REDUCED BASIS METHOD

Transcription

1 REDUCED BASIS METHOD approximation of PDE s, interpolation, a posteriori estimates Yvon Maday Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie, Paris, France, Institut Universitaire de France and Division of Applied Maths, Brown University CEMRACS SUMMER SCHOOL Luminy, July 2016

2 This talk deals with linear constructive approximation methods specifically tailored to approximate the functions in F (or in a set close to F see PBDW).

3 Small Kolomogorov manifold This talk deals with linear constructive approximation methods specifically tailored to approximate the functions in F (or in a set close to F see PBDW).

4 Small Kolomogorov manifold Let F be a compact subset of a functional Banach space X This talk deals with linear constructive approximation methods specifically tailored to approximate the functions in F (or in a set close to F see PBDW).

5 Small Kolomogorov manifold Let F be a compact subset of a functional Banach space X An example can be a set of parametric functions F = {u(., µ),µ2 D} This talk deals with linear constructive approximation methods specifically tailored to approximate the functions in F (or in a set close to F see PBDW).

6 Small Kolomogorov manifold Let F be a compact subset of a functional Banach space X An example can be a set of parametric functions F = {u(., µ),µ2 D} This talk deals with linear constructive approximation methods specifically tailored to approximate the functions in F (or in a set close to F see PBDW). For this, we consider approximations on appropriate n-dimensional spaces X n of relatively small dimension under the hypothesis that the Kolmogorov n-width of F is small.

7

8 The Kolmogorov n-width of F in X, is defined as d n (F, X )= inf X n X dim(xn)applen sup u2f inf ku vk X, ( v2x n And the assumption is that d n (F, X ) is small.

9 The Kolmogorov n-width of F in X, is defined as d n (F, X )= inf X n X dim(xn)applen sup u2f inf ku vk X, ( v2x n And the assumption is that d n (F, X ) is small. In order to satisfy the constraint on the computing time, we consider continuous linear approximations J n : X! X n on appropriate n-dimensional spaces X n of relatively small dimension. A first bound on the approximation error is then supku u2f J n [u]k X apple (1 + n )sup u2f inf ku vk X, (1) v2x n where n =sup '2X kj n [']k X k'k X (2) is the norm of the operator J n : X! X n and is known as the Lebesgue constant.

10 The Kolmogorov n-width of F in X, is defined as d n (F, X )= inf X n X dim(xn)applen sup u2f inf ku vk X, ( v2x n And the assumption is that d n (F, X ) is small. What is the proper definition of X n In order to satisfy the constraint on the computing time, we consider continuous linear approximations J n : X! X n on appropriate n-dimensional spaces X n of relatively small dimension. A first bound on the approximation error is then supku u2f J n [u]k X apple (1 + n )sup u2f inf ku vk X, (1) v2x n where n =sup '2X kj n [']k X k'k X (2) is the norm of the operator J n : X! X n and is known as the Lebesgue constant.

11 EIM/GEIM In 2004 with M. Barrault, N. C. Nguyen, and A. T. Patera, we proposed n generic approach : the Empirical Interpolation Method: Application to cient Reduced-Basis Discretization Of Partial Di erential Equations, that as proven successful This approach allows to determine an empirical optimal set of interpolation points and/or set of interpolating functions. In 2013, with Olga Mula, we have generalized it (GEIM) to include more general output from the functions we want to interpolate : not only pointwize values but also some moments.

12 The first generating function is ' 1 = arg max '2F k'( )k L1 ( ), the associated interpolation point satisfies x 1 = arg max x2 ' 1 (x), we then set q 1 = ' 1 ( )/' 1 (x 1 ) and B11 1 = 1. Then the construction proceeds by induction : assume the nested sets of interpolation points M 1 = {x 1,...,x M 1 },M apple M max, and the associated nested sets of basis functions {q 1,...,q M 1 } are given 1. We first solve the interpolation problem : Find such that I M 1 ['( )] = 1) where Mmax M is some given upper bound fixed a priori MX 1 j=1 M 1,j [']q j, I M 1 ['( )](x i )='(x i ), i =1,...,M 1, that allows to define the M 1,j ['], 1 apple j apple M 1, as it can be proven indeed that the (M 1) (M 1) matrix of running entry q j (x i ) is invertible, actually it is lower triangular with unity diagonal.

13 We then set (note indeed that h M i (x j )= ij ). 8' 2 F, " M 1 (') =k' I M 1 [']k L1 ( ), and define and x M = arg max x2 ' M = arg max '2F " M 1('), ' M (x) J M 1 [' M ](x), we finally set r M (x) =' M (x) J M 1 [' M (x)], q M = r M /r M (x M ) and B M ij = q j (x i ), 1 apple i, j apple M The Lagrangian functions that can be used to build the interpolation operator I M in X M = span {' i, 1 apple i apple M} = span {q i, 1 apple i apple M} over the set of points M = {x i, 1 apple i apple M} verify for any given M, I M [u( )] = MX u(x i )h M i ( ) i=1 where h M i ( )= MX q j ( )[B M ] j=1 1 ji

14 For GEIM, we assume now that we do not have access to the values of ' 2 F at points in easily, but, on the contrary, that we have a dictionary of linear forms 2 assumed to be continuous in some sense, e.g. in L 2 ( ) with norm 1 the application of which over each ' 2 F is easy. Our extension consists in defining ' 1, ' 2,..., ' M and a family of associated linear forms 1, 2,..., M such that the following generalized interpolation process (our GEIM) is well defined : J M ['] = MX j=1 j ' j, such that 8i =1,...,M, i(j M [']) = i (') (1) Note that the GEIM reduces to the EIM when the dictionary is composed of dirac masses, defined in the dual space of C 0 ( ).

15

16 Note that obviously everything has to be implemented on a computer nd thus discretized!!

17 Note that obviously everything has to be implemented on a computer nd thus discretized!! Note also that for some reasons you may want to use your proper set f basis/interpolating function in your preferredspace X N that may come from ntuition, previous knowledge, or POD/SVD.

18 Note that obviously everything has to be implemented on a computer nd thus discretized!! Note also that for some reasons you may want to use your proper set f basis/interpolating function in your preferredspace X N that may come from ntuition, previous knowledge, or POD/SVD.

19 Note that obviously everything has to be implemented on a computer nd thus discretized!! Note also that for some reasons you may want to use your proper set f basis/interpolating function in your preferredspace X N that may come from ntuition, previous knowledge, or POD/SVD. Journal: Communications on Pure and Applied Analysis - A general multipurpose interpolation procedure: the magic points Yvon Maday - Ngoc Cuong Nguyen - Anthony T. Patera - George S. H. Pau Pages: , Volume 8, Issue 1, January 2009 doi: /cpaa

20

21 The important thing is to measure to which extent the EIM/GEIM misses the optimality of the best approximation in the best optimal discrete space suggested by the definition of the Kolmogorov n-width.

22 Formula suggests that n plays an important role in the result and it is therefore important to discuss its behavior as n increases. First of all, n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that n =1/ n,where n = inf '2X n sup 2Span{ 0,..., n 1 } h', i X,X 0 k'k X k k X 0.

23 Formula suggests that n plays an important role in the result and it is therefore important to discuss its behavior as n increases. First of all, n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that n =1/ n,where n = inf '2X n sup 2Span{ 0,..., n 1 } h', i X,X 0 k'k X k k X 0. GEIM interpreted as an oblic projection

24

25

26

27 We are now working on the justification of this nice behavior.

28 The next point is to justify the choice of the interpolating functions. It can be proven that Lemma For any n 1, the nth interpolating function ' n verifies k' n P n (' n )k X 1+ n max f2f kf P n(f)k X. Which allows to use the frame weak greedy of the papers by P. Binev, A. Cohen, W. Dahmen, R.A. DeVore, G. Petrova, and P. Wojtaszczyk, and R. A. DeVore, G. Petrova, and P. Wojtaszczyk, to analyse the convergence properties of our algorithm Work done with O. Mula and G. Turinici (SINUM (2016)

29 In a nutshell, in the case where we have a Hilbert framework, our result states that Theorem If ( n ) 1 n=1 is a monotonically increasing sequence then i) if d n apple C 0 n for any n 1, then n apple C 0 nn,with n := n, if n 2. ii) if d n apple C 0 e c 1n for n 1 and C 0 1, then n apple C 0 ne c 2n,with n := p 2 n, if n 2. Work done with O. Mula and G. Turinici (SINUM (2016)

30 Another application : fluid flow

31 Another application : fluid flow

32

33

34 Application of EIM for Reduced Basis Methods for non affine and nonlinear PDE: see Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations MA Grepl, Y Maday, NC Nguyen, AT Patera ESAIM (2007) or the two recent books Certified Reduced Basis Methods for Parametrized Partial Differential Equations Authors: Hesthaven, Jan S, Rozza, Gianluigi, Stamm, Benjamin Reduced Basis Methods for Partial Differential Equations An Introduction Authors: Quarteroni, Alfio, Manzoni, Andrea, Negri, Federico

35 More about (G)EIM

36 In a recent paper, with J.P. Argaud, B. Bouriquet, H. Gong and O. Mula, we have introduced the GEIM to monitor nuclear reactor. The challenge here is that we do not have access inside the core of the reactor and thus have only the possibility to place the captors inside the surrounding region.

37 The two group di usion equation in matrix notation reads A(µ)' = 1 k eff F (µ)' Where µ is the parameters set, e.g. D,, f. A and F are 2 2 matrix and ' is a 2-element column vector: r D A(µ) = 1 r +( 1 a + 1!2 s ) 0 1!2 s r D 2 r + 2 a 1 1 f 1 2 f F (µ) = 2 1 f 2 2 f '1 ' = Where D i, i =1, 2 is called the di usion coe cient of each group; i a, i = 1, 2 is the absorption cross section of each group; ' i,i=1, 2 is the neutron flux of each group; 1!2 s is called the removal cross section from group 1 to group 2; 1 f, i =1, 2 is the fission source term of each group; i, i =1, 2 is called the fission spectrum of each group; finally k eff is the e ective multiplication factor, also the eigenvalue of equation. ' 2

38 In 1D, this looks like

39 In order you are convinced that the Kolmogorov dimension is small

40 First classical EIM

41

42 Then with a restriction on the position of the points

43

44 Similar results in two dimensions The first twenty interpolation points distribution (constrained in fuel region)

45 Similar results in two dimensions

46 Similar results in two dimensions

47 What about noisy data We clearly see the effect of the Lebesgue constant

48 What about noisy data : how to minimize the effect of the Lebesgue constant

49 Application of GEIM in data assimilation and monitoring

50 Application of GEIM in data assimilation and monitoring

51 Application of GEIM in data assimilation and monitoring

52 Incorporating the model error : Parametrized-Background Data-Weak (PBDW) formulation with A.T. Patera, J. D. Penn and M. Yano The PBDW formulation integrates a parametrized mathematical model and M experimental observations associated with the configuration C to estimate the true field u true [C] as well as any desired output l out (u true [C]) 2 C for given output functional l out. We first introduce a sequence of background spaces that reflect our (prior) best knowledge, Z 1 Z Nmax U; here the second ellipsis indicates that we may consider the sequence of length N max as resulting from a truncation of an infinite sequence. Our goal is to choose the background spaces such that In words, we choose the background spaces such that the most dominant physics that we anticipate to encounter for various system configurations is well represented for a relatively small N.

53 Incorporating the model error : Parametrized-Background Data-Weak (PBDW) formulation with A.T. Patera, J. D. Penn and M. Yano Concerning the form of the noises (e m ) m, we make the following three assumptions:

54 We first associate with each observation functional `om 2 U 0 an observable function,

55

56 Analysis

57 Analysis

58 Experimental settings

59

60

61 Data-Driven Empirical Enrichment of the Background Space

62

63

64 Conclusion

65 Conclusion

66 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method

67 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method From its initial magic point statement to empirical interpolation we have been now able (thanks to the work initiated by devore on greedy algorithms) to state it on more firm ground

68 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method From its initial magic point statement to empirical interpolation we have been now able (thanks to the work initiated by devore on greedy algorithms) to state it on more firm ground There has been some nice generalizations like the matrix form in the group of A. Quartering and G. Rozza or operator interpolation M. Drohmann, B. Haasdonk, M. Ohlberger

69 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method From its initial magic point statement to empirical interpolation we have been now able (thanks to the work initiated by devore on greedy algorithms) to state it on more firm ground There has been some nice generalizations like the matrix form in the group of A. Quartering and G. Rozza or operator interpolation M. Drohmann, B. Haasdonk, M. Ohlberger There is still some missing point on the Lebesgue constant behavior.. on which we are passive but it works still empirically (see however the interpretation of the greedy algorithm in this respect)

70 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method From its initial magic point statement to empirical interpolation we have been now able (thanks to the work initiated by devore on greedy algorithms) to state it on more firm ground There has been some nice generalizations like the matrix form in the group of A. Quartering and G. Rozza or operator interpolation M. Drohmann, B. Haasdonk, M. Ohlberger There is still some missing point on the Lebesgue constant behavior.. on which we are passive but it works still empirically (see however the interpretation of the greedy algorithm in this respect) The pure interpolation process can be enriched by more data as in the PBDW method with A.T. Patera, J. Penn and M. Yano

71 Conclusion EIM and GEIM have been used now extensively in order to implement easily the reduced basis method From its initial magic point statement to empirical interpolation we have been now able (thanks to the work initiated by devore on greedy algorithms) to state it on more firm ground There has been some nice generalizations like the matrix form in the group of A. Quartering and G. Rozza or operator interpolation M. Drohmann, B. Haasdonk, M. Ohlberger There is still some missing point on the Lebesgue constant behavior.. on which we are passive but it works still empirically (see however the interpretation of the greedy algorithm in this respect) The pure interpolation process can be enriched by more data as in the PBDW method with A.T. Patera, J. Penn and M. Yano

72 The PBDW formulation is endowed with the following characteristics: Weak formulation. Actionable a priori theory. The weak formulation facilitates the construction of a priori error estimates Background space that best reflects our (prior) best knowledge of the phenomenon under consideration Design of quasi-optimal set of observations from a library of experimentally realizable observations in order to maximize the stability of the data assimilation. Correction of unmodeled physics with uncertainty. Online computational cost is O(M). We may realize real-time state estimation Simple non-intrusive implementation and generality. The mathematical model appears only in the offline stage. Recently a work of P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, P. Wojtaszczyk has analyzed this approach in terms of optimal recovery and our algorithm is optimal in this frame. They have extended it to the Multi-Space Case.

73 Reduced Basis method for a bifurcation study of a thermal convection problem Henar Herrero, Yvon Maday, Francisco Pla

74 RB-4-RB : Formulation of the problem A sketch of the domain and the physical situation is shown here The domain is a rectangle of depth d and width L. The domain contains a fluid that is heated from below, so that on the bottom plate a temperature T0 is imposed and on the upper plate the temperature is where T 1 = T 0 T = T 0 d is the vertical temperature gradient.

75 Formulation of the problem The equations governing the system are the incompressible Navier-Stokes equations with the Boussinesq approximation coupled with a heat equation

76 Bifurcation diagram

77 Stationary equations : Legendre collocation method

78 Greedy procedure In this calculations we have considered expansions of order n=35 in the x-direction and m=13 in the z-direction.

79 Greedy procedure

80 Greedy procedure

81 Greedy procedure

82 Galerkin procedure for each branch

83 Galerkin procedure for each branch

84 Galerkin procedure for each branch

85 A new concept of reduced basis approximation for convection dominated problems Nicolas Cagniart, Yvon Maday, Benjamin Stamm

86 And if the Kolmogorov n-width is not small? Try to better look at the set of solutions

87 A case where the dimension is not small 1D viscous Burger equation Snapshots of the solution to the unsteady viscous burger equation with u 0 = sin(x), =4, =0.04

88 A case where the dimension is not small 1D viscous Burger equation Snapshots of the solution to the unsteady viscous burger equation with u 0 = + sin(x), =4, =0.04

89 Nevertheless all solutions look alike what we have done above is to center the solutions these are the u(x n,t n )

90 and the dimension diminishes largely!! Eigenvalues of the POD decomposition of the original set of snapshots (in red) and of the centered set of snapshots (in green)

91 and the dimension diminishes largely!! Reconstruction of a snapshot (blue) using 3 POD modes. Left figure is in the centered case. Right figure is the uncentered case

92 The discrete scheme we want to mimic u n+1 (,µ)=u n (,µ) dt u n (,µ)u n x(,µ)+dt u n xx(,µ) leads to At each time step, we are looking for a set of reduced coordinates { i n} and a translation parameter n such that our true solution u N (,µ,t n )iswell apprxoximated by : u N (,µ,t n )= X i n i (µ) i ( n ) (1) We replace the search of the real and s as in the usual galerkin method. That is, we are trying to minimize the residual. min n+1 2 min n+1 2R X i n+1 i i( n+1 ) u n dt u n u n x + dt u n xx 2 (2) Complexity.. how to compute and solve??

93 We have assumed periodic boundary conditions (for the sake of simplicity) thus we need to compute online the following terms : 8 R 8, 8i, j, >< i ( ) j ( ) R 8, 8i, j, p >: i ( ) j ( )( p ) x ( ) R 8, 8i, j, ( i) x ( )( j ) x ( ) for unknown value (to be optimized) of For a su ciently small time step, we expect to be of order t c where c is some local characteristic velocity. We have chosen the following method: precompute the scalar products for a predefined small set of discrete values of in [ t c max, t c max ], where c max is the maximum expected shock speed during the simulation. using the regularity of the scalar product, we use spline interpolation to get and approximate value for any in [ t c max, t c max ]

94 The RB discrete scheme thus iterates between the evaluation of the proper best translation definition of the coefficient n+1 i n+1 and the proper X i n+1 i i( n+1 ) Mean reconstruction error w.r.t number of POD basis used

95 extensions.. what needs to be done - non periodic (superposition of basic space and convection space) - higher dimensions (POD representation of the translations ) see 1) - better fitting (add the derivatives of the POD functions) - better fitting (replace least square with L 1, see 2) approximation) 1) Iollo, A., Lombardi, D.: Advection modes by optimal mass transfer. Physical Review E 89(2), (2014) 2) Technical paper: Roxana Crisovan, Rémi Abgrall, David Amsallem Robust Model Reduction by L1-norm Minimization and Approximation via Dictionaries: Application to Linear and Nonlinear Hyperbolic Problems