Greedy algorithms for high-dimensional eigenvalue problems V. - PowerPoint PPT Presentation

Greedy algorithms for high-dimensional eigenvalue problems V. Ehrlacher Joint work with E. Canc` es et T. Leli` evre Financial support from IPAM is acknowledged. CERMICS, Ecole des Ponts ParisTech & MicMac project-team, INRIA. V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 1 / 32

Motivation High-dimensional problems are ubiquitous: quantum mechanics, kinetic models, molecular dynamics, uncertainty quantification, finance, multiscale models etc. How to compute u ( x 1 , · · · , x d ) with d potentially large? The bottom line of deterministic approaches is to represent solutions as linear combinations of tensor products of small-dimensional functions (parallelepipedic domains): � r 1 k ( x 1 ) r 2 k ( x 2 ) · · · r d u ( x 1 , · · · , x d ) = k ( x d ) k ≥ 1 � r 1 k ⊗ r 2 k ⊗ · · · ⊗ r d = � � ( x 1 , x 2 , · · · , x d ) . k k ≥ 1 V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 2 / 32

Curse of dimensionality Classical approach: Galerkin method using standard finite element discretization with N degrees of freedom per variate. � λ i 1 , ··· , i d φ 1 i 1 ⊗ · · · ⊗ φ d u ( x 1 , · · · , x d ) ≈ i d ( x 1 , · · · , x d ) , ( i 1 , ··· , i d ) ∈{ 1 , ··· , N } d � � φ j where the basis functions 1 ≤ i ≤ N , 1 ≤ j ≤ d are chosen a priori and the real i numbers ( λ i 1 , ··· , i d ) 1 ≤ i 1 , ··· , i d ≤ N are to be computed. DIM = N d This is the so-called curse of dimensionality ( [Bellman, 1957] ) V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 3 / 32

Greedy algorithms Progressive Generalized Decomposition : Here, we consider an approach proposed by: Ladev` eze 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; Nouy et al in the context of uncertainty quantification. They are related to the so-called greedy algorithms introduced in nonlinear approximation theory: ( [Temlyakov, 2008], Cohen, Dahmen, DeVore, Maday... ) The idea is to look iteratively for the “best tensor product”. At the n th iteration of the algorithm, an approximation u n of the function u is given by: n � r 1 k ⊗ r 2 k ⊗ · · · ⊗ r d u ( x 1 , · · · , x d ) ≈ u n ( x 1 , . . . , x d ) = k ( x 1 , · · · , x d ) . k =1 u n ( x 1 , · · · , x d ) = u n − 1 ( x 1 , · · · , x d ) + r 1 n ⊗ r 2 n ⊗ · · · ⊗ r d n ( x 1 , · · · , x d ) . DIM = n × Nd V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 4 / 32

Existing results on greedy algorithms Theoretical results for convex unconstrained minimization problems: [Le Bris, Leli` evre, Maday, 2008], [Canc` es, VE, Leli` evre, 2011], [Nouy, Falco, 2012] A greedy algorithm has been proposed in ( [Chinesta, Ammar, 2010] ) for eigenvalue problems, but no analysis. Here, we propose two new greedy algorithms for eigenvalue problems and provide some theoretical convergence results for these. V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 5 / 32

Outline Algorithms and theoretical convergence results 1 Numerical examples 2 V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 6 / 32

Outline Algorithms and theoretical convergence results 1 Numerical examples 2 V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 7 / 32

Prototypical example Ω = ( − L 1 , L 1 ) × · · · × ( − L d , L d ) where for all 1 ≤ i ≤ d , X i = ( − L i , L i ) is a bounded open interval of R . We wish to compute the lowest eigenvalue µ and an associated eigenvector odinger operator − 1 2 ∆ + Φ on L 2 (Ω): u ( x 1 , · · · , x d ) of the Schr¨ − 1 2∆ u + Φ u = µ u , where Φ( x 1 , · · · , x d ) ∈ L q (Ω) with q = 2 if d ≤ 3, q > 2 for d = 4 and q = d / 2 for d ≥ 5. Weak formulation of the eigenvalue problem: H := L 2 (Ω) , V := H 1 0 (Ω) , � ∀ v , w ∈ H , � v , w � H = Ω vw , a ( v , w ) := 1 � � ∀ v , w ∈ V , Ω ∇ v · ∇ w + Ω Φ vw , 2 V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 8 / 32

Prototypical example The function u ( x 1 , · · · , x d ) and the eigenvalue µ are then solutions of: a ( v , v ) a ( v , v ) ∀ v ∈ V , a ( u , v ) = µ � u , v � H , µ = min , u = argmin . � v � 2 � v � 2 v ∈ V , v � =0 v ∈ V , v � =0 H H At each iteration of the algorithm, only low-dimensional functions are computed, for instance pure tensor product functions r 1 ⊗ r 2 ⊗ · · · ⊗ r d , r 1 ∈ H 1 0 ( X 1 ) , · · · , r d ∈ H 1 � � Σ := 0 ( X d ) . (1) V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 9 / 32

General setting and main assumptions Let V , H be separable Hilbert spaces such that (AV) V ⊂ H is dense and the injection V ֒ → H is compact (i.e. the weak convergence in V implies the strong convergence in H ). Let �· , ·� H denote the scalar product on H . Let a : V × V → R be a continuous symmetric bilinear form such that (AA) ∃ η ≥ 0, such that the bilinear form �· , ·� a defined by ∀ v , w ∈ V , � v , w � a := a ( v , w ) + η � v , w � H defines a scalar product on V whose associated norm � · � a is equivalent to the original norm on V . Let Σ ⊂ V satisfying (A1) Σ is a non-empty cone of V i.e. 0 ∈ Σ and ∀ ( z , c ) ∈ Σ × R , cz ∈ Σ; (A2) Σ is weakly closed in V ; (A3) Span (Σ) is dense in V . V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 10 / 32

Eigenvalue problem in the general framework All the previous assumptions are satisfied in our prototypical example! We wish to compute the lowest eigenvalue µ of the bilinear form a ( · , · ) and an associated H -normalized eigenvector u ∈ V , which satisfy a ( v , v ) a ( v , v ) µ = min , u = argmin . � v � 2 � v � 2 v ∈ V , v � =0 v ∈ V , v � =0 H H In particular, we have ∀ v ∈ V , a ( u , v ) = µ � u , v � H . The greedy algorithm computes iteratively a sequence ( z n ) n ∈ N ⊂ Σ and the approximation u n of u given at the n th iteration of the algorithms satisfies u n ∈ Span { z 0 , z 1 , · · · , z n } . V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 11 / 32

Three greedy algorithms Rayleigh Greedy algorithm ( [Canc` evre, 2013] ); es, VE, Leli` Residual Greedy algorithm ( [Canc` evre, 2013] ); es, VE, Leli` Explicit Greedy algorithm ( [Chinesta, Ammar, 2010] ); All these algorithms begin with some initial guess u 0 ∈ V . The initial guess u 0 ∈ V is defined as follows: Initialization n = 0 : find z 0 ∈ Σ such that a ( z , z ) z 0 ∈ argmin ; (2) � z � 2 z ∈ Σ , z � =0 H z 0 set u 0 := � z 0 � H and λ 0 := a ( u 0 , u 0 ). V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 12 / 32

Pure Rayleigh Greedy algorithm � a ( v , v ) H if v � = 0 , � v � 2 Rayleigh quotient: ∀ v ∈ V , J ( v ) := + ∞ if v = 0 . The Rayleigh Greedy Algorithm reads: Iteration n ≥ 1 : find z n ∈ Σ such that z n ∈ argmin J ( u n − 1 + z ) . (3) z ∈ Σ u n − 1 + z n Set u n = � u n − 1 + z n � H , λ n := a ( u n , u n ) and n = n + 1. V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 13 / 32

Residual Greedy algorithm The Residual Greedy Algorithm reads: Iteration n ≥ 1 : find z n ∈ Σ such that 1 2 � u n − 1 + z � 2 z n ∈ argmin a − ( λ n − 1 + η ) � u n − 1 , z � H . (4) z ∈ Σ u n − 1 + z n Set u n = � u n − 1 + z n � H , λ n := a ( u n , u n ) and n = n + 1. Why is it called Residual? (4) is equivalent to 1 2 � R n − 1 − z � 2 z n ∈ argmin a , z ∈ Σ where R n − 1 is the element of V such that ∀ v ∈ V , � R n − 1 , v � a = λ n − 1 � u n − 1 , v � H − a ( u n − 1 , v ) . V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 14 / 32

Euler equations for the Residual algorithm on a very simple case In the previous prototypical example, with d = 2 and Φ = 0 (In this case, we can take η = 0). r 1 ⊗ r 2 , r 1 ∈ H 1 0 ( X 1 ) , r 2 ∈ H 1 � � Σ = 0 ( X 2 ) , If z n = r 1 n ⊗ r 2 n ∈ Σ, the Euler equations associated to the previous mininmization problem read �� n | 2 � �� n | 2 �  X 1 | r 1 ( − ∆ x 2 r 2 X 1 |∇ x 1 r 1 r 2 n ( x 2 )) + n ( x 2 )    � X 1 [ − ∆ x 1 , x 2 u n − 1 ( x 1 , x 2 ) − λ n − 1 u n − 1 ( x 1 , x 2 )] r 1  = n ( x 1 ) dx 1 ,    �� n | 2 � �� n | 2 �  X 2 | r 2 ( − ∆ x 1 r 1 X 2 |∇ x 2 r 2 r 1 n ( x 1 )) + n ( x 1 )     X 2 [ − ∆ x 1 , x 2 u n − 1 ( x 1 , x 2 ) − λ n − 1 u n − 1 ( x 1 , x 2 )] r 2  = � n ( x 2 ) dx 2 ,  These equations leads to a system of coupled nonlinear equations, which are solved through an alternating direction method (fixed-point procedure). V. Ehrlacher (CERMICS) Greedy algorithm CEMRACS 2013, 31st July 2013 15 / 32