Partitioned Tensor Representations





P_I1

n=i12⁻ⁿχ{ν∈∇ : |ν|=n}

⊗ χ{ν∈∇ : i2≤|ν|≤I2}, i₁≥ I2, χ_{{ν∈∇ : i1≤|ν|≤I1}}⊗

PI2

n=i22⁻ⁿχ{ν∈∇ : |ν|=n}

, i₂≥ I1,

and applying this on each S∈ Jmax, we obtain an expansion of ¯s|^∇max intoO(jmax) separable terms with disjoint supports.

We again use a separate tensor representation for each section v|S and ¯Av|S, S ∈ Jmax; note that the number of parts in this subdivision is of order O(jmax), as opposed to O(jmax² ) as in (5.26). We thus find that the multilinear rank of each ¯Av|S, S ∈ Jmax, is of order O(jmaxr), and the complexity of applying hosvd for all S can be estimated by

O jmax× (jmax³ |r|³∞) + j_max² log(j_max)|r|²∞max

i=1,2#Λ_i
. (5.30)

As we shall see, for the analogous three-dimensional problem, we can proceed similarly to obtain O((jmaxlog j_max) × (jmax⁴ log⁴j_max|r|⁴∞) + j_max³ log²j_max|r|²∞max_i=1,2#Λ_i). More generally, we shall see in the following subsection that a construction analogous to (5.29) in d dimensions yields a subdivision into O(jmaxlog^d−2j_max) parts, as opposed to O(jmax^d ) in the higher-dimensional version of the subdivision in (5.26). Besides a further improvement in (5.30) by a factor j_max – up to the logarithmic terms – compared to (5.27), we thus also obtain a substantial reduction in the scaling of the number of subdivision elements with respect to the space dimension d.

It should be stressed that the simplified worst-case estimates considered in the preceding dis-cussion will in general be very pessimistic. In particular, our previous considerations concerning possible further gains due to wavelet compressibility apply to the above construction based on ¯s as well. Furthermore, it will generally be useful to apply intermediate recompression steps to partial sums that need to be computed for ¯Av|S, instead of applying hosvd only to the final result as described above for simplicity.

5.2.3 Partitioned Tensor Representations

For d∈ N, we define the operator Sd: `2(∇<sup>d</sup>)→ `2(∇^d) by

S_dv:= (2^{− max|ν|}v_ν)_ν∈∇d, v∈ `2(∇<sup>d</sup>) . (5.31) We now give a recursive characterization of a higher-dimensional generalization of the level parti-tionsJ<sub>`,{1,2}</sub>⁽²⁾ described in the previous subsection. The resulting partitionsJ`,{1,...,d}<sup>(d)</sup> for d > 2 have the property that for an accordingly subdivided tensor representation of a vector v, application of S<sub>d</sub>leaves ranks unchanged, that is, we obtain the same ranks for S<sub>d</sub>v as for v. We shall consider this point in more detail after discussing the definition ofJ`,{1,...,d}^(d) .

For d∈ N and for any D ⊆ {1, . . . , d}, we define the binary vectors b^(d)D ∈ {0, 1}^d by b^(d)_D,i= 1 if

5.2 Tensor Structures for Wavelet Coordinates

i∈ D, and b^(d)_D,i= 0 otherwise. For `∈ N0, let

J<sub>`,{}</sub><sup>(d)</sup> :={0, . . . , 2<sup>`</sup>− 1}^d , d≥ 2 . (5.32) For the case d = 2, in our present notation the subdivision defined in (5.29) reads

J_0,{1,2}⁽²⁾ :=J_0,{}⁽²⁾ ,

J_{`+1,{1,2}</sub><sup>(2)</sup> :=J<sub>`,{1,2}}⁽²⁾ ∪ 2<sup>`</sup>b<sup>(2)</sup><sub>{1}</sub>+J<sub>`,{}</sub>⁽²⁾
∪ 2<sup>`</sup>b<sup>(2)</sup><sub>{2}</sub>+J<sub>`,{}</sub>⁽²⁾
∪ 2<sup>`</sup>b<sup>(2)</sup><sub>{1,2}</sub>+J<sub>`,{1,2}</sub>⁽²⁾
. (5.33) The definition of the higher-dimensional version of this subdivision is recursive both in d and in `.

For d > 2, we set

J0,{1,...,d}^(d) :=J_0,{}^(d) ,

J`+1,{1,...,d}<sup>(d)</sup> :=J`,{1,...,d}^(d) ∪^d−1[

k=1

[

D⊂{1,...,d}

#D=k

2<sup>`</sup>b<sup>(d)</sup><sub>D</sub> +J<sub>`,D</sub>^(d)


∪ 2<sup>`</sup>b<sup>(d)</sup><sub>{1,...,d}</sub>+J`,{1,...,d}^(d)
. (5.34)

To complete the definition (5.34), we still need to define J_`,D^(d) for all D ⊂ {1, . . . , d} with #D ∈ {1, . . . , d − 1} for d > 2. In the case #D = 1, we set

J_{`,{i}</sub><sup>(d)</sup> :=J<sub>`,{}}^(d) , i∈ {1, . . . , d} . (5.35) In the case #D∈ {2, . . . , d − 1}, a recursion with respect to d comes into play: we define

J_`,D^(d):=n \

n∈D

ⁿ⁻¹¡

i=1

{0, . . . , 2^`− 1} × Sn×

¡d i=n+1

{0, . . . , 2<sup>`</sup>− 1} : S∈ J`,{1,...,#D}^(#D) , S = ¡

n∈D

S_n with S_n⊂ N0

o. (5.36)

Note that for each `, all elements of J<sub>`,{1,2}</sub>⁽²⁾ can be written as Cartesian products, and since the recursion steps defined above preserve this property, each element ofJ`,{1,...,d}^(d) can be written as a Cartesian product for d > 2 as well. This justifies the use of such a representation in (5.36).

Example 5.5. Expanding (5.34) in the case d = 3, we obtain J`+1,{1,2,3}<sup>(3)</sup> =J<sub>`,{1,2,3}</sub>⁽³⁾

∪ (2<sup>`</sup>, 0, 0) +J<sub>`,{}</sub>⁽³⁾
∪ (0, 2<sup>`</sup>, 0) +J<sub>`,{}</sub>⁽³⁾
∪ (0, 0, 2<sup>`</sup>) +J<sub>`,{}</sub>⁽³⁾

∪ (2^{`</sup>, 2<sup>`}, 0) +J_{`,{1,2}</sub><sup>(3)</sup>
∪ (2<sup>`</sup>, 0, 2<sup>`</sup>) +J<sub>`,{1,3}}⁽³⁾
∪ (0, 2^{`</sup>, 2<sup>`}) +J_`,{2,3}⁽³⁾

∪ (2^{`</sup>, 2<sup>`}, 2<sup>`</sup>) +J<sub>`,{1,2,3}</sub>⁽³⁾
, where

J<sub>`,{1,2}</sub><sup>(3)</sup> =S1× S2× {0, 2<sup>`</sup>− 1}: S1× S2∈ J_`,{1,2}⁽²⁾

with analogous expressions forJ_{`,{1,3}</sub><sup>(3)</sup> andJ<sub>`,{2,3}}⁽³⁾ . The setJ_3,{1,2,3}⁽³⁾ is illustrated in Figure 5.2.

Proposition 5.6. For eachd≥ 2, the set J`,{1,...,d}<sup>(d)</sup> is a partition of{0, . . . , 2<sup>`</sup>− 1}^dwithO(`<sup>d−2</sup>2<sup>`</sup>) elements, and for each S ∈ J`,{1,...,d}^(d) there exists ¯d ∈ {1, . . . , d} such that for all j ∈ S, we have max j = jd¯.

5 Adaptive Wavelet Schemes for Tensor Representations

Figure 5.2. Structure of the partitionsJ`,{1,2,3}<sup>(3)</sup> for ` = 2, 3.

The proof is given in Appendix A.3. Note that the statement of Proposition 5.6 concerning the cardinality of J`,{1,...,d}^(d) can be rephrased as follows: for given J ∈ N, a partition of {0, . . . , J}^d generated by (5.34) hasO(J log^d−2J) elements.

Remark 5.7. For d = 2, 3, 4, we have

#J<sub>`,{1,2}</sub><sup>(2)</sup> = 3· 2<sup>`</sup>− 2 ,

#J<sub>`,{1,2,3}</sub><sup>(3)</sup> = <sup>9</sup><sub>2</sub>` 2^{`</sup>− 2<sup>`+1}+ 3 ,

#J`,{1,...,4}<sup>(4)</sup> = <sup>9</sup><sub>2</sub>`²2<sup>`</sup>+ ` 2^{`−1</sup>+ 5· 2<sup>`}− 4 for `∈ N0.

On the basis of the partitions J`,{1,...,d}<sup>(d)</sup> of{0, . . . , 2<sup>`</sup>− 1}^d, we define a partition of Z^d_j0 by J˜^(d)= [

`∈N

j₀+J`,{1,...,d}^(d)
, (5.37)

where we use thatJ`,{1,...,d}<sup>(d)</sup> ⊂ J`+1,{1,...,d}^(d) , and additionally choose an enumeration of the countable set ˜J^(d) to obtain the vector (Jn^(d))_n∈N.

The above partitioning can in principle be used for any dimension parameter d, but for what follows we specialize the construction to the two cases that are relevant for the problems we will consider. We define sets of wavelet indices corresponding to ˜J⁽³⁾ by

Λ¯_3,n=ν ∈ ∇³: |ν| ∈ Jn⁽³⁾ , (5.38a) Λ¯6,n=ν = (ν1, ν2, ν3)∈ (∇²)³: max|ν1|, max |ν2|, max |ν3|
∈ Jn⁽³⁾ . (5.38b) The sets ¯Λ_d,n, for d = 3, 6 and n∈ N, are pairwise disjoint andS

nΛ¯_d,n =∇^d. Furthermore, each Λ¯_d,n can be written as a Cartesian product, and we denote by ¯Λ⁽ⁱ⁾_d,n⊂ ∇^d/3, i∈ {1, 2, 3}, the unique

5.2 Tensor Structures for Wavelet Coordinates

lower-dimensional index sets that satisfy ¯Λ_d,n=‘₃

i=1Λ¯⁽ⁱ⁾_d,n.

With the above preparations, we can define the class of tensor representations that we will rely on in what follows: for d = 3, 6 and r = (rn)_n∈N with rn∈ N³0, let

Td(r) :=u ∈ `2(∇<sup>d</sup>) : # supp u <∞ and rank(u|Λ<sup>¯</sup>d,n)≤ rn for all n∈ N . (5.39) Correspondingly, for u∈ `2(∇^d) we define the sequence of multilinear ranks

rank(u) := rank(u|Λ^¯d,n)

n∈N. (5.40)

The following proposition, applied to f (j) = 2^−j, shows that S_dmapsTd(r) toTd(r) for given r, that is, applying S_d to elements of Td(r) does not increase their ranks as in (5.40).

Proposition 5.8. Letf : Zj0 → R and r = (rn)_n∈N withrn∈ N^d0. If u∈ Td(r), then f (max|ν|) uν

ν∈∇^d ∈ Td(r) , d = 3, 6 , and for eachn, there exist S⁽ⁱ⁾_d,n such that S_d|Λ^¯d,n = S⁽¹⁾_d,n⊗ S⁽²⁾_d,n⊗ S⁽³⁾_d,n.

Proof. The statement follows from the observation made in Proposition 5.6 that by the construction of ¯Λ_d,n, for each n∈ N, there exist i3, i₆∈ {1, 2, 3} such that

max|ν| = |νi3| for ν ∈ ¯Λ_3,n, max|˜ν| = max |˜νi6| for ν˜∈ ¯Λ_6,n.

We assume without loss of generality that i₃, i₆ = 1 to simplify notation. Let f := f (max|ν|)

ν∈∇^d

and

f⁽¹⁾:=

( f (|ν1|)

ν∈∇, d = 3 , f (max|ν1|)

ν∈∇², d = 6 . In both cases, we thus obtain the representation

f u|Λ^¯d,n =X

k

a_n,k(f⁽¹⁾U⁽¹⁾_n,k

1)⊗ U⁽²⁾_n,k2⊗ U⁽³⁾_n,k3, d = 3, 6 .

Note that the proof of Proposition 5.8 exposes a scheme for evaluating S_du for u ∈ Td(r) by rescaling one of the mode frames of each u|Λ^¯d,n.

We have thus constructed a class of tensor representations of wavelet coefficients that, by impos-ing additional structure, enables us to efficiently perform rescalimpos-ing operations required by iterative solvers for (5.9). The additional fixed subdivision by wavelet levels will in general lead to approx-imations which are slightly more expensive than those possible by a direct representation in the Tucker format (5.14) without further constraints. This approach can therefore be regarded as a compromise between approximation efficiency and feasibility of computational schemes. It should be noted, however, that the output u ∈ Td(r) of a computational scheme operating on separate tensor representations for each ¯Λ_d,ncan immediately be rewritten as a single tensor representation in the Tucker format (5.14) on all of ∇^d, with formal multilinear rank ˆr := P

nr_n. The actual multilinear rank required for approximating this single combined representation within the target accuracy can typically be expected to be smaller than ˆr, and such an approximate representation with smaller rank can be found by recursively combining the pieces u|Λ^¯d,nby hosvd with appropriate truncation tolerances.

Remark 5.9. Let us now juxtapose the above constructed partitioned tensor representation to a direct tensor representation of wavelet coefficients for u∈ `2(∇^d),

u=X

k

akU_k1 ⊗ Uk2 ⊗ Uk3. (5.41)

5 Adaptive Wavelet Schemes for Tensor Representations

Note that S_d has the following separable expansion: Let P_≤j denote the coordinate projection onto S

i≤j∇i, then

S_d=

∞

X

j=j0

2^−j−1

d

O

i=1

P_≤j. (5.42)

In principle, this could be used directly to operate on vectors of wavelet coefficients represented as in (5.41). As we have seen in Subsection 5.2.2, the resulting bounds on the rank increase, in terms of the maximum arising wavelet level jmax, are substantially less favorable than for the subdivided tensor representation constructed above. However, although a higher power of jmax enters in the worst-case estimates for the direct application of (5.42), there is no dependence on d in terms of log jmax as in the subdivided case.

Remark 5.10. The two cases of, on the one hand, a single tensor representation for all coefficients as in (5.41), and on the other hand, a subdivided representation such that S_d leaves all ranks unchanged, may also be regarded as extreme cases of a more general class of subdivisions. One can group several ¯Λ_d,n such that their union is again a Cartesian product, and use a single tensor representation on each such group. Then S_d does no longer have rank one, but on each group has some bounded rank depending only on the respective group.

In document Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (Page 86-90)