ i ; µ) = Ω = M m=1 m=1 g na (x; µ) ζ i ϕ M m (µ) }{{} EIM Ω ϕ M m (µ) q m (x) ζ i, Ω q m (x) }{{} Collateral RB g na M (x; µ) ζ i M. Grepl (RWTH Aachen) 23

60 Empirical Interpolation Method Main Idea: g na (x; µ) g na (x; µ) = M M Then: b(ζ i ; µ) = Ω = M m=1 m=1 g na (x; µ) ζ i ϕ M m (µ) }{{} EIM Ω ϕ M m (µ) q m (x) ζ i, Ω q m (x) }{{} Collateral RB g na M (x; µ) ζ i Issues How do we choose X g M = span{q m (x), m = 1,..., M}? M. Grepl (RWTH Aachen) 23

61 Empirical Interpolation Method Main Idea: g na (x; µ) g na (x; µ) = M M Then: b(ζ i ; µ) = Issues Ω = M m=1 How do we choose X g M m=1 g na (x; µ) ζ i ϕ M m (µ) }{{} EIM Ω ϕ M m (µ) q m (x) ζ i, Ω q m (x) }{{} Collateral RB g na M (x; µ) ζ i = span{q m (x), m = 1,..., M}? Can we compute the coefficients ϕ M m (µ) efficiently? M. Grepl (RWTH Aachen) 23

62 Empirical Interpolation Method Main Idea: g na (x; µ) g na (x; µ) = M M Then: b(ζ i ; µ) = Issues Ω = M m=1 How do we choose X g M m=1 g na (x; µ) ζ i ϕ M m (µ) }{{} EIM Ω ϕ M m (µ) q m (x) ζ i, Ω q m (x) }{{} Collateral RB g na M (x; µ) ζ i = span{q m (x), m = 1,..., M}? Can we compute the coefficients ϕ M m (µ) efficiently? How do we quantify the (interpolation) error introduced? M. Grepl (RWTH Aachen) 23

63 Empirical Interpolation Method Greedy Procedure Initialize: Choose µ g 1 D and set x T 1 = arg max x Ω g(x; µg 1 ), q 1 = g(x; µg 1 ) g(x T 1 ; µg 1 ). [MNPP07] M. Grepl (RWTH Aachen) 24

64 Empirical Interpolation Method Greedy Procedure Initialize: Choose µ g 1 D and set x T 1 For 1 M M max 1: = arg max x Ω g(x; µg 1 ), q 1 = g(x; µg 1 ) g(x T 1 ; µg 1 ). [MNPP07] M. Grepl (RWTH Aachen) 24

65 Empirical Interpolation Method Greedy Procedure Initialize: Choose µ g 1 D and set x T 1 For 1 M M max 1: Sample Set = arg max x Ω g(x; µg 1 ), q 1 = g(x; µg 1 ) g(x T 1 ; µg 1 ). µ g M +1 arg max g( ; µ) g µ Ξ g M ( ; µ) L (Ω) train [MNPP07] M. Grepl (RWTH Aachen) 24

66 Empirical Interpolation Method Greedy Procedure Initialize: Choose µ g 1 D and set x T 1 For 1 M M max 1: Sample Set = arg max x Ω g(x; µg 1 ), q 1 = g(x; µg 1 ) g(x T 1 ; µg 1 ). µ g M +1 arg max g( ; µ) g µ Ξ g M ( ; µ) L (Ω) train Interpolation Points x T = arg max r M +1 M +1(x), x Ω where r M +1 (x) = g(x; µ g M +1) g M (x; µ g M +1) [MNPP07] M. Grepl (RWTH Aachen) 24

67 Empirical Interpolation Method Greedy Procedure Initialize: Choose µ g 1 D and set x T 1 For 1 M M max 1: Sample Set = arg max x Ω g(x; µg 1 ), q 1 = g(x; µg 1 ) g(x T 1 ; µg 1 ). µ g M +1 arg max g( ; µ) g µ Ξ g M ( ; µ) L (Ω) train Interpolation Points x T = arg max r M +1 M +1(x), x Ω where r M +1 (x) = g(x; µ g M +1) g M (x; µ g M +1) EIM Space q M +1 (x) = r M +1(x) r M +1 (x T M +1 ) [MNPP07] M. Grepl (RWTH Aachen) 24

68 Empirical Interpolation Method Interpolation Points, Samples, and Spaces: T g M = {x T 1 Ω,..., xt M Ω}, S g M {µ g 1 D,..., µg M D}, and associated X g M = span{q 1,..., q M }, 1 M M max, chosen by greedy procedure. M. Grepl (RWTH Aachen) 25

69 Empirical Interpolation Method Interpolation Points, Samples, and Spaces: T g M = {x T 1 Ω,..., xt M Ω}, S g M {µ g 1 D,..., µg M D}, and associated X g M = span{q 1,..., q M }, 1 M M max, chosen by greedy procedure. Approximation: for given µ D where M m=1 g(x; µ) g M (x; µ) = M m=1 ϕ M m (µ) q m (x), q m (x T n ) ϕ M m(µ) = g(x T n ; µ), 1 n M. Note: Bnm M = q m(x T n ) lower triangular and invertible. M. Grepl (RWTH Aachen) 25

70 Empirical Interpolation Method Interpolation Points, Samples, and Spaces: T g M = {x T 1 Ω,..., xt M Ω}, S g M {µ g 1 D,..., µg M D}, and associated X g M = span{q 1,..., q M }, 1 M M max, chosen by greedy procedure. Approximation: for given µ D where M m=1 g(x; µ) g M (x; µ) = M m=1 ϕ M m (µ) q m (x), q m (x T n ) ϕ M m(µ) = g(x T n ; µ), 1 n M. Note: Bnm M = q m(x T n ) lower triangular and invertible. How do we quantify the (interpolation) error? M. Grepl (RWTH Aachen) 25

71 Empirical Interpolation Method Next Point Estimator Given an approximation g M (x; µ) for M M max 1, we define ˆε M (µ) g(x T ; µ) g M +1 M(x T ; µ) M +1 and, if g( ; µ) X g M +1, we have g( ; µ) g M ( ; µ) L (Ω) ˆε M (µ). M. Grepl (RWTH Aachen) 26

72 Empirical Interpolation Method Next Point Estimator Given an approximation g M (x; µ) for M M max 1, we define ˆε M (µ) g(x T ; µ) g M +1 M(x T ; µ) M +1 and, if g( ; µ) X g M +1, we have g( ; µ) g M ( ; µ) L (Ω) ˆε M (µ). Note The estimator ˆε M (µ) is very cheap to evaluate M. Grepl (RWTH Aachen) 26

73 Empirical Interpolation Method Next Point Estimator Given an approximation g M (x; µ) for M M max 1, we define ˆε M (µ) g(x T M +1 ; µ) g M(x T M +1 ; µ) and, if g( ; µ) X g M +1, we have Note g( ; µ) g M ( ; µ) L (Ω) ˆε M (µ). The estimator ˆε M (µ) is very cheap to evaluate In general g( ; µ) X g M +1, and hence our estimator ˆε M (µ) is indeed a lower bound M. Grepl (RWTH Aachen) 26

74 Empirical Interpolation Method Next Point Estimator Given an approximation g M (x; µ) for M M max 1, we define ˆε M (µ) g(x T M +1 ; µ) g M(x T M +1 ; µ) and, if g( ; µ) X g M +1, we have Note g( ; µ) g M ( ; µ) L (Ω) ˆε M (µ). The estimator ˆε M (µ) is very cheap to evaluate In general g( ; µ) X g M +1, and hence our estimator ˆε M (µ) is indeed a lower bound An a posteriori error bound exists [EGP10], but does not extend to the nonlinear case M. Grepl (RWTH Aachen) 26

75 EIM-RB-Galerkin Approximation Given µ D, we evaluate s N,M (µ) = l(u N,M (µ); µ) where u N,M (µ) X N satisfies a(u N,M, v; µ) + g nl,u N,M M (x; µ) v = f(v), v X N. and with M j=1 Ω g nl,u N,M M (x; µ) M m=1 ϕ M m (µ)q m (x), Bij M ϕ M j(µ) = g nl (u N,M (x T i ; µ); xt i ; µ), 1 i M. M. Grepl (RWTH Aachen) 27

76 EIM-RB-Galerkin Approximation Given µ D, we evaluate s N,M (µ) = l(u N,M (µ); µ) where u N,M (µ) X N satisfies a(u N,M, v; µ) + g nl,u N,M M (x; µ) v = f(v), v X N. and with M j=1 Ω g nl,u N,M M (x; µ) M m=1 ϕ M m (µ)q m (x), Bij M ϕ M j(µ) = g nl (u N,M (x T i ; µ); xt i ; µ), 1 i M. Admits offline/online treatment: online cost (per Newton iteration) O(M 2 + MN 2 + N 3 ) M. Grepl (RWTH Aachen) 27

77 EIM-RB-Galerkin Approximation Given µ D, we evaluate s N,M (µ) = l(u N,M (µ); µ) where u N,M (µ) X N satisfies a(u N,M, v; µ) + g nl,u N,M M (x; µ) v = f(v), v X N. and with M j=1 Ω g nl,u N,M M (x; µ) M m=1 ϕ M m (µ)q m (x), Bij M ϕ M j(µ) = g nl (u N,M (x T i ; µ); xt i ; µ), 1 i M. Admits offline/online treatment: online cost (per Newton iteration) O(M 2 + MN 2 + N 3 ) How do we quantify the (RB+EIM) error? M. Grepl (RWTH Aachen) 27

78 Error Estimation Define the dual norm ϑ q M = sup v X Ω q M+1 v v X and the residual, for all v X, r N,M (v; µ) := f(v) a(u N,M, v; µ) Ω g nl,u N,M M (x; µ) v. M. Grepl (RWTH Aachen) 28

79 Error Estimation Define the dual norm ϑ q M = sup v X Ω q M+1 v v X and the residual, for all v X, r N,M (v; µ) := f(v) a(u N,M, v; µ) Ω g nl,u N,M M (x; µ) v. If g nl (u N,M ; x; µ) X g M +1, the error, e N,M := u u N,M, is bounded by 1 ( e N,M (µ) X rn,m ( ; µ) X + ˆε M (µ)ϑ q ) M. α LB (µ) M. Grepl (RWTH Aachen) 28

80 Error Estimation Define the dual norm ϑ q M = sup v X Ω q M+1 v v X and the residual, for all v X, r N,M (v; µ) := f(v) a(u N,M, v; µ) Ω g nl,u N,M M (x; µ) v. If g nl (u N,M ; x; µ) X g M +1, the error, e N,M := u u N,M, is bounded by 1 ( e N,M (µ) X rn,m ( ; µ) X + ˆε M (µ)ϑ q ) M. α LB (µ) Note Admits offline/online treatment: online cost O(M 2 N 2 ) M. Grepl (RWTH Aachen) 28

81 Error Estimation Define the dual norm ϑ q M = sup v X Ω q M+1 v v X and the residual, for all v X, r N,M (v; µ) := f(v) a(u N,M, v; µ) Ω g nl,u N,M M (x; µ) v. If g nl (u N,M ; x; µ) X g M +1, the error, e N,M := u u N,M, is bounded by 1 ( e N,M (µ) X rn,m ( ; µ) X + ˆε M (µ)ϑ q ) M. α LB (µ) Note Admits offline/online treatment: online cost O(M 2 N 2 ) In general g nl (u N,M ; x; µ) X g M +1, and hence the bound is more like an estimator M. Grepl (RWTH Aachen) 28

82 Nonlinear Parabolic Problems We wish to compute, for any µ D, s(t; µ) = l(u(t; µ)) where u(t; µ) Y, satisfies u(0; µ) = 0 m( u(t; µ), v; µ) + a(u(t; µ), v; µ) + g nl (u(t; µ); x; µ) v = b(v)u(t), v Y. Key Ingredients: Ω M. Grepl (RWTH Aachen) 29

83 Nonlinear Parabolic Problems We wish to compute, for any µ D, s(t; µ) = l(u(t; µ)) where u(t; µ) Y, satisfies u(0; µ) = 0 m( u(t; µ), v; µ) + a(u(t; µ), v; µ) + g nl (u(t; µ); x; µ) v = b(v)u(t), v Y. Key Ingredients: FE[x]-FD[t] truth approximation Ω M. Grepl (RWTH Aachen) 29

84 Nonlinear Parabolic Problems We wish to compute, for any µ D, s(t; µ) = l(u(t; µ)) where u(t; µ) Y, satisfies u(0; µ) = 0 m( u(t; µ), v; µ) + a(u(t; µ), v; µ) + g nl (u(t; µ); x; µ) v = b(v)u(t), v Y. Ω Key Ingredients: FE[x]-FD[t] truth approximation POD/Greedy algorithm to construct X N and X g M M. Grepl (RWTH Aachen) 29

85 Nonlinear Parabolic Problems We wish to compute, for any µ D, s(t; µ) = l(u(t; µ)) where u(t; µ) Y, satisfies u(0; µ) = 0 m( u(t; µ), v; µ) + a(u(t; µ), v; µ) + g nl (u(t; µ); x; µ) v = b(v)u(t), v Y. Key Ingredients: FE[x]-FD[t] truth approximation Ω POD/Greedy algorithm to construct X N and X g M Error bound/estimator for space-time energy norm M. Grepl (RWTH Aachen) 29

86 Nonlinear Parabolic Problems We wish to compute, for any µ D, s(t; µ) = l(u(t; µ)) where u(t; µ) Y, satisfies u(0; µ) = 0 m( u(t; µ), v; µ) + a(u(t; µ), v; µ) + g nl (u(t; µ); x; µ) v = b(v)u(t), v Y. Key Ingredients: FE[x]-FD[t] truth approximation Ω POD/Greedy algorithm to construct X N and X g M Error bound/estimator for space-time energy norm Online cost: O(M 2 + MN 2 + N 3 ) plus O(KM 2 N 2 ). Cost per Newton iteration per timestep. M. Grepl (RWTH Aachen) 29

87 Extensions & References Empirical Interpolation Method: methodology [BMNP04], [GMNP07], [MNPP07],... error bounds [EGP10], [EGPR13],... generalized EIM [MM13], [MMPY15],... variations [CS10],... Related Methods: best points interpolation [NPP08],... adaptive cross approximation [Beb00], [Beb11],... gappy POD [ES95], [BTDW04], [Wil06], [GFWG10],... missing point estimation [AWWB08],... For a comparison of these methods: [BMS14] M. Grepl (RWTH Aachen) 30

88 Extensions & References RB for nonaffine problems: elliptic [BMNP04], [GMNP07], [Ngu07], [Roz09],... parabolic [GMNP07], [Gre12a], [KGV12],... RB for nonlinear problems: elliptic [GMNP07], [NP08], [CTU09],... parabolic [GMNP07], [Gre12a], [DHO12], [Gre12b],... Issues Convergence and stability of EIM-RB approximation Greedy EIM algorithm requires (truth) solution at all µ D s Simultaneous EIM + RB [DP15] M. Grepl (RWTH Aachen) 31

89 Model Problem Given µ = (µ 1, µ 2 ) D [0.01, 10] 2, evaluate Ω =]0, 1[ 2 s k (µ) = u k (µ) where u k (µ) X, 1 k K, satisfies u 0 (µ) = 0 1 t m(uk (µ) u k 1 (µ), v) + a(u k (µ), v) + g nl (u k (µ); x; µ) v = b(v) sin(2πt k ), v X, Ω with g nl (u k (µ); x; µ) = µ 1 e µ 2 u k (µ) 1 µ 2. Truth Approximation: Space: X X e H0 1 (Ω) with dimension N = 2601; Time: Ī = (0, 2], t = 0.01, and thus K = 200. M. Grepl (RWTH Aachen) 32 Ω

90 Sample Results Truth solution u(t k ; µ) at time t k = 25 t and µ = (0.01, 0.01) µ = (10, 10) Solution for µ = (0.01,0.01), t k = 25 t Solution for µ = (10,10), t k = 25 t y k (µ) x (2) 0 0 x (1) y k (µ) x (2) 0 0 x (1) b(v) = 100 Ω v sin(2πx 1) sin(2πx 2 ) M. Grepl (RWTH Aachen) 33

91 Convergence: Energy Norm ε N,M,max,rel y M = 10 M = M = 40 M = M = 80 M = N N,M,max,rel y M = 20 M = M = 70 M = M = 130 M = N Results for sample Ξ test D of size 225. Plateau in curves for M fixed. Knees reflect balanced contribution of both error terms. Sharp bounds require conservative choice of M. M. Grepl (RWTH Aachen) 34

92 Convergence: Energy Norm & Output N M ɛ y N,M,max,rel y N,M,max,rel η y N,M E E E E E E E E E E N M ɛ s N,M,max,rel s N,M,max,rel η N,M s E E E E E E E E E E Results for sample Ξ test D of size 225. M. Grepl (RWTH Aachen) 35

93 Online Computational Times N M s N,M (µ, t k ) s N,M (µ, tk ) s(µ, t k ) E E E E E E E E E E 04 1 Average CPU times for sample Ξ test D of size 225. Computational savings O(10 3 ) for s N,M,max,rel < 1%. But offline stage much more expensive than for linear case. M. Grepl (RWTH Aachen) 36

94 Nonlinear Reaction-Diffusion Systems General formulation: y(x, t; µ) = (D(µ) y(x, t; µ)) + f(y(x, t; µ); µ) t Self-ignition of a coal stockpile [Gre12b] T (x,t) t = 2 T (x, t) + β Φ 2 (c(x, t) + 1) e γ/(t (x,t)+1), c(x,t) t = Le 2 c(x, t) Φ 2 (c(x, t) + 1) e γ/(t (x,t)+1), where γ : Arrhenius number, β : Prater temperature, Le : Lewis number, Φ : Thiele modulus. 1 s 1 s 2 T (x, t) c(x, t) x Nonlinearity not monotonic: a posterior error bounds not valid. M. Grepl (RWTH Aachen) 37

95 Nonlinear Reaction-Diffusion Systems Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 Output Temperature and Concentration: γ = 12.0 Outputs, µ = 12 s 1 (t;µ): Temperature s 2 (t;µ): Concentration Time t Concentration s 2 (t;µ) Phase Plot: µ = Temperature s 1 (t;µ) M. Grepl (RWTH Aachen) 38

96 Nonlinear Reaction-Diffusion Systems Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 Output Temperature and Concentration: γ = 12.5 Outputs, µ = 12.5 s 1 (t;µ): Temperature s 2 (t;µ): Concentration Time t Concentration s 2 (t;µ) Phase Plot: µ = Temperature s 1 (t;µ) M. Grepl (RWTH Aachen) 38

97 Nonlinear Reaction-Diffusion Systems Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 Output Temperature and Concenctration: γ = Outputs, µ = s 1 (t;µ): Temperature s 2 (t;µ): Concentration Time t Concentration s 2 (t;µ) Phase Plot: µ = Temperature s 1 (t;µ) M. Grepl (RWTH Aachen) 38

98 Nonlinear Reaction-Diffusion Systems Rel. Error in s 1 (t;µ) Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 N T = N c = 16, M = 36: t RB /t FEM = 5.12 E 03 Output s 1 (t;µ) FEM solution and RB approximation: γ = 12.6 Temperature, µ = 12.6 s 1 (t;µ) s 1,N (t;µ) Time t Time t Output s 2 (t;µ) Rel. Error in s 2 (t;µ) Concentration, µ = s 2 (t;µ) 0.6 s (t;µ) 2,N Time t Time t M. Grepl (RWTH Aachen) 38

99 Nonlinear Reaction-Diffusion Systems Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 N T = N c = 16, M = 36: t RB /t FEM = 5.12 E 03 FEM, RB, and output interpolation: γ = 12.5 Rel. Error in s 1 (t;µ) Output s 1 (t;µ) Temperature, µ = Time t s 1 (t;µ) s 1,N (t;µ) s 1,int (t;µ) 10 6 RB Int. Time t Output s 2 (t;µ) Rel. Error in s 2 (t;µ) Concentration, µ = Time t s 2 (t;µ) s 2,N (t;µ) s 2,int (t;µ) RB Int. Time t M. Grepl (RWTH Aachen) 38

100 Nonlinear Reaction-Diffusion Systems Very complex dynamic behavior (depending on parameters). Here: µ γ [12, 12.6], all other parameters fixed, N = 501 N T = N c = 16, M = 36: t RB /t FEM = 5.12 E 03 FEM, RB, and output interpolation: γ = 12.5 Output s 1 (t;µ) Temperature, µ = 12.5 s 1 (t;µ) s 1,N (t;µ) s 1,int (t;µ) Time t Output s 2 (t;µ) Concentration, µ = s 2 (t;µ) 0.6 s (t;µ) 2,N 0.4 s 2,int (t;µ) Time t Rel. Error in s 1 (t;µ) RB Int Time t Rel. Error in s 2 (t;µ) RB Int Time t M. Grepl (RWTH Aachen) 38

101 References I [AWWB08] [BBL + 10] [BBM + 09] [BCD + 11] P. Astrid, S. Weiland, K. Willcox, and T. Backx. Missing point estimation in models described by proper orthogonal decomposition. IEEE Transactions on Automatic Control, 53(10): , S. Boyaval, C. Le Bris, T. Lelièvre, Y. Maday, N.C. Nguyen, and A.T. Patera. Reduced basis techniques for stochastic problems. Arch. Comput. Method. E., 17: , S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen, and A.T. Patera. A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable robin coefficient. Comput. Methods Appl. Mech. Engrg., 198(41-44): , P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal., 43(3): , [Beb00] M. Bebendorf. Approximation of boundary element matrices. Numer. Math., 86(4): , [Beb11] M. Bebendorf. Adaptive cross approximation of multivariate functions. Constr. Approx., 34(2): , [BMNP04] [BMP + 12] [BMS14] [BRR80] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera. An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math., 339(9): , A. Buffa, Y. Maday, A.T. Patera, C. Prud homme, and G. Turinici. A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Math. Model. Num., 46(03): , M. Bebendorf, Y. Maday, and B. Stamm. Comparison of some reduced representation approximations. In A. Quarteroni and G. Rozza, editors, Reduced Order Methods for Modeling and Computational Reduction, volume 9 of MS&A - Modeling, Simulation and Applications, pages Springer International Publishing, F. Brezzi, J. Rappaz, and P. A. Raviart. Finite dimensional approximation of nonlinear problems. Numer. Math., 38(1):1 30, /BF M. Grepl (RWTH Aachen) 39

102 References II [BTDW04] [BTWG08] [BTWGvBW07] [CHMR09] [CS10] [CTU09] [Dep08] [DHO12] [DL12] [DP15] T. Bui-Thanh, M. Damodaran, and K. Willcox. Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA Journal, 42(8): , T. Bui-Thanh, K. Willcox, and O. Ghattas. Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput., 30: , October T. Bui-Thanh, K. Willcox, O. Ghattas, and B. van Bloemen Waanders. Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput. Phys., 224(2): , Y. Chen, J.S. Hesthaven, Y. Maday, and J. Rodríguez. Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d maxwell s problem. ESAIM: Math. Model. Num., null: , S. Chaturantabut and D.C. Sorensen. Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput., 32(5): , C. Canuto, T. Tonn, and K. Urban. A-posteriori error analysis of the reduced basis method for non-affine parameterized nonlinear pde s. SIAM J. Numer. Anal., 47: , S. Deparis. Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM Journal of Numerical Analysis, 46(4): , M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34(2):A937 A969, S. Deparis and A.E. Løvgren. Stabilized reduced basis approximation of incompressible three-dimensional Navier-Stokes equations in parametrized deformed domains. J. Sci. Comput., 50: , /s C. Daversin and C. Prud homme. Simultaneous empirical interpolation and reduced basis method for non-linear problems. Technical report, arxiv: [math.ap], M. Grepl (RWTH Aachen) 40

103 References III [DPW12] [DPW14] [EGP10] [EGPR13] [EPR10] R. DeVore, G. Petrova, and P. Wojtaszczyk. Greedy algorithms for reduced bases in Banach spaces. Constr. Approx., W. Dahmen, C. Plesken, and G. Welper. Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM: Mathematical Modelling and Numerical Analysis, 48: , J.L. Eftang, M.A. Grepl, and A.T. Patera. A posteriori error bounds for the empirical interpolation method. C. R. Math., 348: , J.L. Eftang, M.A. Grepl, A.T. Patera, and E.M. Rønquist. Approximation of parametric derivatives by the empirical interpolation method. Foundations of Computational Mathematics, 13(5): , J.L. Eftang, A.T. Patera, and E.M. Rønquist. An hp certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput., 32: , [ES95] R. Everson and L. Sirovich. Karhunen-Loeve procedure for gappy data. JOSA A, 12(8): , [GFWG10] [GMNP07] [GP05] [Gre12a] [Gre12b] D. Galbally, K. Fidkowski, K. Willcox, and O. Ghattas. Non-linear model reduction for uncertainty quantification in large-scale inverse problems. Int. J. Numer. Meth. Engng, 81(12): , M.A. Grepl, Y. Maday, N.C. Nguyen, and A.T. Patera. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math. Model. Num., 41(3): , M.A. Grepl and A.T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Math. Model. Num., 39(1): , M.A. Grepl. Certified reduced basis methods for nonaffine linear time-varying and nonlinear parabolic partial differential equations. M3AS: Mathematical Models and Methods in Applied Sciences, 22(3):40, M.A. Grepl. Model order reduction of parametrized nonlinear reaction-diffusion systems. Computers and Chemical Engineering, 43(0):33 44, M. Grepl (RWTH Aachen) 41

104 References IV [GV12] A.-L. Gerner and K. Veroy. Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput., 34(5):A2812 A2836, [Haa13] B. Haasdonk. Convergence rates of the pod-greedy method. ESAIM: Math. Model. Num., 47: , [HKC + 10] [HKP11] [HO08] [HRS15] [HRSP07] [KGV12] [LR10] D. B. P. Huynh, D. J. Knezevic, Y. Chen, J.S. Hesthaven, and A.T. Patera. A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Engrg., 199(29-32): , D. P. B. Huynh, D. J. Knezevic, and A. T. Patera. A laplace transform certified reduced basis method; application to the heat equation and wave equation. C. R. Math., 349: , B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Math. Model. Num., 42(02): , J.S. Hesthaven, G. Rozza, and B. Stamm. Certified Reduced Basis Methods for Parametrized Problems. Springer Briefs in Mathematics. Springer, D. B. P. Huynh, G. Rozza, S. Sen, and A. T. Patera. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math., 345(8): , D. Klindworth, M.A. Grepl, and G. Vossen. Certified reduced basis methods for parametrized parabolic partial differential equations with non-affine source terms. Comput. Methods Appl. Mech. Engrg., (0): , T. Lassila and G. Rozza. Parametric free-form shape design with pde models and reduced basis method. Comput. Methods Appl. Mech. Engrg., 199(23-24): , M. Grepl (RWTH Aachen) 42

105 References V [MM13] [MMPY15] [MNPP07] [MPR02] [MPT02] [MPU14] [Ngu07] [Ngu08] [NP08] Y. Maday and O. Mula. A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In Analysis and Numerics of Partial Differential Equations, volume 4 of Springer INdAM Series, pages Springer Milan, Y. Maday, O. Mula, A.T. Patera, and M. Yano. The generalized empirical interpolation method: Stability theory on hilbert spaces with an application to the stokes equation. Computer Methods in Applied Mechanics and Engineering, 287(0): , Y. Maday, N.C. Nguyen, A.T. Patera, and G.S.H. Pau. A general, multipurpose interpolation procedure: the magic points. Communications on Pure and Applied Analysis (CPAA), 8: , Y. Maday, A. T. Patera, and D. V. Rovas. A blackbox reduced-basis output bound method for noncoercive linear problems. Studies in Mathematics and its Applications, D. Cioranescu and J. L. Lions, eds., Elsevier Science B.V., 31: , Y. Maday, A.T. Patera, and G. Turinici. Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Math., 335(3): , M.Yano, A.T. Patera, and K. Urban. A space-time hp-interpolation-based certified reduced basis method for burgers equation. Mathematical Models and Methods in Applied Sciences, 0(0):1 33, N. C. Nguyen. A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations. J. Comput. Phys., 227: , December N.C. Nguyen. A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys., 227(23): , N. C. Nguyen and J. Peraire. An efficient reduced-order modeling approach for non-linear parametrized partial differential equations. Int. J. Numer. Methods Eng., 76:27 55, M. Grepl (RWTH Aachen) 43

106 References VI [NPP08] [PR07] [PRV + 02] [PRVP02] [QRM11] [RG12] [RHP08] [RMM06] [Rov03] N. C. Nguyen, A. T. Patera, and J. Peraire. A best points interpolation method for efficient approximation of parametrized functions. Int. J. Numer. Methods Eng., 73(4): , Anthony T. Patera and Einar M. Ronquist. Reduced basis approximation and a posteriori error estimation for a boltzmann model. Comput. Methods Appl. Mech. Engrg., 196(29-30): , C. Prud homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluid. Eng., 124(1):70 80, C. Prud homme, D.V. Rovas, K. Veroy, and A.T. Patera. A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM: Math. Model. Num., 36: , A. Quarteroni, G. Rozza, and A. Manzoni. Certified reduced basis approximation for parametrized partial differential equations and applications. Journal of Mathematics in Industry, 1(3):1 49, M. Rasty and M.A. Grepl. Efficient reduced basis solution of quadratically nonlinear diffusion equations. In Proceedings of 7th Vienna Conference on Mathematical Modelling MATHMOD 2012, accepted. G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3): , D. V. Rovas, L. Machiels, and Y. Maday. Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal., 26(3): , D.V. Rovas. Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. PhD thesis, Massachusetts Institute of Technology, February M. Grepl (RWTH Aachen) 44

107 References VII [Roz09] [RV07] [UP14] [VP05] [VPP03] [VPRP03] [VRP02] [Wil06] [Yan14] G. Rozza. Reduced basis methods for stokes equations in domains with non-affine parameter dependence. Computing and Visualization in Science, 12:23 35, /s G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg., 196(7): , K. Urban and A.T. Patera. An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comp., 83: , K. Veroy and A. T. Patera. Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Methods Fluids, 47: , K. Veroy, C. Prud homme, and A.T. Patera. Reduced-basis approximation of the viscous burgers equation: rigorous a posteriori error bounds. C. R. Math., 337(9): , K. Veroy, C. Prud homme, D. V. Rovas, and A. T. Patera. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, AIAA Paper K. Veroy, D.V. Rovas, and A.T. Patera. A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: convex inverse bound conditioners. ESAIM: Contr. Optim. Ca., 8: , Special Volume: A tribute to J.L. Lions. K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Computers and Fluids, 35(2): , M. Yano. A space-time petrov galerkin certified reduced basis method: Application to the boussinesq equations. SIAM J. Sci. Comput., 36(1):A232 A266, M. Grepl (RWTH Aachen) 45

108 Thank you for your attention! For questions or comments: M. Grepl (RWTH Aachen) 46

109 Part III Extra Slides M. Grepl (RWTH Aachen) 47

110 Offline/Online Decomposition: Error Bounds Crucial ingredient: Dual norm of residual r N ( ; µ) X We expand u N (µ) = N j=1 u N j (µ) ζ j and obtain from the definition of the residual and affine dependence r N (v; µ) = f(v) a(u N (µ), v; µ) ( N ) = f(v) a u N n (µ)ζ n, v; µ n=1 = f(v) N u N n (µ) a ( ζ n, v; µ ) n=1 = f(v) N n=1 u N n (µ) Q a q=1 θ q a (µ) aq( ζ n, v ) For simplicity, we assume here that f(v) does not depend on µ. M. Grepl (RWTH Aachen) 48

111 Offline/Online Decomposition: Error Bounds Riesz representation: (ê(µ), v) X = r N (v; µ) = f(v) Q a N q=1 n=1 θ q a (µ)u N n(µ)a q( ζ n, v ), M. Grepl (RWTH Aachen) 49

112 Offline/Online Decomposition: Error Bounds Riesz representation: (ê(µ), v) X = r N (v; µ) Linear Superposition: where = f(v) ê(µ) = C + Q a N q=1 n=1 Q a N q=1 n=1 (C, v) X = f(v), v X ; (A q n, v) X = a q (ζ n, v), v X, θ q a (µ)u N n(µ)a q( ζ n, v ), θ q a (µ)u N n(µ)a q n 1 n N, 1 q Q a. M. Grepl (RWTH Aachen) 49

113 Offline/Online Decomposition: Error Bounds Thus ê(µ) 2 X = (C + Q a N q=1 n=1 θ q a (µ)u N n(µ)a q n, ) X = M. Grepl (RWTH Aachen) 50

114 Offline/Online Decomposition: Error Bounds Thus ê(µ) 2 X = (C + Q a N q=1 n=1 = (C, C) X + Q a 2(C, A q n ) X + θ q a (µ)u N n(µ)a q n, ) N q=1 n=1 Q a q =1 n =1 { θa q(µ)u N n(µ) N X } θa q (µ)u N n (µ)(aq n, Aq n ) X M. Grepl (RWTH Aachen) 50

115 Offline/Online Decomposition: Error Bounds Offline: once, parameter independent Compute C, A q n, 1 n N max, 1 q Q a, from (C, v) X = f(v), v X ; (A q n, v) X = a q (ζ n, v), v X, 1 n N, 1 q Q a. M. Grepl (RWTH Aachen) 51

116 Offline/Online Decomposition: Error Bounds Offline: once, parameter independent Compute C, A q n, 1 n N max, 1 q Q a, from (C, v) X = f(v), v X ; (A q n, v) X = a q (ζ n, v), v X, 1 n N, 1 q Q a. Form/Store (C, C) X, (C, A q n ) X, (A q n, Aq n ) X, 1 n, n N max, 1 q, q Q a. Complexity depends on N, Q a, and N. M. Grepl (RWTH Aachen) 51

117 Offline/Online Decomposition: Error Bounds Online: many times, for each new µ Evaluate ê(µ) 2 X = (C, C) X + Q a 2(C, A q n ) X + N q=1 n=1 Q a q =1 n =1 (and associated solution u N (µ)) { θa q(µ)u N n(µ) N } θa q (µ)u N n (µ)(aq n, Aq n ) X O(Q 2 a N 2 ) Complexity depends on N, Q a, but not N. M. Grepl (RWTH Aachen) 52

118 Offline/Online Decomposition: Error Bounds Summary of computational cost: OFFLINE O(Q a N max N ) + O(Q 2 a N 2 max N ) solve Poisson problems form µ-independent inner products ; ONLINE O(Q 2 a N 2 ) evaluate ê(µ) X -sum ; Online cost is independent of N. M. Grepl (RWTH Aachen) 53