Greedy algorithms for high-dimensional non-symmetric problems

Transcription

1 Greedy algorithms for high-dimensional non-symmetric problems V. Ehrlacher Joint work with E. Cancès et T. Lelièvre Financial support from Michelin is acknowledged. CERMICS, Ecole des Ponts ParisTech & MicMac project-team, INRIA. V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

2 Outline 1 Greedy algorithms for symmetric coercive problems 2 Problem in the non-symmetric case 3 Other algorithms V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

3 Outline 1 Greedy algorithms for symmetric coercive problems 2 Problem in the non-symmetric case 3 Other algorithms V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

4 Curse of dimensionality We wish to approximate a function u(x, ξ 1,, ξ d ) where ξ 1,, ξ d are random variables with d very large. x X, ξ 1 Ξ 1,, ξ d Ξ d Standard (Galerkin methods): a priori fixed basis functions (φ i (x)) 1 i N, (ψ j1 (ξ 1 )) 1 j1 N,..., (ψ jd (ξ d )) 1 jd N u(x, ξ 1,, ξ d ) 1 i,j 1,,j d N λ i,j1,,j d φ i (x)ψ j1 (ξ 1 ) ψ jd (ξ d ). DIM = N d V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

5 Separated variable representation Separated variable representation (also called canonical format) The solution is represented as linear combinations of tensor products of small-dimensional functions to avoid the curse of dimensionality [Bellman, 1957]: u(x, ξ 1,, x d ) = n s k (x)rk 1 (ξ 1)... rk d (ξ d) k=1 n k=1 ( ) s k rk 1... r k d (x, ξ 1,..., ξ d ). DIM = nnd V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

6 Symmetric problem u(x, ξ 1,, ξ d ) V where V = V x V ξ1 V ξd. V x L 2 (X ), V ξ1 L 2 (Ξ 1 ),, V ξd L 2 (Ξ d ), V L 2 (X X 1 Ξ d ) v V, a(u, v) = l(v) (1) where a is a symmetric, coercive continuous bilinear form on V V ; l is a continuous linear form on V. Typical example: { Find u V = H 1 (X ) L 2 p(ξ), E [ X xu(x, ξ) x v(x, ξ) + u(x, ξ)v(x, ξ) dx ] = E [ X with f L 2 (X ) L 2 p(ξ). V x = H 1 (X ), V ξ = L 2 p(ξ). f (x, ξ)v(x, ξ) dx] (2) V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

7 Greedy algorithm - Progressive Generalized Decomposition We consider an approach proposed by: Ladevèze et al. to do time-space variable separation Chinesta et al. to solve high-dimensional Fokker-Planck equations in the context of kinetic models for polymers [Ammar et al., 2006] Nouy et al in the context of UQ. [Nouy, 2009] In the symmetric coercive setting, problem (1) can be rewritten as an optimization problem u = argmin E(v) v V where E(v) = 1 2a(v, v) l(v). V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

8 Definition of the greedy algorithm The idea is to look iteratively for the best tensor product. (x, ξ) X Ξ u(x, ξ) = k 1 s k(x)r k (ξ). V V x V ξ where the solution u is the unique global minimizer of the functional E : V R. Greedy algorithm [Temlyakov, 2008]: We define recursively (r n, s n ) argmin (r,s) V t V x E ( n 1 ) r k s k + r s k=1 (3) Let us denote u n = n k=1 r k s k. Question: Does u n converge towards u? Yes! V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

9 Convergence results [Le Bris, Lelièvre, Maday, 2009], [Cancès, VE, Lelièvre, 2011] or [Nouy, Falco, 2011] Σ = {r s, (r, s) V x V ξ } Span(Σ) dense in V ; Σ is weakly closed in V ; E is differentiable and E is Lipschitz continuous on bounded sets; E is elliptic, i.e. there exists α > 0 and s > 1 such that v, w V, E (v) E (w), v w V α v w s V ; u n n u Remark: E does not necessarily need to be a quadratic functional for the algorithm to converge. V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

10 Euler equations How are computed the (s n, r n ) V x V ξ in practice? By solving the Euler equations: a(u n 1 + s n r n, s n δr + δs r n )= l(s n δr + δs r n ), (δs, δr) V x V ξ. For problem (5), ( X E [ r n (ξ) 2] ( x s n (x) + s n (x)) = E [(f (x, ξ) + x u n 1 (x, ξ))r n (ξ)], ) x s n (x) 2 + s n (x) 2 dx r n (ξ) = (f (x, ξ) + x u n 1 (x, ξ))s n (x) dx. X V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

11 Outline 1 Greedy algorithms for symmetric coercive problems 2 Problem in the non-symmetric case 3 Other algorithms V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

12 Non-symmetric problem where v V, a(u, v) = l(v) (4) a = a s + a as where a s is a symmetric, coercive continuous bilinear form on V V and a as an antisymmetric continuous bilinear form on V V ; l is a continuous linear form on V. There is no minization problem formulation of the problem as in the symmetric case!! How can we define the PGD/greedy algorithm? Naive idea: By solving the Euler equations: a(u n 1 + s n r n, s n δr + δs r n )= l(s n δr + δs r n ), (δs, δr) V x V ξ. V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

13 Convection-diffusion example Let us denote by A the operator on L 2 (X Ξ), with domain D(A) such that u, v D(A), Au, v L 2 (X Ξ) = a(u, v). Typical example: A = x + b x + 1 = A x I ξ, D(A) = H 2 (X ) L 2 p(ξ), with A x = x + b x + 1 on L 2 (X ) and I is the identity operator on L 2 p(ξ). Find u V = H 1 (X ) L 2 p(ξ), E [ X xu(x, ξ) x v(x, ξ) + b u(x, ξ)v(x, ξ) + u(x, ξ)v(x, ξ) dx ] = E [ X f (x, ξ)v(x, ξ) dx], (5) with f L 2 (X ) L 2 p(ξ). V x = H 1 (X ), V ξ = L 2 p(ξ). V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

14 Euler algorithm For problem (5), E [ r n (ξ) 2] A x s n (x) = E [(f (x, ξ) + Au n 1 (x, ξ))r n (ξ)], ( ) (A x s n (x))s n (x) dx r n (ξ) = (f (x, ξ) + Au n 1 (x, ξ))s n (x) dx. X X Problem: There are cases where the only solutions of these Euler equations are (r n, s n ) = (0, 0) even if u n 1 u n!! V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

15 Outline 1 Greedy algorithms for symmetric coercive problems 2 Problem in the non-symmetric case 3 Other algorithms V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

16 Symmetrize the problem: minimization of the L 2 residual [Falco et al, 2011] The idea is then to symmetrize the problem by minimizing the L 2 residual. In the convection-diffusion case, perform the symmetric greedy algorithm on E(v) = Av f L 2 (X Ξ). V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

17 Euler equations: minimization of the L 2 residual When f is regular enough for the convection diffusion problem E [ r n (ξ) 2] A xa x s n (x) = E [(A (f (x, ξ) + Au n 1 (x, ξ)))r n (ξ)], ( ) A xa x s n, s n L 2 (X ) dx r n (ξ) = (A (f (x, ξ) + Au n 1 (x, ξ)))s n (x) dx. X X The Euler equations are the ones associated to the problem A Au = A f The conditining of the discretized problems scales quadratically with the conditioning of the original problem!! V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

18 Symmetrize the problem: minimization of the H 1 residual The idea is then to symmetrize the problem by minimizing the H 1 residual. In the convection-diffusion case, perform the symmetric greedy algorithm on E(v) = Av f H 1 (X ) L 2 (Ξ). V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

19 Euler equations: minimization of the L 2 residual E [ r n (ξ) 2] A x( x ) 1 A x s n (x) = E [ (A ( x ) 1 (f (x, ξ) + Au n 1 (x, ξ)))r n (ξ) ], ( ) A x( x ) 1 A x s n, s n L 2 (X ) dx r n (ξ) X = (A ( x ) 1 (f (x, ξ) + Au n 1 (x, ξ)))s n (x) dx. X The Euler equations are the ones associated to the problem A ( x ) 1 Au = A ( x ) 1 f The conditining of the discretized problems scales linearly with the conditioning of the original problem!! However, this method needs to solve a lot of small-dimensional Poisson problems. It takes more time (although it can be done in parallel) and more memory. V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

20 Explicitation of the antisymmetric part Idea: Perform the greedy algorithm with the symmetric part a s of the bilinear form a and update the right-hand side at each iteration. where (r n, s n ) argmin (r,s) V x V ξ E n 1 (r s), E n 1 (r s) = 1 2 a s(u n 1 + r s, u n 1 r s) l(r s) a as (u n 1, r s) with u n 1 = n 1 k=1 r k s k. In other words, at each iteration, one performs one greedy iteration on the problem v V, a s (u, v) = l(v) a as (u n 1, v). V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

21 Partial convergence results Of course, such an algorithm is expected to converge only if the antisymmetric part is small enough. Proposition If V x and V ξ are finite-dimensional, there exists κ > 0 such that if a as L(V,V ) κ a s L(V,V ), then the algorithm converges strongly in V. Problem: The rate κ depends on the dimension of V x and V ξ. Numerically, the rate κ seems not to depend on the dimension... V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

22 Use a symmetric formulation of the antisymmetric problem Idea: use a symmetric version of the antisymmetric problem [Cohen et al., 2011] For the convection diffusion problem: ( 0 A ) ( u A y ) = ( 0 f ) This is a symmetric problem whose solution is (v, y) = (u, 0). But it is not a coercive problem!!! V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

23 Other algorithms in the literature Minimax algorithm [Nouy, 2010] Dual algorithm, X-Greedy algorithm (Lozinski, based on ideas of Temlyakov) Good numerical results but no theoretical proof of convergence. V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

24 References R.E. Bellman. Dynamic Programming. Princeton University Press, A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech., 139: , E. Cancès, V.E., Tony Lelièvre. Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, M3AS, A. Nouy and A. Falco. Proper Generalized Decomposition for Nonlinear Convex Problems in Tensor Banach Spaces, Numerische Mathematik,2011. C. Le Bris, Tony Lelièvre and Y. Maday. Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constructive Approximation 30(3): , A. Nouy, Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations, Archives of Computational Methods in Engineering 16: , A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, preprint, V.N. Temlyakov. Greedy approximation. Acta Numerica, 17: , A. Falco, L. Hilario, N. Montés, M.C. Mora. Numerical Strategies for the Galerkin Proper Generalized Decomposition Methods A. Cohen, W. Dahmen, G. Welper. Adaptivity and Variational Stabilization for Convection-Diffusion Equations, V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25

25 Thank you for your attention! V. Ehrlacher (CERMICS) Greedy non-symmetric MASCOT-NUM, May / 25