Lecture 2

More documents

Recommendations

Info

Historical remarks – hierarchical Tucker approx. Physics: (I do not have suff. complete knowledge) The ideas are well established in many body quantum physics, quantum optics and quantum information theory 1. DMRG: S. White (91) 2. MPS: Roemmer & Ostlund (94), Vidal (03), Verstraete et al. 3. (tree) tensor networks : Verstrate, Cirac, Schollwöck, Wolf, Eisert .... (?) 4. MERA, PEPS, ..... 5. Chemistry: Multilevel MCTDH � HT : Meyer et al. (2000) Mathematics: Completely new in tensor product approximation 1. Khoromskij et al. (2006), remark about multi level Tucker 2. Lubich (book) (2008): review about Meyers paper 3. HT - Hackbusch (2009) et al.: Introduction of the subspace approximation!!!! 4. Grasedyck (2009): HT -sequential SVD 5. Oseledets & Tyrtishnikov (2009): TT and sequential SVD
Successive SVD for e.g. TT tensors - Vidal (2003), Oseledets (2009), Grasedyck (2009) 1. Matricisation - unfolding F(x1, . . . , x d) � F x2,...,x d x1 low rank approximation up to an error ɛ1, e.g. by SVD. r1� = , F x2,...,x d x1 k1=0 u k1 x1 vx2,...,x d k1 U1(x1, k1) := u k1 x1 , k1 = 1, . . . , r1 . 2. Decompose V (k1, x2, . . . , xd) via matricisation up to an accuracy ɛ2, = r2� V x3,...,x d k1,x2 k2 . u k2 k1,x2 vx3,...,x d k2 U2(k1, x2, k2) := u k2 k1,x2 . 3. � repeat with � V (k2, x3, . . . xd) until one ends with vkd−1,x ↦→ Ud(k d d−1, xd) .
Page 1 and 2: Tensor networks, TT (Matrix Product
Page 3 and 4: Contracted tensor representations E
Page 5 and 6: Contracted tensor representations D
Page 7 and 8: Tensor networks Diagramm: u [x] Dia
Page 9 and 10: Tensor formats ⊲ Hierarchical Tuc
Page 11 and 12: Tensor formats ⊲ Hierarchical Tuc
Page 13 and 14: TT approximations of Friedman data
Page 15 and 16: Hierarchical Tucker as subspace app
Page 17 and 18: Orthonormal basis and recursive des
Page 19 and 20: Hierarchical Tucker tensors ⊲ Can
Page 25: HOSVD bases (Hackbusch) 1) Assume r
Page 29 and 30: Successive SVD for e.g. TT tensors
Page 31 and 32: Closedness A tree T D is characteri
Page 33 and 34: Summary For Tucker and HT redundanc

Lecture 2

Create successful ePaper yourself

Delete template?

Save as template?