The Proper Generalized Decomposition: A Functional Analysis Approach

Transcription

1 The Proper Generalized Decomposition: A Functional Analysis Approach Méthodes de réduction de modèle dans le calcul scientifique

2 Main Goal Given a functional equation Au = f It is possible to construct u n = n i=1 ui 1 ui d such that u u n is small enough? Here r r u(x 1,..., x d ) u j 1 (x 1) u j d (x d) = u j 1 uj d (x 1,..., x d ) The best r-term approximation Given u in a normed tensor product space there exists a best approximation given by the sum of elementary tensors v 1 v 2 v d?

3 Some definitions 1 d Let V := a i=1 V i be the algebraic tensor product space, where V i d is a vector space for 1 i d. Here v a i=1 V i implies for some n N. v = n v1 i vd. i i=1 2 Assume that is a norm over V. Then we take the completion of the normed space V obtained a Banach space: V = a 3 ϕ V iff ϕ : V R linear i=1 V i := i=1 4 ϕ V iff ϕ : (V, ) R linear and continuous. V i

4 An approximating format is the tensor subspace or Tucker format u = i I a i b (j) i j, (1) where I = I 1... I d is a multi-index set with I j = {1,..., r j }, r j dim(v j ), b (j) i j V j (i j I j ) are (usually orthonormal) basis vectors, and a i R. Here, i j are the components of i = (i 1,..., i d ). The data size is determined by the numbers r j collected in the tuple r := (r 1,..., r d ). The set of all tensors representable by (1) with fixed r is T r := { v V : there are subspaces U j V j such that dim(u j ) = r j and v U := a d U j. Here, the vectors b (j) i are replaced by the generated subspace U j = span{b (j) i : i I j }. Note that T r is neither a subspace of V nor a convex set. } (2)

5 Proposition Let V be a Banach space. Assume that F : R(T r ) V R satisfy: P1 R(T r ) is weakly closed. P2 For each sequence (u n ) R(T r ) weakly convergent to u it holds that F (u) lim n F (u n ). P3 F (u) as u, u R(T r ). Then F is bounded below and has a minimum in R(T r ). Example: Given u 0 in a reflexive Banach space V, consider F (u) = u u 0 for a weakly closed set R it satisfies P2 and P3.

6 Some remarks dim U = r 1 r 2 r d. Thus an aproximating vector û T r has r 1, r 2 r d -terms, that is, û = r 1 r d i=1 u i 1 u i d. If r 1 = r 2 = = r d = 1 we have the rank-one approximating tensors, that is, û = û 1 û d. Question Under that conditions is the set T r weakly closed in a Banach space V = V i? i=1

7 Example For I j R (1 j d) and 1 p <, the Sobolev space H N,p (I j ) consists of all function f from L p (I j ) with bounded norm ( N f N,p;Ij := n=0 I j d n p dx n f 1/p, dx) (3a) whereas H N,p (I) for I = I 1 I 2... I d R d is endowed with the norm ( 1/p f N,p := n f dx) p (3b) 0 n N with n N d 0 being a multi-index of length n := d n j. It is well-known that H N,p (I) and H N,p (I j ) are reflexive and separable Banach spaces. I

8 Moreover, for p = 2 the Sobolev spaces H N (I j ) := H N,2 (I j ) and H N (I) := H N,2 (I) are Hilbert spaces. Since 0,2 L 2 (I j ) = 0,2 a L 2 (I j ) = L 2 (I), then it can be show H N (I) = N,2 H N (I j ). (4) On the other hand, it can be prove that L p (I) = αp L p (I j ) holds for the p-nuclear norm αp.

9 The 0,2 on L 2 (I) satisfies f j (x j ) = 0,2 d f j (x j ) 0,2;Ij and then it is a crossnorm, whereas the norm N,2 for N > 1 on H N (I) not is a crossnorm. Indeed, 0,2 is more than a crossnorm it is a reasonable crossnorm that is: its associated dual norm is also a crossnorm.

10 Grothendiek named the following norm the injective norm. Definition Let V i be a Banach space with norm i for 1 i d. Then for d v V = a V j define by v := sup { ( ϕ (1) ϕ (2)... ϕ (d)) (v) d ϕ(j) j : 0 ϕ (j) V j, 1 j d }. Here ϕ j : V j R is a linear and continuous map and over elementary tensors ( ϕ (1) ϕ (2)... ϕ (d)) (v 1 v d ) = It uses the generalized Rayleight quotient. d ϕ j (v j ).

11 Proposition The following statements hold. (a) The injective norm is the weakest reasonable crossnorm over V, that is, if is a reasonable crossnorm over V, then. (b) For any norm over V satisfying, it holds ( d a V j d a j) V. Note that the tensor product of Banach spaces contructed with the, satisfy V = V i V = i=1 i=1 V i

12 Theorem Let V i be a Banach space with norm i for 1 i d and assume that is a norm on V satisfying. Then the set T r weakly closed in V. Example 1: It can be applied for a tensor product product of Hilbert spaces with an inner product induced by the individual ones, L 2 (I) with 0,2 Example 2: Any Banach space with a reasonable crossnorm. Question And H N (I) with N,2 for N 1?

13 Fix N > 0 and consider N := { n N d 0 with n N } (5) Define V n := H n1,p (I 1 ) a a H n d,p (I d ) for each n N, and then V N,alg = n N V n = V (N,...,N) = H N,p (I 1 ) a a H N,p (I d ). The choice of the norm over V n would be ( f n := k f 1/p, dx) p (6) and over V N := n N V n n = n N k n I H n1,p (I 1 ) a a H n d,p (I d ) n. we take f N := ( n N f p n )1/p, which is equivalent to the usual norm N,p.

14 Proposition The set V N,alg is dense in V N with respect the N -topology. Then, by using this Proposition, N,p H N,p (I j ) = H n1,p (I 1 ) a a H n d,p (I d ) n. }{{} V N,alg N n N } {{ } V N (7)

15 Fix integers N j and denote the j-th scale by V j = V (0) j V (1) j... V (N j ) j with dense embedding, (8) which means that V (n) j is a dense subspace of (V (n 1) j, j,n 1 ) for n = 1,..., N j. This fact implies that the corresponding norms satisfy j,n j,m for N j n m 0 on V (n) j. (9) Thus, without loss of generality, we assume that (after suitable scaling) j,n j,m for N j n m 0 and 1 j d, holds.

16 Definition Let N N d 0 be a subset of admissible d-tuples satisfying n N n j N j, (10a) 0 := (0,..., 0) N, (10b) N j := (0,..., 0, N }{{} j, 0,..., 0) = N }{{} j e j N. (10c) j 1 d j The Banach space induced by the intersection of the set {V (n) : n N } of tensor products of Banach spaces is defined by V N := n N V (n) n or an equivalent one. with the intersection norm v N := max n N v n (11)

17 The condition used before has to be adapted to the situation of the intersection space. Consider the tuples N j = (0,..., 0, N j, 0,..., 0) N from (10c) and the corresponding tensor product of Banach spaces V (N j ) = V 1... V j 1 V (N j ) j V j+1... V d+1 endowed with the norm Nj. From now one we denote by (Nj ) the injective norm over the tensor product of Banach spaces V (N j ). Theorem Assume that V N is a Banach space induced by the intersection of the set {V (n) : n N } of tensor products of Banach spaces satisfying (10a-c) and (Nj ) N j for all 1 j d. (12) Then the set T r weakly closed in V.

18 Thank you for your attention!