Two-Electron Coulomb Potentials

2^{−(|ν|+|µ|)}(ψ_νψ_µ)|x=0



. 2^{−12(|µ|+|ν|)}.

For a given accuracy parameter ` ∈ N, a compressed approximation of this infinite matrix can be defined by setting to zero all entries with|µ| + |ν| > `. The error in spectral norm can be estimated using Lemma 6.8, with weight sequence ω_j = 1, by

X

j>max{j0,`−|ν|}

2^{−12(j+|ν|)}. 2^−12`.

The number of nonzero entries per row and column in the compressed matrix is of order O(`), and hence δ₀ is s^∗-compressible for any s^∗ > 0. This essentially corresponds to the limiting case of the construction in Theorem 6.20 asj_α → ∞.

6.4.2 Two-Electron Coulomb Potentials

In what follows, for i = (i₁, i₂) ∈ Z², we use the notation max i = max{i1, i₂} and min i = min{i1, i₂}. Note that if {2^−|ν|ψ_ν}ν∈∇is a Riesz basis of H¹(R), then by (3.23),{2^{− max|ν|}Ψ_ν}ν∈∇²

is a Riesz basis of H¹(R²).

For the statement of the following theorem, recall the definition of hα in (6.21). In addition, for i, j∈ Z² we introduce the abbreviation

m(i, j) := max{i1, i₂, j₁, j₂} − max {i1, i₂, j₁, j₂ } \ max{i1, i₂, j₁, j₂}
, that is, the difference between the largest and the second largest value in i and j.

Theorem 6.27. Let α > 0 and ψ∈ W^p∞(R) for a p∈ N, then for (aνµ)_ν,µ∈∇² with entries a_νµ=

Z

e^{−α(x1−x2)2}Ψ_µ(x) Ψ_ν(x) dx , ν, µ∈ ∇², (6.49)

6.4 Wavelet Compression of Approximate Potentials

we have

k(|aνµ|)ν,µ∈∇²k`2(∇<sup>2</sup>)→`2(∇²) . (max{1, jα− j0})², (6.50) and for` > 0 there exists a symmetric infinite matrix (˜a<sup>(`)</sup>_νµ) satisfying

√α2− max|µ|−max|ν||aνµ− ˜a^(`)νµ|

ν,µ∈∇²

`2(∇<sup>2</sup>)→`2(∇²). 2^−`, (6.51) where the number of entries in row and column corresponding toν ∈ ∇² can be estimated by

C 1 +p` + (ln α)+
min 1 + (ln α)<sub>+</sub>
22(`−max |ν|)−||ν1|−|ν2||

, max{1, 2^{− max|ν|}√

α}2^(p+14)−1` , (6.52) and hence the maximum number of entries per row and column is bounded by

C 1 +˜ p` + (ln α)+
min 1 + (ln α)<sub>+</sub>
2<sup>2`</sup>, 1 +√

α
2<sup>(p+14)−1`</sup> , (6.53) withC, ˜C > 0 independent of ` and α. In particular, ˜a^(`)_νµ = 0 for ν, µ∈ ∇² if

max|ν| + max|µ| −¹₂(min|ν| + min|µ|)

+ p maxm(|µ|, |ν|) , hα(|ν1|, |µ1|)

++ h_α(|µ2|, |ν2|)

+ > ` . (6.54) Remark 6.28. Theorem 6.27 can be applied to the compression of terms in exponential sum approximations in the same manner as described in Remark 6.21, which yields

√

α 2− max |µ|−max |ν|Y³

i=1

a_νiµi−

3

Y

i=1

˜ a_νiµi

µ,ν∈(∇²)³

`2(∇<sup>6</sup>)→`2(∇⁶)

. max{1, jα− j0}⁴2^−`.

This type of estimate is relevant for the methods considered in Chapter 5.

Remark 6.29. In the proof of the theorem, we restrict ourselves to an integral-order differentiability assumption of ψ. Analogously to the one-electron case treated in Remark 6.24, the result can be extended by interpolation theory to also make use of fractional-order differentiability ofψ.

Proof of Theorem 6.27. For given ν, µ∈ ∇², let S_ν, S_νµ ⊂ R² be the smallest Cartesian products of closed intervals such that supp Ψ_ν ⊆ Sν and supp Ψ_νΨ_µ ⊆ Sνµ. In addition, we introduce β(ν, µ), γ(ν, µ)∈ ∇² with

β_i(ν, µ) =

(µ_i, |µi| < |νi|

ν_i, |µi| ≥ |νi| , γ_i(ν, µ) =

(ν_i, β_i= µ_i

µ_i, β_i= ν_i, i = 1, 2 , (6.55) which we will abbreviate as β, γ in what follows; in other words, γ comprises the wavelet indices on the higher levels, β those on the lower levels for each coordinate direction.

Estimates for matrix entries: By Remark 3.4, ψ has at least p + 1 vanishing moments. We thus obtain, for 0≤ p1, p2 ≤ p,

|aνµ| . kΨγkL∞2^{−p1|γ1|−p2|γ2|} Z

Sγ

D^p_x¹₁D^p_x²₂e^{−α(x1−x2)2}Ψ_β(x₁, x₂)

dx . (6.56)

6 Approximation of Operators

Using Lemma 6.23 we can estimate (6.56) by

2^{−p1|γ1|−p2|γ2|}2^{12(|γ1|+|γ2|)} In summary, with constants depending on ψ and p, similarly to the proof of Theorem 6.20 we obtain

Note that without the use of vanishing moments, a direct application of H¨older’s inequality leads to d = 1, 2, we instead use (6.60). Combining this, we obtain

X

A further estimate can be derived from the observation that a_νµ=

6.4 Wavelet Compression of Approximate Potentials

which, using properties of the convolution, can be estimated further by

. 2^{−(p+12)|ν1|}

p

X

i=0

2^{(p−i+12)|µ1|}

e^−α(·)2 ∗ Dⁱ(ψ_ν2ψ_µ2)

L∞(S_ν2µ2). Note that

sup

x1∈S_ν1µ1

    Z

e^{−α(x1−x2)2}Dⁱ(ψ_ν2ψ_µ2) dx₂   

. 2^{12(|µ2|+|ν2|)}2^{i max{|µ2|,|ν2|}} sup

x1∈S_ν1µ1

Z

S_ν2µ2

e^{−α(x1−x2)2}dx₂,

and since by our assumption on|ν1|, we have |Sν1µ1| . |Sν2µ2| if supp ΨνΨ_µ6= ∅, we obtain X

ν2∈∇

sup

x1∈S_ν1µ1

Z

S_ν2µ2

e^{−α(x1−x2)2}dx₂. α⁻¹².

In summary, for j∈ Z²j0 we obtain the additional estimate X

µ∈∇²

|aνµ| . 2^{− max|ν|}2^−jα2^{12(|ν1|+|ν2|+j1+j2)}2−p m(|ν|,|µ|), (6.62)

which complements (6.61).

Proof of (6.50): Let ω_j = 2^−12(j1+j2)for j∈ Z²j0. Expanding the different cases in (6.61) similarly to step 2 in the proof of Theorem 6.20, we find

ω_|ν|⁻¹X

µ

ω_µ|aνµ| . (max{1, jα− j0})². (6.63)

Construction of compressed matrices and proof of (6.51): Let Θ ∈ W2, and for i, j ∈ Z²j0 and S⊆ R², let

F_s^(i,j)(S) :=

(sup_x∈Se^{−α(x1−x2)2}, i₁, i₂, j₁, j₂≤ jα, sup_x∈Se^{−α2(x1−x2)2}, otherwise.

For t∈ R, we define

Λˆ_s(t) :=(ν, µ) ∈ (∇²)²: F_s^(|ν|,|µ|)(S_νµ) < Θ_|ν|,|µ|2^−t . (6.64) For i, j∈ Z², we define the abbreviations

g(i, j) := max i + max j−1

2(min i + min j) as well as

c_s(i, j) := g(i, j) + p (h_α(j₁, i₁))₊+ (h_α(j₂, i₂))₊
,

¯

c_s(i, j) := g(i, j) + p max{m(i, j) , (hα(j₁, i₁))₊+ (h_α(j₂, i₂))₊} .

Note that ¯c_s is precisely the expression appearing in (6.54). In addition, we define the index sets Λˆ<sup>(`)</sup><sub>s,i,j</sub>:= ˆΛ<sub>s</sub> `− cs(i, j) + (j_α− min{min i, min j})+
.

With this notation, for ` > 0, the compressed matrix is defined by

˜ a^(`)_νµ=

(0 , c¯_s(|ν|, |µ|) > ` or (ν, µ) ∈ ˆΛ<sup>(`)</sup>_s,|ν|,|µ|,

a_νµ, otherwise. (6.65)

6 Approximation of Operators

In other words, entries are dropped from (a_νµ) if it can be ensured that their modulus is small enough either due to the combination of the wavelet levels, or because the Gaussian coefficient function is sufficiently small on the support of the wavelet product. In what follows, we use the simplified notation ˜a_νµ for ˜a^(`)_νµ.

Let now ω_j = 2^{−12max j} for j ∈ Z²j0. From (6.61) and (6.62), we obtain on the one hand 2^jαω_|ν|⁻¹ω_j X

µ∈∇²_j

2− max|ν|−max j|aµν| . 2^−g(|ν|,j)2−p max{m(|ν|,j),P

dhα(|νd|,jd)+}= 2^{−¯cs(|ν|,j)}

for ν ∈ ∇², j∈ Z²_j0. On the other hand, using (6.59), 2^jαω_|ν|⁻¹ω_j X

{µ∈∇²_j: (ν,µ)∈ ˆΛ^(`)_s,|ν|,|µ|}

2− max|ν|−max j|aµν|

. 2^{jα−min|ν|}2^{−cs(|ν|,j)}Θ_|ν|,j2^{−`+cs(|ν|,j)−(jα</sup>−min{min|ν|,min j})+ ≤ Θ|ν|,j2<sup>−`}. Combining these estimates and proceeding as for (6.41) in the proof of Theorem 6.20 yields

2^jαω_|ν|⁻¹ X

µ∈∇²

ω_|µ|2− max |ν|−max j|aνµ− ˜aνµ|

. X

{j : ¯cs(|ν|,j)>`}

2^{−¯cs(|ν|,j)}+ X

j∈Z²_j0

Θ_|ν|,j2^{−`</sup> ≤ CΘ2<sup>−`},

which by Lemma 6.8 implies (6.51).

Estimates for the number of matrix entries: We shall use the abbreviation dα := (jα− j0)+. For ν ∈ ∇², let

n_`(ν, j) := #µ ∈ ∇²:|µ| = (j1, j₂) , ˜a_νµ6= 0 . Without loss of generality, for what follows we assume |ν1| ≥ |ν2|.

By considering only the support sizes of the basis functions, we immediately obtain n_`(ν, j) . 2^{(j1−|ν1|)+}2^{(j2−|ν2|)+}.

In the case j₂ >|ν2|, we improve this estimate by taking the second compression condition in (6.65) into account, which is related to the decay of the Gaussian coefficient. This is done similarly as in the derivation of the condition (6.43) in the proof of Theorem 6.20.

Recall the definition of L in (6.20) as a bound on the support size of the basis functions on level zero. In addition to ν, we fix a µ₁ ∈ ∇ with |µ1| = j1 and S_ν1µ1 6= ∅. Let j2 > |ν2|. We now estimate the number of µ₂ ∈ ∇ with |µ2| = j2 such that (ν, µ) /∈ ˆΛ^(`)_s,|ν|,|µ|.

To this end, note that for ε > 0, the condition sup_x∈Sνµe^{−α(x1−x2)2} < ε is ensured by max

|2^−j2|k(µ2)| − 2^−j1|k(µ1)|| − (2^−j1 + 2^−j2)L , 

|2^−j2|k(µ2)| − 2^−|ν1||k(ν1)|| − (2^−|ν1|+ 2^−j2)L

 & 2^−jαp|ln ε| . Consequently, the number of such µ2 can be estimated up to a constant by

2^j2−jαp|ln ε| + (1 + min{2^j2−j1, 2^j2−|ν1|})L . In summary, we arrive at

n<sub>`</sub>(ν, j) . 2<sup>(j1−|ν1|)+</sup>min1 + 2<sup>j2−jα</sup>p` + d_α+ 2^{j2−max{j1,|ν1|}}, 2^{(j2−|ν2|)+} . (6.66)

6.4 Wavelet Compression of Approximate Potentials

It remains to estimate the sum over all n<sub>`</sub>(ν, j) with j = (j<sub>1</sub>, j<sub>2</sub>) satisfying ¯c<sub>s</sub>(|ν|, j) ≤ ` by a constant multiple of (6.52).

For givenJ ⊂ Z²j0, we introduce the abbreviation N (J ) :=X

j∈J

n`(|ν|, j) .

At several points we will make use of the fact that for any ˜c : Z²_j0 × Z²j0 → Z with ˜c ≤ cs ≤ ¯cs and anyJ ⊂ Z²j0,

N {j ∈ J : ˜c(|ν|, j) < `}

≥ N {j ∈ J : cs(|ν|, j) < `}

≥ N {j ∈ J : ¯cs(|ν|, j) < `}
. In particular, from (6.66) it can be seen that replacing ¯c_sby the lower bound c_sdoes not change the asymptotic behaviour of the estimate for the number of nonzero entries. As illustrated by Example 6.30 below, the quantitative difference is of practical importance. In the following estimates for the asymptotics, however, we only consider c_s.

We first treat the case |ν1| ≥ |ν2| > jα, where (6.66) implies

n_`(ν, j) . 2^{(j1−|ν1|)++(j2−|ν2|)+}. (6.67) We consider first the summation over the corresponding subset of J1 = {j ∈ Z²_j0: j1, j2 > jα}.

Note that for j∈ J1, we have c_s(|ν|, j) = max j −¹₂min j +|ν1| −¹₂|ν2| + p(||ν1| − j1| + ||ν2| − j2|).

We subdivide the summation further into

N (J1) = N (J1∩ {j1<|ν1|, j2 <|ν2|}) + N(J1∩ {j1 ≥ |ν1|, j2 <|ν2|})

+ N (J1∩ {j1 <|ν1|, j2≥ |ν2|}) + N(J1∩ {j1 ≥ |ν1|, j2 ≥ |ν2|}) , (6.68) and treat each term on the right hand side separately. By (6.66),

N (J1∩ {j1 <|ν1|, j2<|ν2|}) . (`/p)².

For j∈ J1∩{j1<|ν1|, j2 ≥ |ν2|}, we have cs(|ν|, j) ≥ (p+¹₂)(j₂−|ν2|)+|ν1| ≥ (p+¹₂)(j₂−|ν2|)+j0

as well as c_s(|ν|, j) ≥ p(|ν1| − j1) + j₀ and thus

N (J1∩ {j1<|ν1|, j2≥ |ν2|}) . ` 2<sup>(p+12)−1(`−j0)</sup>.

Similarly, for j ∈ J1 ∩ {j1 ≥ |ν1|, j2 < |ν2|}, we have cs(|ν|, j) ≥ (p + ¹₂)(j₁− |ν1|) + |ν1| and c_s(|ν|, j) ≥ p(|ν2| − j2) + j₀, and consequently also

N (J1∩ {j1≥ |ν1|, j2<|ν2|}) . ` 2<sup>(p+12)−1(`−j0)</sup>.

Finally, for j∈ J2 :=J1∩{j1≥ |ν1|, j2 ≥ |ν2|}, we have cs(|ν|, j) ≥ (p+¹₂)(j₁−|ν1|)+p(j2−|ν2|) =:

˜

c₁(|ν|, j) if j1 ≥ j2, and c_s(|ν|, j) ≥ p(j1− |ν1|) + (p +¹₂)(j₂− |ν2|) =: ˜c2(|ν|, j) if j1< j₂. Hence we obtain

N (J2) . X

j∈J2∩{j1≥j2}

˜

c1(|ν|,j)≤`

2^j1−|ν1|2^j2−|ν2|+ X

j∈J2∩{j1<j2}

˜

c2(|ν|,j)≤`

2^j1−|ν1|2^j2−|ν2|

.

s1

X

j1=|ν1| s2(j1)

X

j2=|ν2|

2^j1−|ν1|2^j2−|ν2| . 2^(p+14)−1`,

where s₁:=b(2p + ¹₂)⁻¹(` + p|ν1| + (p +<sup>1</sup><sub>2</sub>)|ν2|)c, s2(j<sub>1</sub>) :=b(p +<sup>1</sup><sub>2</sub>)<sup>−1</sup> `− p(j1− |ν1|)
+ |ν2|c.

We have thus already completely covered the case j_α< j₀, and therefore assume for the following

6 Approximation of Operators

that j_α ≥ j0.

If |ν1| ≥ |ν2| > jα and j_α ≥ j0, we additionally need to sum over J3 := {j ∈ Z²j0: j₁, j₂ ≤ j_α, c_s(|ν|, j) ≤ `}, which is empty unless ` ≥ (jα+ j₀)/2; since for j ∈ J3 we have n<sub>`</sub>(|ν|, j) . 1, we obtain N (J3) . `². By estimates completely analogous to those for (6.68), we also find

N ({j1 > j_α, j₂≤ jα, c_s(|ν|, j) ≤ `}), N({j2≤ jα, j<sub>2</sub> > j<sub>α</sub>, c<sub>s</sub>(|ν|, j) ≤ `}) . `2<sup>(p+1)−1`</sup>, which concludes the treatment of the case|ν1| ≥ |ν2| > jα.

We next consider the case |ν1| > jα ≥ |ν2|, where we have cs(|ν|, j) = g(|ν|, j) + p||ν1| − max{j1, j_α}| + p(j2 − jα)₊ and n_`(|ν|, j) . 2^{(j1−|ν1|)+}2^(j2−jα)+(1 +√

` + d_α) + 2^j2−|ν1|. Noting that j₂ − |ν1| < j2 − jα, we can proceed analogously to the case of |ν1| ≥ |ν2| > jα above, by distinguishing cases depending on the signs of j₁− |ν1| and j2− jα, to likewise obtain

N ({j ∈ Z²j0: c_s(|ν|, j) ≤ `}) . 2<sup>(p+14)−1`</sup>(1 +p` + dα) for|ν1| > jα≥ |ν2|.

If |ν1|, |ν2| ≤ jα, the number of nonzero entries can be estimated by N (J4) . X

j∈J4

(2^j1−|ν1|+ 2^j2−|ν1|+ 2^{j1+j2−|ν1|−jα}p` + d_α) , (6.69)

whereJ4 =j ∈ Z²_j0: max j +|ν1| −¹₂(min j +|ν2|) + p(j1− jα)₊+ p(j₂− jα)₊≤ ` .

For estimating the right hand side of (6.69) further, we consider two cases: First, if ` ≤ ¹₂jα+

|ν1| −¹₂|ν2|, the summation extends only over certain j1, j₂ ≤ jα, and (6.69) can be estimated by

X

j: max j−¹₂min j

≤`−|ν1|+¹₂|ν2|

2^{max j−|ν1|}(2 + 2^{min j−jα}p` + dα) .

s2

X

j2=j0

(1 + 2^j2−jαp` + dα)

s1(j2)

X

j1=j0

2^j1−|ν1|

. 2<sup>2`−3|ν1|+|ν2|</sup>(1 +p` + d_α2^{2`−jα−2|ν1|+|ν2|})

≤ 2<sup>2`−3|ν1|+|ν2|</sup>(1 +p` + dα) ,

where s₂ :=b2` − 2|ν1| + |ν2|c, s1(j<sub>2</sub>) :=b` +¹₂j₂− |ν1| + ¹₂|ν2|c; note that by the assumption on

`, we have 2<sup>2`−3|ν1|+|ν2|</sup>≤ 2^jα−|ν1|, and we thus have the sought estimate of the number of matrix entries by (6.52) in this case.

Second, in case that ` > ¹₂j_α +|ν1| − ¹₂|ν2|, the partial sum for j1, j₂ ≤ jα can be estimated similarly by

(d_α+p` + d<sub>α</sub>)2<sup>jα−|ν1|</sup>≤ (dα+p` + d_α)2^{4p+14 `−4p+54p+1|ν1|+4p+12 |ν2|+4p−14p+1jα},

where we have added (p +¹₄)⁻¹(`−¹₂jα+|ν1| −¹₂|ν2|) > 0 in the exponent to obtain the right hand side, which in turn can be bounded by a constant multiple of (6.53), that is,

. (1 +p` + d<sub>α</sub>) min(1 + d<sub>α</sub>)2<sup>2`−3|ν1|+|ν2|</sup>, 2^jα2^{(p+14)−1`−|ν1|} .

6.4 Wavelet Compression of Approximate Potentials

For the partial sum over max j > j_α, min j≤ jα, we obtain

X

j: max j−¹₂min j +p(max j−jα)

≤`−|ν1|+¹₂|ν2|

2^{max j−|ν1|}(1 + 2^{min j−jα}p` + dα) =

bjαc

X

j2=j0

(1 + 2^j2−jαp` + dα)

s1(j2)

X

j1=jα

2^j1−|ν1|

. 2^(p+1)−1`+

2p+1

2p+2jα−^p+2

p+1|ν1|+_2(p+1)¹ |ν2|

(1 +p` + dα) . 2<sup>jα</sup>2<sup>(p+14)−1`−|ν1|</sup>(1 +p` + d_α)

with s1(j2) := b(p + 1)⁻¹(` + ¹₂j2 + pjα − |ν1| + ¹₂|ν2|)c. Note that in this case, because jα <

2`− 2|ν1| + |ν2| we still have

2^{p+1` +2p+12p+2jα−p+2p+1|ν1|+2(p+1)1 |ν2|</sup>< 2<sup>2`−3|ν1|+|ν2|}.

For the partial sum over j₁, j₂ > j_α, the condition on c_s reads (p + 1) max j + (p− ¹₂) min j ≤

`− |ν1| +¹₂|ν2| + 2pjα, which leads to an estimate by

(1 +p` + dα)

s2

X

j2=jα

s1(j2)

X

j1=j2

2^j1−|ν1|2^j2−jα ≤ 2<sup>(p+14)−1`−4p+54p+1|ν1|+4p+12 |ν2|+4p−14p+1jα</sup>(1 +p` + dα) . 2^jα2<sup>(p+14)−1`−|ν1|</sup>(1 +p` + dα)

with s₁(j₂) :=b(p+1)⁻¹(`−|ν1|+<sup>1</sup><sub>2</sub>|ν2|+2pjα−(p−<sup>1</sup><sub>2</sub>)j<sub>2</sub>)c, s2:=b(2p+<sup>1</sup><sub>2</sub>)<sup>−1</sup>(`−|ν1|+¹₂|ν2|+2pjα)c.

Again, from ` > ¹₂j_α+|ν1| − ¹₂|ν2| it follows that

2^{4p+14 `−4p+54p+1|ν1|+4p+12 |ν2|+4p−14p+1jα</sup> < 2<sup>2`−3|ν1|+|ν2|},

which completes our analysis for the case|ν1|, |ν2| ≤ jα. Note that the maximum number of entries arises in the case|ν| = (j0, j0).

Example 6.30. As in Example 6.25, we compare the estimates for matrix entries in the proof of Theorem 6.27 to the numerical observation. We consider

L_j1,j2 :=√

α 2^{− max{j1,j2}} X

µ∈∇²_(j1,j2)

    Z

e^{−α(x1−x2)2}Ψ_µΨ_ν0dx     with ν₀ = ((0, 0, 0), (0, 0, 0))∈ ∇² and α = 10², 10⁴, corresponding to j_α= 3.82, 7.14.

In view of the estimates in the proof of Theorem 6.20, for j₁, j₂≥ 0 we expect

L_j1,j2 ≤ C2^{−12|j1−j2|}2^{−p max{|j1−j2|,(j1−jα)++(j2−jα)+}} (6.70) with some C > 0 and with p depending on the wavelet basis. Here we use the same wavelet as in Example 6.30.

As can be seen from Figures 6.4 and 6.5, the estimate (6.70) captures the essential qualitative behaviour. However, similarly to Example 6.25, the predicted values for jα and p yield an overesti-mate. The lines with markers show the actual values of L_j1,j2 for the two values of α. The dashed grey lines show the right hand side of (6.48) with C = 80, p = 4.3, and j_α as above, whereas the dashed black lines show these reference values with C = 350, p = 5.4, and jα= 2.5, 6.0. The latter reproduce the observed decay more accurately.

Remark 6.31. There is a similar interpretation for the resulting compressibility as in Remark

6 Approximation of Operators

10⁻¹⁰ 10⁻⁵ 10⁰

0, 0 0, 1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 0, 8 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 2, 2 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 4, 4 4, 5 4, 6 4, 7 4, 8 5, 5 5, 6 5, 7 6, 6

Figure 6.4. Actual values and estimates of Lj1,j2 as in Example 6.30 with α = 10².

10⁻¹⁰ 10⁻⁵ 10⁰

0, 0 0, 1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 0, 8 0, 9 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, 9 2, 2 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 2, 9 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, 9 4, 4 4, 5 4, 6 4, 7 4, 8 4, 9 5, 5 5, 6 5, 7 5, 8 5, 9 6, 6 6, 7 6, 8 6, 9 7, 7 7, 8 7, 9 8, 8 8, 9

Figure 6.5. Actual values and estimates of Lj₁,j₂ as in Example 6.30 with α = 10⁴.

6.26. For large α,

√π⁻¹α Z

R²

e^{−α(x1−x2)2}Ψ_µ(x) Ψ_ν(x) dx≈ Z

R

Ψ_µ(x, x) Ψ_ν(x, x) dx =: m_νµ.

Let ν ∈ ∇² and j ∈ Z²j0, then P

µ∈∇²_j|mνµ| . 2^{− max |ν|}2^{12(|ν1|+|ν2|+j1+j2)}, and the total number of nonzero entries for the level combination (|ν|, j), with this fixed ν, is of order 2(max j−max |ν|)+.

If we now compress M := (2− max|ν|−max|µ|m_νµ) by setting to zero all entries for which max|µ| + max|ν| − ¹₂(min|µ| + min |ν|) > `, we thus find with Lemma 6.8, using the weight sequence ωj = 2^{−12max j}, that M is s^∗-compressible with s^∗ = ¹₂. This value corresponds to the first term in the minimum in (6.53); the second term, however, can yield a better compressibility depending on α, which will be considered next.

From Theorem 6.27, we can derive a result concerning the compressibility of the full six-dimensional Coulomb interaction potential as well. To this end, we additionally estimate the arising parameters α.

Corollary 6.32. For a tensor product wavelet basis{Ψν}ν∈∇⁶ constructed from a univariate

wave-6.4 Wavelet Compression of Approximate Potentials

of the two-electron Coulomb potential considered as a multiplication operator H¹(R⁶) → H⁻¹(R⁶) iss^∗-compressible with

s^∗ = 4p + 1 4p + 5

 1 3.

Proof. In this proof, let C denote a generic positive constant. For given δ > 0, Theorem 4.17 yields an exponential sum approximation of t7→ t^−1/2with error in supremum norm on [1,∞) bounded by δ, with N .|ln δ|² terms. Let ω_δ,k, α_δ,k, k = 1, . . . , N , denote the corresponding coefficients. Using Theorem 6.9, which here can be applied with R =∞, we obtain an exponential sum approximation with coefficients ˆω_k:= r⁻¹ω_δ,k, ˆα_k:= r⁻²α_δ,kfor the two-electron Coulomb potential|x−y|⁻¹with 6.28, we obtain an approximation for each term in the exponential sum approximation with error bounded by C(ˆω_k/√

Remark 6.33 (Resulting compressibility of factor matrices for relevant choices of α). Taking the maximum size of the parameterα required for a certain error in the exponential sum approximation of the Coulomb potential into account, Theorem 6.27 can yield better compressibility thans^∗ = ¹₂ for the factor matrices (√

α2− max|µ|−max|ν|a_νµ)_ν,µ∈∇², with a_νµ as in (6.49). For an exponential sum approximation based on Theorem 4.17, the same argument as in the proof of Corollary 6.32 yields s^∗ = (4p + 1)/(4p + 5) for these lower-dimensional components – in other words, one can come arbitrarily close to s^∗ = 1 for large p. Note that this is substantially better than the worst case, for generalα, as considered in Remark 6.31.

In document Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (Page 124-133)