Bounded condition numbers

Chapter IV: Proving exponential decay of Cholesky factors

4.5 Bounded condition numbers

In this section, we will bound the condition numbers of𝐵^(𝑘) based on the following condition, which we will show to be satisfied for Examples1and2.

Condition 2. Let𝐻 ∈ (0,1), 𝐶_Φ ≥ 1be constants such that for1 ≤ 𝑘 < 𝑙 ≤ 𝑞, 𝜆_min Θ^(𝑘)

≥ 1 𝐶_Φ

𝐻^2𝑘, (4.25)

𝜆_max Θ^(𝑞)

𝑙 ,𝑙 −Θ^(𝑞)

𝑙 ,1:𝑘Θ^(𝑞),−1

1:𝑘 ,1:𝑘Θ^(𝑞)

1:𝑘 ,𝑙

≤ 𝐶_Φ𝐻^2𝑘. (4.26) Theorem 5. Condition2implies that, for all1≤ 𝑘 ≤ 𝑞,

𝐶⁻¹

Φ 𝐻^{−2(𝑘−1)}Id≺ 𝐵^(𝑘) ≺ 𝐶_Φ𝐻^−2𝑘Id, (4.27)

and, for𝜅 B 𝐻⁻²𝐶²

Φ,

cond 𝐵^(𝑘)

≤ 𝜅 . (4.28)

Proof. The lower bound in (4.27) follows from (4.26) and 𝐵^(𝑘) = Θ^(𝑞)

𝑘 , 𝑘−Θ^(𝑞)

𝑘 ,1:(𝑘−1)Θ^(𝑞),−1

1:𝑘 ,1:𝑘Θ^(𝑞)

1:(𝑘−1), 𝑘

⁻1

. (4.29)

The upper bound in (4.27) follows from (4.25) and 𝐵^(𝑘) = Θ^(𝑘)−1

𝑘 , 𝑘.

The following theorem shows that (4.26) is a Poincaré inequality closely related to the accuracy of numerical homogenization basis functions [120, 163, 191] and (4.25) is an inverse Sobolev inequality related to the regularity of the discretization ofL:

Theorem 6. Condition2holds true if the constants𝐶_Φ ≥ 1and𝐻 ∈ (0,1)satisfy (1) _𝐶¹

Φ

𝐻^2𝑘 ≤ ^k𝜙k∗2

|𝛼|², for𝛼 ∈R^𝐼

(𝑘)

and𝜙=Í

𝑖∈𝐼^(𝑘) 𝛼_𝑖𝜙_𝑖; and (2) min𝜑∈span(𝜙𝑖)

𝑖∈𝐼(𝑘−1)

k𝜙−𝜑k_∗²

|𝛼|² ≤ 𝐶_Φ𝐻^2(𝑘−1), for 𝛼 ∈ R^𝐽

(𝑙), 𝑘 < 𝑙 ≤ 𝑞, and 𝜙 =Í

𝑖∈𝐽^(𝑙) 𝛼_𝑖𝜙_𝑖.

Proof. Inequality (4.25) is a direct consequence of the first assumption of the the- orem, whereas (4.26) follows from the variational property [258, Theorem 5.1] of

the Schur complement:

𝛼^>

Θ𝑙 ,𝑙−Θ^(𝑞)

𝑙 ,1:𝑘Θ^(𝑞),−1

1:𝑘 ,1:𝑘Θ^(𝑞)

1:𝑘 ,𝑙

𝛼= inf

𝛽∈R^𝐼(𝑘)

(𝛼−𝛽)^>Θ^(𝑞)(𝛼− 𝛽) (4.30)

= min

𝜑∈span{𝜙_𝑖|𝑖∈𝐼^(𝑘)}

k𝜙−𝜑k_∗² ≤ 𝐶_Φ𝐻^2𝑘|𝛼|². (4.31) We will now show that Examples1and2satisfy the conditions of Theorem6. For simplicity, for ˜Ω ⊂ Ω and 𝜙 ∈ 𝐻^−𝑠(Ω), we still write 𝜙 for the unique element

˜

𝜙 ∈ 𝐻^−𝑠(Ω)˜ such that [𝜙, 𝑢˜ ] = [𝜙, 𝑢] for 𝑢 ∈ 𝐻^𝑠

0(Ω)˜ . The following Fenchel conjugate identity [40, Ex. 3.27, p. 93] will be useful throughout this section.

k𝜙k²

𝐻^−𝑠(Ω) = sup

𝑣∈𝐻^𝑠

0(Ω)

2[𝜙, 𝑣] − k𝑣k²

𝑣∈𝐻^𝑠

0(Ω). (4.32)

The first condition can be verified in a similar way as is done in [189].

Lemma 3. LetΘbe given as in Examples1and2. Then there exists a constant𝐶 depending only on𝛿,𝑠, and𝑑, such that

1 𝐶_Φ

ℎ^{2𝑠 𝑘} ≤ k𝜙k_∗²

|𝛼|² , (4.33)

for𝐶_Φ= kL k𝐶,𝛼 ∈R^𝐼

(𝑘)

, and𝜙=Í

𝑖𝛼_𝑖𝜙_𝑖.

Proof. The proof can be found in Section.1.

In order to verify the second condition in Theorem 6, we will construct a 𝜑 such that𝜙−𝜑integrates to zero against polynomials of order at most𝑠−1 on domains of size ℎ^𝑘. Then an application of the Bramble–Hilbert lemma [65] will yield the desired factorℎ^{𝑘 𝑠}. To avoid scaling issues, we define, for 1 ≤ 𝑘 ≤ 𝑞and𝑖 ∈ 𝐼^(𝑘),

𝜙⁽

𝑘)

𝑖 B











𝜹𝑥_𝑖, in Example1, 1_𝜏^(𝑘)

𝑖

/|𝜏^(𝑘)

𝑖 |, in Example2,

(4.34)

noting that span{𝜙^(𝑘)

𝑖 | 𝑖 ∈ 𝐼^(𝑘)} = span{𝜙_𝑖 | 𝑖 ∈ 𝐼^(𝑘)}. To obtain estimates independent of the regularity ofΩ, for the simplicity of the proof and without loss of generality, we will partially work in the extended space R^𝑑 (rather than onΩ).

We write𝑣for the zero extension of𝑣 ∈ 𝐻^𝑠

0(Ω)to𝐻^𝑠(R^𝑑)and𝜙⁽

𝑘)

𝑖 for the extension of 𝜙⁽

𝑘)

𝑖 ∈ 𝐻^−𝑠(Ω) to an element of the dual space of𝐻^𝑠

loc(R^𝑑). We introduce new

54 measurement functions in the complement of Ω as follows. For 1 ≤ 𝑘 ≤ 𝑞, we consider countably infinite index sets ˜𝐼^(𝑘) ⊃ 𝐼^(𝑘). We choose points (𝑥_𝑖)_{𝑖∈𝐼˜}(𝑞)\𝐼^(𝑞)

satisfying sup

𝑥∈R^𝑑\Ω

min

𝑖∈𝐼˜^(𝑘)

dist(𝑥_𝑖, 𝑥) ≤𝛿⁻¹ℎ^𝑘, min

𝑖≠𝑗∈𝐼˜^(𝑘)\𝐼^(𝑘)

dist(𝑥_𝑖, 𝑥_𝑗∪𝜕Ω) ≥𝛿 ℎ^𝑘. (4.35) We then define, for 1 ≤ 𝑘 ≤ 𝑞 and 𝑖 ∈ 𝐼˜^(𝑘), 𝜙^(𝑘)

𝑖 B 𝛿_𝑥

𝑖 for Example 1, and 𝜙⁽

𝑘)

𝑖 B

1𝐵 𝛿 ℎ𝑘(𝑥𝑖)

|𝐵

𝛿 ℎ𝑘(𝑥_𝑖) | for Example2. LetP^𝑠−1denote the linear space of polynomials of degree at most𝑠−1 (onR^𝑑).

Lemma 4. Let Θ be as in Example 1 or Example 2. Given 𝜌 ∈ (2,∞) and 1≤ 𝑘 < 𝑙 ≤ 𝑞, let𝑤 ∈R^𝐽

(𝑙)×𝐼˜^(𝑘)

be such that

∫

𝐵

𝜌 ℎ𝑘(𝑥𝑖)

©

­

«

𝜙_𝑖− Õ

𝑗∈𝐼˜^(𝑘)

𝑤_{𝑖 𝑗}𝜙⁽

𝑘) 𝑗

ª

®

¬

(𝑥)𝑝(𝑥)d𝑥 =0, for all𝑝 ∈ P^𝑠−1and𝑖 ∈𝐽^(𝑙) (4.36) and𝑤_{𝑖 𝑗} ≠ 0 ⇒ supp

𝜙⁽

𝑘) 𝑗

⊂ 𝐵

𝜌 ℎ^𝑘(𝑥_𝑖). Then, for𝛼 ∈ R^𝐽

(𝑙)

, 𝜙 B Í

𝑖∈𝐽^(𝑙) 𝛼_𝑖𝜙_𝑖 and 𝜑 BÍ

𝑖∈𝐽^(𝑙), 𝑗∈𝐼^(𝑘) 𝛼_𝑖𝑤_{𝑖 𝑗}𝜙^(𝑘)

𝑗 satisfy

k𝜙−𝜑k_∗² ≤ kL⁻¹k𝐶(𝑑 , 𝑠)𝜌^𝑑+2𝑠 𝛿^𝑑

1+ℎ^{−𝑙 𝑑}𝜔²

𝑙 , 𝑘

ℎ^{2𝑠 𝑘}|𝛼|², (4.37) with 𝜔_{𝑙 , 𝑘} B sup_𝑖∈𝐽(𝑙)Í

𝑗∈𝐼˜^(𝑘) |𝑤_{𝑖 𝑗}| and k𝜙k_∗ B sup_𝑢∈𝐻^𝑠

0(Ω)[𝜙, 𝑢]/[L𝑢, 𝑢]¹² as in (4.7).

We proceed by proving Lemma 4 in the setting of Example 1. The proof in the setting of Example 2can be found in Section .1. For 𝑢 ∈ 𝐻^𝑠(Ω), write D⁰𝑢 B 𝑢 and for 1 ≤ 𝑘 ≤ 𝑠, write D^𝑘𝑢for the vector of partial derivatives of𝑢of order𝑘, i.e.

D^𝑘𝑢 B

𝜕^𝑘𝑢

𝜕𝑖 1···𝜕𝑖

𝑘

𝑖₁,...,𝑖𝑘=1,...,𝑑

. The proof of Lemma4will use the following version of the Bramble–Hilbert lemma:

Lemma 5 ([65]). Let Ω ⊂ R^𝑑 be convex and let 𝜙 be a sublinear functional on 𝐻^𝑠(Ω)for 𝑠∈Nsuch that

(1) there exists a constant𝐶˜such that, for all𝑢 ∈ 𝐻^𝑠(Ω),

|𝜙(𝑢) | ≤𝐶˜

𝑠

Õ

𝑘=0

diam(Ω)^𝑘kD^𝑘𝑢k_𝐿2(Ω); (4.38)

(2) and𝜙(𝑝)=0for all 𝑝 ∈ P^𝑠−1.

Then, for all𝑢 ∈𝐻^𝑠(Ω),

|𝜙(𝑢) | ≤𝐶 𝐶˜ (𝑑 , 𝑠)diam(Ω)^𝑠kD^𝑠𝑢k_𝐿2(Ω). (4.39) The following lemma is obtained from Lemma5:

Lemma 6. For 1 ≤ 𝑘 < 𝑙 ≤ 𝑞 and 𝑖 ∈ 𝐽^(𝑙), let 𝜙_𝑖, 𝑤_{𝑖 𝑗} be as in Lemma 4 and Example2 and define 𝜑_𝑖 B Í

𝑗∈𝐼^(𝑘)𝑤_{𝑖 𝑗}𝜙^(𝑘)

𝑗 . Then there exists a constant𝐶(𝑑 , 𝑠) such that, for all𝑣 ∈𝐻^𝑠

0(Ω),

∫

𝐵 𝜌 ℎ𝑘(𝑥𝑖)

(𝜙_𝑖−𝜑_𝑖) (𝑥)𝑣(𝑥)d𝑥

≤𝐶(𝑑 , 𝑠)𝜌^{𝑠−𝑑/2}ℎ^{(𝑠−𝑑/2)𝑘}©

­

«

ℎ^{𝑙 𝑑/2}+ Õ

𝑗∈𝐼˜^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬ kD^𝑠𝑣k

𝐿²

𝐵 𝜌 ℎ𝑘(𝑥𝑖).

(4.40)

Proof. We apply Lemma5to the linear functional𝑢 ↦→∫

𝐵

𝜌 ℎ𝑘

(𝜙_𝑖−𝜑_𝑖)𝑢. Since the second requirement of Lemma 5is fulfilled by definition, it remains to bound ˜𝐶. We only execute the proof for Example1; the proof for Example2is analogous. We first note that while the sum in the definition of𝜑_𝑖only ranges over 𝑗 ∈ 𝐼^(𝑘), we can increase it to run over all of 𝑗 ∈ 𝐼˜^(𝑘), since for 𝑗 ∈ 𝐼˜^(𝑘) \𝐼^(𝑘), the support of𝜙⁽

𝑘) 𝑗 is disjoint from that of𝑣 ∈𝐻^𝑠

0(Ω). Let𝑢 ∈ 𝐻^𝑠(Ω). Writing𝐶(𝑑 , 𝑠)for the continuity constant of the embedding of𝐻^𝑠(𝐵₁(0))into𝐶_𝑏(𝐵₁(0)), the inequalities

max

𝐵

𝜌 ℎ𝑘(𝑥𝑖)|𝑢( · ) |= max

𝑥∈𝐵₁(0)

𝑢

𝜌 ℎ^𝑘(𝑥−𝑥_𝑖)

≤𝐶(𝑑 , 𝑠)

𝑠

Õ

𝑚=0

(𝜌 ℎ^𝑘)^𝑚

[D^𝑚𝑢] 𝜌 ℎ^𝑘( · −𝑥_𝑖) _𝐿2(𝐵₁(0))

and

[D^𝑚𝑢] 𝜌 ℎ^𝑘( · −𝑥_𝑖)

𝐿²(𝐵₁(0)) = (𝜌 ℎ^𝑘)^−𝑑/2kD^𝑚𝑢k_𝐿2(𝐵

𝜌 ℎ𝑘(𝑥_𝑖))

imply that

|𝜙_𝑖(𝑢) −𝜑_𝑖(𝑢) | ≤©

­

«

ℎ^{𝑙 𝑑/2}+ Õ

𝑗∈𝐼˜^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬ max

𝑥∈𝐵 𝜌 ℎ𝑘(𝑥_𝑖)

|𝑢(𝑥) | (4.41)

≤𝐶(𝑑 , 𝑠)𝜌^−𝑑/2ℎ^{−𝑘 𝑑/2}©

­

«

ℎ^{𝑙 𝑑/2}+ Õ

𝑗∈𝐼˜^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬

𝑠

Õ

𝑚=0

(𝜌 ℎ^𝑘)^𝑚kD^𝑚𝑢k𝐿2(𝐵

𝜌 ℎ𝑘(𝑥_𝑖)).

(4.42)

Therefore the first condition of Lemma5holds with

˜

𝐶 =𝐶(𝑑 , 𝑠)𝜌^−𝑑/2ℎ^{−𝑘 𝑑/2}©

­

«

ℎ^{𝑙 𝑑/2}+ Õ

𝑗∈𝐼˜^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬

, (4.43)

and we conclude the proof by writing𝐶(𝑑 , 𝑠)for any constant depending only on𝑑

and𝑠.

56 We can now conclude the proof of Lemma4.

Proof of Lemma4. Write 𝜑 B Í

𝑖∈𝐽^(𝑙)𝛼_𝑖𝜑_𝑖 and 𝜑_𝑖 B Í

𝑗∈𝐼^(𝑘) 𝑤_{𝑖 𝑗}𝜙⁽

𝑘)

𝑗 . Equa- tion (4.32) implies that

k𝜙−𝜑k²

𝐻^−𝑠(Ω) = sup

𝑣∈𝐻^𝑠

0(Ω)

Õ

𝑖∈𝐽^(𝑙)

2𝛼_𝑖

∫

𝐵

𝜌 ℎ𝑘(𝑥𝑖)

(𝜙_𝑖−𝜑_𝑖) (𝑥)𝑣(𝑥)d𝑥

!

− k𝑣k²

𝐻^𝑠

0(Ω). (4.44) The packing inequality Í

𝑖∈𝐽^(𝑙) kD^𝑠𝑣k²

𝐿2 𝐵

𝜌 ℎ𝑘(𝑥_𝑖) ≤ 𝐶(𝑑) ℎ^𝑘−𝑙𝜌/𝛿^𝑑 k𝑣k²

𝐻^𝑠

0(Ω) to- gether with Lemma6yields

k𝜙−𝜑k²_𝐻−𝑠(Ω)≤ sup

𝑣∈𝐻₀^𝑠(Ω)

Õ

𝑖∈𝐽^(𝑙)

"

2|𝛼_𝑖|𝐶(𝑑 , 𝑠)𝜌^𝑠−

𝑑 2ℎ^(𝑠−

𝑑 2)𝑘©

­

« ℎ

𝑙 𝑑

2 + Õ

𝑗∈𝐼^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬

kD^𝑠𝑣k_𝐿2(𝐵 𝜌 ℎ𝑘(𝑥𝑖))

(4.45)

− (𝐶(𝑑))⁻¹ ℎ^𝑘−𝑙𝜌/𝛿

−𝑑

kD^𝑠𝑣k²

𝐿²

𝐵 𝜌 ℎ𝑘(𝑥𝑖)

#

. (4.46)

Applying the inequality 2𝑎𝑥−𝑏𝑥² ≤ 𝑎²/𝑏to each summand yields

k𝜙−𝜑k_𝐻²−𝑠(Ω) ≤𝐶(𝑑) ℎ^𝑘−𝑙𝜌/𝛿

^𝑑 Õ

𝑖∈𝐽^(𝑙)

©

­

«

𝛼_𝑗𝐶(𝑑 , 𝑠)𝜌^𝑠−

𝑑 2ℎ^(𝑠−

𝑑 2)𝑘©

­

« ℎ

𝑙 𝑑

2 + Õ

𝑗∈𝐽^(𝑘)

|𝑤_{𝑖 𝑗}|ª

®

¬ ª

®

¬

2

(4.47)

≤𝐶(𝑑 , 𝑠)𝜌^2𝑠 𝛿^𝑑

1+ℎ^{−𝑙 𝑑}𝜔²

𝑙 , 𝑘

ℎ^{2𝑠 𝑘}|𝛼|². (4.48)

Since, for all 𝑓 ∈𝐻^−𝑠(Ω),

k𝑓k²_∗ =[𝑓 ,L⁻¹𝑓] ≤ k𝑓k𝐻^−𝑠(Ω)kL⁻¹𝑓k𝐻^𝑠

0(Ω) ≤ kL⁻¹k k𝑓k²

𝐻^−𝑠(Ω), (4.49) we havek𝜙−𝜑k_∗ ≤ p

kL⁻¹k k𝜙−𝜑k𝐻^−𝑠(Ω), and this completes the proof.

The following geometric lemma shows that the assumption (4.36) of Lemma4can be satisfied with a uniform bound on the value of 𝜌and the norm of weights𝑤_{𝑖, 𝑗}. Lemma 7. There exist constants 𝜌(𝑑 , 𝑠) and𝐶(𝑑 , 𝑠, 𝛿) such that for all 1 ≤ 𝑘 <

𝑙 ≤ 𝑞, there exist weights 𝑤 ∈ R^𝐽

(𝑙)×𝐼˜^(𝑘)

satisfying(4.36)and (with𝜔_{𝑙 , 𝑘} defined as in Lemma4)

𝜔²

𝑙 , 𝑘 ≤ ℎ^{𝑙 𝑑}𝐶(𝑑 , 𝑠, 𝛿). (4.50)

Proof. For Example1, (4.36) is equivalent to ℎ^{𝑙 𝑑/2}𝑝(𝑥_𝑖) = Õ

𝑗∈𝐼˜_𝜌^(𝑘)

𝑤_{𝑖 𝑗}𝑝(𝑥_𝑗),∀𝑝 ∈ P^𝑠−1, (4.51)

where ˜𝐼^(𝑘)

𝜌 B {𝑗 ∈ 𝐼˜^(𝑘) | 𝑥_𝑗 ∈ 𝐵(𝑥_𝑖, 𝜌 ℎ^𝑘)}.

Fix𝑖 ∈ 𝐽^(𝑙), let𝜆 >0, and write𝑥^𝜆

𝑗 B

𝑥𝑗−𝑥𝑖

𝜆 . Write0 B (0, . . . ,0) ∈R^𝑑. Since the function 𝑝( · ) ↦→ 𝑝(·−𝑥𝑖

𝜆 )is surjective onP^𝑠−1, (4.51) is satisfied if ℎ^{𝑙 𝑑/2}𝑝(0) = Õ

𝑗∈𝐼˜_𝜌^(𝑘)

𝑤_{𝑖 𝑗}𝑝(𝑥^𝜆

𝑗),∀𝑝 ∈ P^𝑠−1. (4.52) For a multiindex 𝑛 = (𝑛₁, . . . , 𝑛_𝑑) ∈ N^𝑑 and a point 𝑧 = (𝑧₁, . . . , 𝑧_𝑑) ∈ R^𝑑, write 𝑧^𝑛 B Î^𝑑

𝑚=1𝑧^𝑛𝑚

𝑚 . Use the convention0^𝑛 =0 if𝑛 ≠ 0and0⁰ =1. To satisfy (4.52), it is sufficient to identify a subset 𝜎 of ˜𝐼⁽

𝑘)

𝜌 and 𝑤_𝑖,· ∈ R^𝐼˜

(𝑘)

such that #𝜎 = 𝑠^𝑑, 𝑤_{𝑖, 𝑗} =0 for 𝑗 ∉𝜎, and

ℎ^{𝑙 𝑑/2}0^𝑛 =Õ

𝑗∈𝜎

𝑤_{𝑖 𝑗}(𝑥^𝜆

𝑗)^𝑛,∀𝑛 ∈ {0, . . . , 𝑠−1}^𝑑. (4.53) LetV^𝜆 ∈R^{{0,1,...,𝑠−1}}

𝑑×𝜎

be the 𝑠^𝑑×𝑠^𝑑matrix defined by V^𝜆𝑛, 𝑗 B

𝑥^𝜆

𝑗

^𝑛

. (4.54)

For a multiindex 𝑛 ∈ N^𝑑 and a point 𝑥 ∈ R^𝑑, 𝑥^𝑛 B Î^𝑑

𝑚=1𝑥^𝑛𝑚. Let w ∈ R^𝜎 be defined byw𝑗 B 𝑤_{𝑖, 𝑗} for 𝑗 ∈𝜎. Equation (4.53) is then equivalent to

ℎ^{𝑙 𝑑/2}e=V^𝜆w, (4.55)

wheree ∈ R^{{0,1,...,𝑠−1}}

𝑑

is defined by e𝑛 B 0^𝑛 for 𝑛 ∈ {0,1, . . . , 𝑠−1}^𝑑. We will now identifyw by inverting (4.55). To achieve this while keeping the norm of w under control, we will seek to identify the subset 𝜎 and𝜆 > 0 such that𝜎_min(V^𝜆) (the minimal singular value ofV^𝜆) is bounded from below by a constant depending only on𝑠and𝑑.

For 𝛼 ≥ 0, let (𝜖_𝑗)_{𝑗∈{0,1,...,𝑠−1}}^𝑑 be elements of R^𝑑 satisfying |𝜖_𝑗| ≤ 𝛼 for all 𝑗 ∈ {0,1, . . . , 𝑠−1}^𝑑. Let1B (1, . . . ,1) ∈R^𝑑and, for 𝑗 ∈ {0,1, . . . , 𝑠−1}^𝑑, let 𝑧_𝑗 B 1+ 𝑗 +𝜖_𝑗. Observe that for 𝛼 = 0, the points 𝑧_𝑗 are on a regular grid. Let V¯^𝛼 ∈ R^{{0,1,...,𝑠−1}}

𝑑×{0,1,...,𝑠−1}^𝑑 be the 𝑠^𝑑×𝑠^𝑑 matrix defined by ¯V^𝛼𝑛, 𝑗 B 𝑧_𝑗^𝑛 . Let 𝑉 be the 𝑠× 𝑠Vandermonde matrix defined by𝑉_{𝑖, 𝑗} = 𝑖^𝑗. Writing𝜎_min(𝑉) for the minimal singular value of𝑉, we have for𝛼=0, by [119, Theorem 4.2.12],

𝜎_min V¯⁰

=(𝜎_min(𝑉))^𝑑. (4.56)

Since univariate polynomial interpolation on 𝑠 points with polynomials of degree 𝑠 −1 is uniquely solvable, we have 𝜎_min(𝑉) > 0 and 𝜎_min(V¯⁰) > 𝐶(𝑑 , 𝑠) > 0.

58 Therefore, the continuity of the minimal singular value with respect to the entries of ¯V^𝛼implies that there exists𝛼^∗, 𝜎^∗ > 0 depending only on𝑠, 𝑑such that 𝛼 ≤ 𝛼^∗ implies 𝜎_min(V¯^𝛼) > 𝜎^∗. Since (by construction) the (𝑥_𝑖)_{𝑖∈𝐼˜}(𝑘) form a covering of R^𝑑 of radius ℎ^𝑘, the (𝑥^𝜆

𝑖)_{𝑖∈𝐼˜}(𝑘) form a covering of R^𝑑 of radius ℎ^𝑘/𝜆 and for each 𝑛 ∈ {0,1, . . . , 𝑠−1}^𝑑, there exists an𝑥^𝜆

𝑗𝑛that is at distance at mostℎ^𝑘/𝜆from𝑛. Let 𝜎 B {𝑗_𝑛 | 𝑛∈ {0,1, . . . , 𝑠−1}^𝑑} ⊂ 𝐼˜^(𝑘) be the collection of corresponding labels.

It follows from |𝑥^𝜆

𝑗_𝑛| ≤ √

𝑑 𝑠+ℎ^𝑘/𝜆that |𝑥_𝑗

𝑛−𝑥_𝑖| ≤ 𝜆

√

𝑑 𝑠+ ℎ^𝑘, and 𝜎 ⊂ 𝐼˜^(𝑘)

𝜌 for

𝜌 > 1+𝜆

√

𝑑 𝑠/ℎ^𝑘. Selecting𝜆 = ℎ^𝑘/𝛼^∗ implies that𝜎_min(V^𝜆) > 𝜎^∗ and𝜎 ⊂ 𝐼˜_𝜌^(𝑘) for 𝜌 >1+√

𝑑 𝑠/𝛼^∗. Defining

𝑤_{𝑖 𝑗} B











(V^𝜆)⁻¹ℎ^{𝑙 𝑑/2}e

𝑛

, if 𝑗 = 𝑗_𝑛∈ 𝜎,

0, otherwise,

(4.57)

the weights𝑤_{𝑖 𝑗} satisfy𝜔_{𝑘 𝑙} ≤𝐶(𝑠, 𝑑)ℎ^{𝑙 𝑑/2}and (4.36) with a𝜌 depending only on𝑠 and𝑑. This concludes the proof for Example1. The proof is similar for Example2 with minor changes (the bound on𝜔 also depends on𝛿).

The following lemma concerns the satisfaction of the second condition of Theorem6:

Lemma 8. In the setting of Examples1and2, there exists some constant𝐶(𝑑 , 𝑠, 𝛿) >

0such that, for2≤ 𝑘 < 𝑙 ≤ 𝑞,𝛼 ∈R^𝐽

(𝑙) and𝜙 =Í

𝑖𝛼_𝑖𝜙_𝑖, min

𝜑∈span(𝜙_𝑖)

𝑖∈𝐼(𝑘−1)

k𝜙−𝜑k_∗²

|𝛼|² ≤𝐶(𝑑 , 𝑠, 𝛿) kL⁻¹kℎ^{2𝑠(𝑘−1)}. (4.58) Proof. Apply Lemma4with the bounds on 𝜌and𝜔 obtained in Lemma7.

The following theorem is a direct consequence of Theorems 6, Lemma 3 and Lemma8.

Theorem 7. In the setting of Examples1and2, there exists a constant𝐶(𝑑 , 𝑠, 𝛿)such that Condition2is fulfilled with𝐶_Φ B max( kL k,kL⁻¹k)𝐶(𝑑 , 𝑠, 𝛿) and𝐻 B ℎ^𝑠. Propagation of exponential decay

We will now derive the exponential decay of the Cholesky factors𝐿 by combining the algebraic identities of Lemma 1with the bounds on the condition numbers of the 𝐵^(𝑘) (implied by Condition2) and the exponential decay of the 𝐴^(𝑘) (specified in Condition1). The core of our proof is based on a combination/extension of the

results of [31,32,34,68,128,144] on decay algebras. The pseudodistance𝑑( ·,· ) appearing in (4.13) is not a pseudometric because it does not satisfy the triangle inequality. However, to prove (4.13), we will only need the following weaker version of the triangle inequality:

Definition 8. A function𝑑: 𝐼×𝐼 −→R⁺is called ahierarchical pseudometricif (1) 𝑑(𝑖, 𝑖) =0, for all𝑖 ∈𝐼;

(2) 𝑑(𝑖, 𝑗) =𝑑(𝑗 , 𝑖), for all𝑖, 𝑗 ∈ 𝐼;

(3) for all1 ≤ 𝑘 ≤ 𝑞, 𝑑( ·, · )restricted to𝐽^(𝑘) ×𝐽^(𝑘) is a pseudometric;

(4) for all 1 ≤ 𝑘 ≤ 𝑙 ≤ 𝑚 ≤ 𝑞 and 𝑖 ∈ 𝐽^(𝑘), 𝑠 ∈ 𝐽^(𝑙), 𝑗 ∈ 𝐽^(𝑚), we have 𝑑(𝑖, 𝑗) ≤ 𝑑(𝑖, 𝑠) +𝑑(𝑠, 𝑗).

Note that the 𝑑( ·,· ) specified in (4.20) for Examples 1 and 2 is a hierarchical pseudometric. For a hierarchical pseudometric𝑑( ·,· )and𝛾 ∈R+, let

𝑐_𝑑(𝛾) B sup

1≤𝑘≤𝑙≤𝑞

sup

𝑗∈𝐽^(𝑙)

Õ

𝑖∈𝐽^(𝑘)

exp(−𝛾 𝑑(𝑖, 𝑗)). (4.59)

The following theorem states the main result of this section:

Theorem 8(Exponential decay of the Cholesky factors). Assume thatΘfulfills Con- ditions1and2 with the constants𝛾 , 𝐶_𝛾, 𝐻 , 𝐶_Φ and the hierarchical pseudometric 𝑑( ·, · ). Then

(chol(Θ))_{𝑖 𝑗}

≤ 2𝐶_Φ𝑐_𝑑(𝛾˜/8)²

(1−𝑟)² 4𝑐_𝑑(𝛾˜/4)𝐶_Φ𝐶_𝛾(𝑐_𝑑(𝛾˜/2))² (1−𝑟)²

!^𝑞 exp

−𝛾˜ 8𝑑(𝑖, 𝑗)

, (4.60) where 𝐶_𝑅 B max

n 1, ²

𝐶𝛾𝐶_Φ 1+𝜅

o

, 𝑟 B ^1−𝜅

−1

1+𝜅⁻¹, 𝛾˜ B ^−log(

𝑟)

1+log(𝑐_𝑑(𝛾/2))+log(𝐶_𝑅)−log(𝑟) 𝛾 2, and 𝜅 =𝐻⁻²𝐶²

Φ is defined as in Theorem5.

The remaining part of this section will present the proof of Theorem8. We will use the following lemma on the stability of exponential decay under matrix multiplica- tion, the proof of which is a minor modification of that of [128].

Lemma 9. Let 𝐼 be an index set that is partitioned as 𝐼 = 𝐽⁽¹⁾ ∪ · · ·𝐽^(𝑞) and let 𝑑: 𝐼×𝐼 →R^≥0satisfy

𝑑(𝑖₁, 𝑖_𝑛+1) ≤

𝑛

Õ

𝑘=1

𝑑(𝑖_𝑘, 𝑖_𝑘+1) for all1 ≤ 𝑛≤ 𝑞−1and𝑖_𝑘 ∈ 𝐽^(𝑘).

60 Let 𝑀^(𝑘) ∈ R^𝐽

(𝑘)×𝐽^(𝑘+1)

be such that |𝑀^(𝑘)

𝑖, 𝑗 | ≤ 𝐶exp(−𝛾 𝑑(𝑖, 𝑗)) for1 ≤ 𝑘 ≤ 𝑞−1 and let

𝑐_𝑑(𝛾/2) B sup

1≤𝑘≤𝑞−1

sup

𝑗∈𝐽^(𝑘+1)

Õ

𝑖∈𝐽^(𝑘)

exp

−𝛾

2𝑑(𝑖, 𝑗)

for𝛾 ∈R⁺. (4.61) Then, for1≤ 𝑛 ≤ 𝑞−1,

𝑛

Ö

𝑘=1

𝑀^(𝑘)

!

𝑖, 𝑗

≤ (𝑐_𝑑(𝛾/2)𝐶)^𝑛exp

−𝛾

2𝑑(𝑖, 𝑗) .

Proof. Set𝑖₁ B𝑖,𝑖_𝑛+1 B 𝑗. Then

𝑛

Ö

𝑘=1

𝑀^(𝑘)

!

𝑖, 𝑗

≤𝐶^𝑛

Õ

𝑖₂,...,𝑖_𝑛∈𝐽⁽²⁾,...,𝐽^(𝑛)

exp −𝛾

𝑛

Õ

𝑘=1

𝑑(𝑖_𝑘, 𝑖_𝑘+1)

!

≤𝐶^𝑛exp

−𝛾

2𝑑(𝑖₁, 𝑖_𝑛+1) Õ

𝑖₂,...𝑖_𝑛∈𝐼

exp −𝛾 2

𝑛

Õ

𝑘=1

𝑑(𝑖_𝑘, 𝑖_𝑘+1)

!

≤ (𝑐_𝑑(𝛾/2)𝐶)^𝑛exp

−𝛾

2𝑑(𝑖, 𝑗) .

The proof of the following lemma (on the stability of exponential decay under matrix inversion for well-conditioned matrices) is nearly identical to that of [128] (we only keep track of constants; see also [68] for a related result on the inverse of sparse matrices).

Lemma 10. Let 𝐴∈R^𝐼×𝐼 be symmetric and positive definite such that for𝐶 , 𝛾 >0 and a metric𝑑( ·,· )on 𝐼we have |𝐴_{𝑖, 𝑗}| ≤𝐶exp(−𝛾 𝑑(𝑖, 𝑗)). It holds true that

(𝐴⁻¹)𝑖 , 𝑗

≤ 4

k𝐴k + k𝐴⁻¹k⁻¹

(1−𝑟)²exp − log(_𝑟¹)

(1+log(𝑐_𝑑(𝛾/2)) +log(𝐶_𝑅)) +log(¹_𝑟) 𝛾 2𝑑(𝑖, 𝑗)

! (4.62)

where 𝑐_𝑑(𝛾/2) B sup𝑗∈𝐼

Í

𝑖∈𝐼exp −^𝛾₂𝑑(𝑖, 𝑗)

, 𝐶_𝑅 B max n

1, ^2𝐶

k𝐴k+k𝐴⁻¹k⁻¹

o

= maxn

1,²

𝐶k𝐴⁻¹k 1+𝜅

o , 𝑟 B

1− ¹

k𝐴k k𝐴−1k

1+ ¹

k𝐴k k𝐴−1k

= ^1−𝜅−1

1+𝜅⁻¹, and 𝜅 B k𝐴k k𝐴⁻¹k is the condition number of 𝐴.

Proof. On a compact set not containing 0, the function𝑥 ↦→ 𝑥⁻¹can be accurately approximated by low-order polynomials in𝑥. Then, the spread of the exponential decay can be controlled by Lemma9. See Section.1for details.

By representing Schur complements as matrix inverses, Lemma 10 can also be used to show that the Cholesky factors of well-conditioned exponentially-decaying matrices are exponentially decaying. The following lemma appears in a similar form in [34] for banded matrices and in [144] without explicit constants.

Lemma 11. Let𝐵 ∈R^𝐼×𝐼 'R^𝑁×𝑁be symmetric and positive definite with condition number 𝜅 and such that

𝐵_{𝑖, 𝑗}

≤ 𝐶exp(−𝛾 𝑑(𝑖, 𝑗)) for some constant 𝐶 > 0 and some metric 𝑑 on 𝐼. Let 𝐿 be the Cholesky factor (in an arbitrary order) of 𝐵⁻¹ (𝐵⁻¹ =𝐿 𝐿^𝑇). Then

𝐿_{𝑖 , 𝑗}

≤ 4p k𝐵k k𝐵k + k𝐵⁻1k⁻¹

(1−𝑟)² exp

log(𝑟)

1+log(𝑐_𝑑(𝛾/2)) +log(𝐶_𝑅) −log(𝑟) 𝛾 2𝑑(𝑖, 𝑗)

(4.63) where 𝑐_𝑑(𝛾/2) B sup𝑗∈𝐼

Í

𝑖∈𝐼exp −^𝛾

2𝑑(𝑖, 𝑗)

, 𝐶_𝑅 B max n

1, ^2𝐶k𝐵

−1k 1+𝜅

o

, and𝑟 B

1−𝜅⁻¹ 1+𝜅⁻¹.

Proof. Lemma2 implies that the Schur complements of 𝐵⁻¹ can be expressed as inverses of sub-matrices of𝐵. The result then follows from Lemma10(see Proof.1

for details).

The last ingredient needed to prove the exponential decay of the Cholesky factors of Θis the following lemma showing the stability of exponential decay under inversion for block-lower-triangular matrices (this operation appears in the definition of ¯𝐿 in (4.14)):

Lemma 12. Let𝐼be an index set that is partitioned as𝐼 =𝐽⁽¹⁾∪ · · ·𝐽^(𝑞)and assume that the matrix𝐿 ∈R^𝐼×𝐼 is block-lower triangular with respect to this partition, with identity matrices as diagonal blocks. If𝑑( ·,· )is a hierarchical pseudometric such that|𝐿_{𝑖 𝑗}| ≤𝐶exp(−𝛾 𝑑(𝑖, 𝑗))(for some𝐶 ≥ 1and𝛾 > 0), then it holds true that

(𝐿⁻¹)𝑖 𝑗

≤ 2^𝑞(𝑐_𝑑(𝛾/2)𝐶)^𝑞exp

−𝛾

2𝑑(𝑖, 𝑗)

(4.64) with𝑐_𝑑(𝛾) Bsup_{1≤𝑘≤𝑙≤𝑞}sup_𝑗∈𝐽(𝑙) Í

𝑖∈𝐽^(𝑘)exp(−𝛾 𝑑(𝑖, 𝑗)).

Proof. The Neumann series of a𝑞×𝑞 block-lower-triangular matrix with identity matrices on the (block) diagonal can be written as

𝐿⁻¹=

𝑞

Õ

𝑘=0

(Id−𝐿)^𝑘 . (4.65)

62 Since the sum terminates in𝑞steps, the thickening of the exponential decay can be

bounded using Lemma9. See Proof.1for details.

By applying the above results to the decomposition obtained in Lemma 1, we conclude the proof of Theorem8. See Proof.1for details.