The Generalized Empirical Interpolation Method: Analysis of the convergence and application to data assimilation coupled with simulation

Transcription

1 The Generalized Empirical Interpolation Method: Analysis of the convergence and application to data assimilation coupled with simulation Y. Maday (LJLL, I.U.F., Brown Univ.) Olga Mula (CEA and LJLL) G. Turinici (CEREMADE) SampTa 2013 Bremen, 01-05/07/2013 Olga MULA GEIM 1/ 26

2 Motivations of the present work Let Ω R d be a bounded spatial domain. Classical Lagrangian interpolation Classical interpolation problem: Approximate f C 0 (Ω) in the finite n-dimensional space X n = span {q i C 0 (Ω)} n i=1 where: the basis functions q i are of polynomial nature (algebraic polynomials, rational functions...). the interpolation points: d = 1: an almost optimal location is provided by the Chebychev-Gauss nodes. d > 1: more complex conditions in order for a polynomial interpolation to be well defined. Answers exist for domains of simple shapes (e.g. those obtained through tensor-product operations). Olga MULA GEIM 2/ 26

3 Motivations of the present work Extension of classical Lagrangian interpolation Let F be a compact subset of X = C 0 (Ω). Hypothesis: F is supposed to be such that any f F is approximable by linear combinations of small size we suppose that the Kolmogorov n-width d n (F,X) of F in X is small. d n (F,X) := inf sup inf x y X X n X x F y X n dim(xn)=n Extended interpolation problem: Approximate any f F by finite dimensional subspaces X n = span {q i F,i = 1,...,n} contained in the span of F. Here: the basis functions q i are very general (the only requirement is that q i F). given X n, the general conditions for which the interpolation points give an unique interpolant is an open problem. Olga MULA GEIM 3/ 26

4 Motivations of the present work Generalized Empirical Interpolation Method (GEIM) Empirical Interpolation Method (EIM) [Maday, Nguyen, Patera, Pau, 2009], [Barrault, Maday, Nguyen, Patera, 2004], [Grepl, Maday, Nguyen, Patera, 2007]: proposes a constructive approach to find the basis functions {q i F,i = 1,...,n} and the interpolation points {x i Ω, i = 1,...,n} through a Greedy algorithm. Generalized EIM: Replaces the set of interpolation points {x i Ω, i = 1,...,n} by continuous linear functions {σ i Σ, i = 1,...,n}, where the dictionary Σ L(F). A Greedy algorithm provides the basis functions and the interpolating linear functions. Unisolvence of Σ: if ϕ F is such that σ(ϕ) = 0 for any σ Σ, then ϕ = 0. Olga MULA GEIM 4/ 26

5 Outline 1 Theoretical properties of GEIM 2 Olga MULA GEIM 5/ 26

6 The generalized interpolation Assume you are given a set {q 1,...,q M } of linearly independent functions coming from a space of small Kolmogorov n-width AND a dictionary of continuous linear forms {σ}. Given a function f that you want to approximate, the problem is: find a family of scalars {α n M } 1 n M such that σ i (f) = M α M n σ i (q n ), for i = {1 n M}. n=1 where the linear forms σ i M are suitably chosen and X M = span {q n, 1 n M}. If σ i = δ( r i ) EIM. Olga MULA GEIM 6/ 26

7 The generalized interpolation Assume you are given a set {q 1,...,q M } of linearly independent functions coming from a space of small Kolmogorov n-width AND a dictionary of continuous linear forms {σ}. Given a function f that you want to approximate, the problem is: find a family of scalars {α n M } 1 n M such that σ i (f) = M α M n σ i (q n ), for i = {1 n M}. n=1 where the linear forms σ i M are suitably chosen and X M = span {q n, 1 n M}. The classical continuity assumption can be relaxed: L 2 (Ω) instead of C 0. Olga MULA GEIM 6/ 26

8 Among the questions raised by GEIM: is there an optimal selection for the linear forms σ i? is there a constructive optimal selection for the functions q i? given a set of linearly independent functions {q i } i [1,M] and a set of continuous linear forms {σ i } i [1,M], does the interpolant exist? is the interpolant unique? how does the interpolation process compares with other approximations (in particular orthogonal projections)? Under what hypothesis can we expect the GEIM approximation to converge rapidly to f? Olga MULA GEIM 7/ 26

9 The Greedy Algorithm for GEIM We are looking for a constructive way of approximating in X. From now on, we assume that X L 2. We propose a Greedy approach for selecting the interpolation linear forms and constructing the discrete spaces X M : X 1 X 2 X M X Olga MULA GEIM 8/ 26

10 The Greedy Algorithm for GEIM The Greedy algorithm for GEIM Constructive way of approximating in X: 1 Find: ϕ 0 = argsup ϕ F ϕ L 2 (Ω) and σ 0 = argsup σ Σ σ(ϕ 0 ) We define: q 0 = ϕ 0/σ 0(ϕ 0) X 1 span {q 0} J 0[ϕ] := σ 0(ϕ)q 0, ϕ F 2 By recursion: given, X N span {q j, j [0,N 1]} and {σ 0,σ 1,...,σ N 1 }, find: ϕ N = argsup ϕ F ϕ J N [ϕ] L 2 (Ω) σ N = argsup σ Σ σ(ϕ N J N [ϕ N ]) We define: q N = ϕ N J N [ϕ N ] σ N (ϕ N J N [ϕ N ]) X N+1 span {q j, j [0,N]} J N+1 : F X N+1 Olga MULA GEIM 9/ 26

11 Well-posedness It has been proven in [Maday, Mula, 2013] : Lemma For any N, the set {q j, j [0,N 1]} is linearly independent and X N is of dimension N. The generalized empirical interpolation procedure is well-posed in L 2 (Ω) and ϕ F, the interpolation error satisfies: ϕ J N [ϕ] L 2 (Ω) (1+Λ N ) inf ψ N X N ϕ ψ N L 2 (Ω) where Λ N is the Lebesgue constant in the L 2 norm: Λ N := sup ϕ F J N [ϕ] L 2 (Ω). ϕ L 2 (Ω) Olga MULA GEIM 10/ 26

12 A priory convergence As an extension of the results presented in [DeVore, Petrova, Wojtaszczyk, 2012], an optimal convergence result (modulo the Lebesgue constant) has been proven in [Maday, Mula, Turinici, 2013] Theorem 1 Assume that d n (F,L 2 (Ω)) C 0 n α for any n 1, then the interpolation error of the GEIM Greedy selection process satisfies for any ϕ F the inequality ϕ J n [ϕ] L 2 (Ω) C 0 (1+Λ n )β n n α, where the parameters γ n and β n are defined as: γ n = 1 1+Λ n β 1 = 2 n 2: β n = β 4l+k := 1 2β l1 l 2 i=1 l 1 = 2l+ 2k 3, l 2 = 2 ( l+ k 4 ) γ 1 l 2 l 1 k 4 +i (2 2) α and Olga MULA GEIM 11/ 26

13 A priory convergence As an extension of the results presented in [DeVore, Petrova, Wojtaszczyk, 2012], an optimal convergence result (modulo the Lebesgue constant) has been proven in [Maday, Mula, Turinici, 2013] Theorem 2 Assume that d n (F,L 2 (Ω)) C 0 e c 1n α for any n 1, then the interpolation error of the GEIM Greedy selection process satisfies for any ϕ F the inequality ϕ J n [ϕ] L 2 (Ω) C 0 (1+Λ n )β n e c 2n α, where β n and c 2 are defined as: c 2 := 2 1 3α c 1 β 1 = 2 for n 2: β n := n 2 i=1 γ 1 1 n 2 n 2 +i 2β n 2. Olga MULA GEIM 11/ 26

14 A priory convergence Corollary If (Λ n ) n 1 is a monotonically increasing sequence, then the sequence (γ n ) n 1 in the GEIM procedure is monotonically decreasing. The following decay rates in the generalized interpolation error can be derived: For any ϕ F, if d n (F,L 2 (Ω)) C 0 n α for any n 1, then the interpolation error of the GEIM Greedy selection process can be bounded as ϕ J n [ϕ] L 2 (Ω) C 0 2 3α+1 (1+Λ n ) 3 n α. For any ϕ F, if d n (F,L 2 (Ω)) C 0 e c 1n α for any n 1, then the interpolation error of the GEIM Greedy selection process can be bounded as ϕ J n [ϕ] L 2 (Ω) C 0 2(1+Λ n ) 3 e c 2n α. Olga MULA GEIM 12/ 26

15 A priory convergence Remark on the Lebesgue constant Our first experiments in the GEIM provide cases where it is uniformly bounded when evaluated in the L(L 2 ) norm. We do not pretend that this is universal, but only shows that the theoretical increasing upper bound is far from being optimal in a class of sets F that have a small Kolmogorov n-width Computed Lebesgue constant L 2 Lebesgue constant H 1 Lebesgue constant Dimension Figure : Estimation of the Lebesgue constant on a simple case (Laplace problem - see numerical example below) Olga MULA GEIM 13/ 26

16 Outline 1 Theoretical properties of GEIM 2 Olga MULA GEIM 14/ 26

17 Reduced basis Let E R p be a compact set and let D µ u = g µ E (1) be a family of parameter dependent PDE s representing a physical experiment. Hypothesis: {u µ, µ E} F where F X is compact and of small Kolmogorov n-width in X. Aim: For a given µ E, we wish to compute u µ in real time (i.e. at the same time as the experiment is taking place). Idea of reduced basis: Offline: We solve (1) for µ 1,...,µ n,... to build an M-dimensional reduced basis X M = span{q j F,j [1,M]}. Online: we approximate the solution by: u µ M a j q j j=1 Olga MULA GEIM 15/ 26

18 Reduced basis Suppose further that the experiment involves captor measurements we have a dictionary Σ L(F). Then, in this reduced basis approach, data from sensors can be taken into account if we apply our generalized interpolation procedure: Offline: Construction of X M = span{q j F,j [1,M]} and selection of {σ i,1 i M} by the Greedy algorithm. Online: u µ I M (u µ ). Olga MULA GEIM 16/ 26

19 Numerical application The Laplace problem Spatial domain: Ω R 2 and a non overlapping partition of it is: Ω = Ω 1 Ω2. Model: the Laplace problem. Hypothesis: only Ω 1 is driving the phenomenon. The rest (Ω 2 ) is the environment. Equation: u µ = f µ in Ω = Ω 1 Ω 2,µ f µ (x,y) = 1+(αsin(x)+βcos(γπy))χ 1 (x,y) Parameters: µ = (α,β,γ,geometry of Ω 2 ). Olga MULA GEIM 17/ 26

20 Aim We will suppose that the set of solutions {u µ Ω2, µ} F and that F is of small Kolmogorov n-width in L 2 (Ω 2 ). Given µ, we wish to: 1 Show that u µ Ω2 can be reconstructed with GEIM. 2 Run a real-time simulation: Ω 2 : Data acquisition + reconstruction with GEIM Ω 1 : Implicit calculation with the boundary conditions provided by the interpolant of Ω 2. Olga MULA GEIM 18/ 26

21 1) Reconstruction of Ω 2 with GEIM Interpolation error We have reconstructed several solutions and computed for 1 M 13: max u I M[u] ; = L 2 (Ω), H 1 (Ω) u reconstructed Interpolation error in the interpolated functions by GEIM L 2 interpolation error H 1 interpolation error Dimension Olga MULA GEIM 19/ 26

22 1) Reconstruction of Ω 2 with GEIM Comparison with POD Our GEIM approach seems to have explored the set of solutions good enough to perform as well as POD Interpolation error in the interpolated functions by GEIM L 2 interpolation error H 1 interpolation error H 1 : POD error L 2 : POD error Dimension Olga MULA GEIM 20/ 26

23 1) Reconstruction of Ω 2 with GEIM The FEM noise Going much beyond M = 13 is of no use since we reach the FEM precision ( 10 4 ) POD errors POD errors H 1 POD errors L Dimension Olga MULA GEIM 21/ 26

24 1) Reconstruction of Ω 2 with GEIM The Lebesgue constant A computational estimation of Λ M has been carried out: I M [u i ] L Λ M = max 2 (Ω) i [1,256] u i L 2 (Ω) Computed Lebesgue constant L 2 Lebesgue constant H 1 Lebesgue constant Dimension Olga MULA GEIM 22/ 26

25 2) Data acquisition coupled with simulation Application to the Laplace problem The interpolant in the environment Ω 2 can give the appropriate boudary conditions to the subdomain that has a big Kolmogorov n-width (Ω 1 ). We compute Ω 1 by FEM with the boundary conditions provided by the GEIM procedure Reconstructed analysis H 1 error in Ω H 1 error in Ω Olga MULA GEIM 23/ 26

26 Current work Treatment of noisy measurements: some developments in [Maday, Mula, 2013] Future work More complex applications (e.g. Stokes, Navier-Stokes, transport equations) Extension to time-dependent problems Olga MULA GEIM 24/ 26

27 For further reading [Maday, Nguyen, Patera, Pau, 2009] Y. Maday, N. Nguyen, A. Patera, and G. Pau, A general multipurpose interpolation procedure: the magic points Commun. Pure Appl. Anal., pp , [Barrault, Maday, Nguyen, Patera, 2004] M. Barrault, Y. Maday, N. Y. Nguyen, and A. Patera, An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations C. R. Acad. Sci. Paris, Serie I., vol. 339, pp , [Grepl, Maday, Nguyen, Patera, 2007] M. Grepl, Y. Maday, N. Nguyen, and A. Patera, Efficient reduced basis treatment of nonaffine and nonlinear partial differential equations Math. Model. Numer. Anal., vol. 41(3), pp , Olga MULA GEIM 25/ 26

28 For further reading [DeVore, Petrova, Wojtaszczyk, 2012] R. DeVore, G. Petrova, and P. Wojtaszczyk Greedy algorithms for reduced bases in Banach spaces Constructive Approximation, pp. 1-12, 2012 [Maday, Mula, 2013] Y. Maday and O. Mula A generalized empirical interpolation method: application of reduced basis techniques to data assimilation Analysis and Numerics of Partial Differential Equations, vol. XIII, pp , 2013 [Maday, Mula, Turinici, 2013] Y. Maday, O. Mula and G. Turinici A priori convergence of the Generalized Empirical Interpolation Method accepted in the proceedings of the 10th International Conference on Sampling Theory and Applications, Olga MULA GEIM 26/ 26