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Preasymptotic estimates for approximation of multivariate Sobolev functions

Thomas K¨ uhn

Universit¨ at Leipzig, Germany

ICERM Semester Program ”High-dimensional Approximation” Brown University, Providence, Rhode Island IBC Workshop – 16 September 2014

joint work with S. Mayer (Bonn), W. Sickel (Jena) and T. Ullrich (Bonn)

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 1 / 18

Approximation numbers

Approximation numbers

• f bounded linear operators T : X → Y between two Banach spaces

an(T : X → Y ) := inf{T − A : rank A < n} Interpretation in IBC and Numerical Analysis Every operator A : X → Y of finite rank k can be written as Ax =

k

• j=1

Lj(x) yj for all x ∈ X with linear functionals Lj ∈ X ∗ and vectors yj ∈ Y .

• A is a linear algorithm using k arbitrary linear informations

T − A = sup

x≤1

Tx − Ax = worst-case error of A.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 2 / 18

For compact operators between Hilbert spaces one has an(T) = sn(T) = n-th singular number of T . General problem in functional analysis or approximation theory: Find the asymptotic behaviour of an(T) as n → ∞ . Typical results are of the form c n−α ≤ an(T) ≤ C n−α for all (or for large) n ∈ N, with certain (often unspecified) constants. More relevant for practical issues, for instance in – tractability problems in IBC – error analysis of numerical algorithms is the preasymptotic behaviour of an(T) i.e. estimates for small n Our aims. It is well known that an(Id : Hs(Td) → L2(Td)) ∼ n−s/d . We will give – explicit constants, in particular asymptotic constants – sharp preasymptotic estimates in the range 2 ≤ n ≤ 2d with special emphasis on the dependence

• n the dimension d and on the chosen norm.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 3 / 18

Isotropic Sobolev spaces – integer smoothness

Td is the d-dimensional torus = [0, 2π]d with identification of opposite points, equipped with the normalized Lebesgue measure (2π)−ddx. Sobolev spaces on Td of integer smoothness m ∈ N Hm(Td) consists of all f ∈ L2(Td) such that Dαf ∈ L2(Td) for all multi-indices α ∈ Nd

0 with |α| ≤ m .

Natural norm (all partial derivatives) f |Hm(Td) :=

|α|≤m

Dαf |L2(Td)21/2 Modified natural norm (only highest derivatives in each coordinate) f |Hm(Td)∗ :=

• f |L2(Td)2 +

d

• j=1
• ∂mf

∂xm

j

• L2(Td)
• 21/2

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 4 / 18

Fourier coefficients – equivalent norms

Fourier coefficients of f ∈ L2(Td) ck(f ) := 1 (2π)d

• Td f (x)e−ikxdx

, k ∈ Zd Parseval’s identity and ck(Dαf ) = (ik)αck(f ) = ⇒ norms in Hm(Td) can be expressed in terms of ck(f ) For the natural norm one has equivalence f |Hm(Td) ∼  

k∈Zd

• 1 +

d

• j=1

|kj|2m |ck(f )|2  

1/2

with equivalence constants independent on d. For the modified natural norm one has even equality f |Hm(Td)∗ =  

k∈Zd

• 1 +

d

• j=1

|kj|2m |ck(f )|2  

1/2

.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 5 / 18

Fractional smoothness s > 0

• Idea. Replace, in der Fourier norm, m ∈ N with a real number s > 0.
• all norms are weighted ℓ2-sums of Fourier coefficients

Hs,p(Td) consists of all f ∈ L2(Td) such that f |Hs,p(Td) :=

k∈Zd

ws,p(k)2|ck(f )|21/2 < ∞ , where the weights are ws,p(k) =

• (1 + |k1|p + . . . + |kd|p)s/p

, 0 < p < ∞ max(1, |k1|, . . . , |kd|)s , p = ∞ For fixed s > 0 and d ∈ N, all these norms are equivalent. The equivalence constants depend heavily on d, but clearly all spaces Hs,p(Td), 0 < p ≤ ∞, coincide as vector spaces. p = 2 ⇐ ⇒ natural norm p = 2s ⇐ ⇒ modified natural norm

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 6 / 18

New results

Existence and computation of the limits lim

n→∞ ns/d an(Id : Hs,p(Td) → L2(Td))

for all s > 0, d ∈ N and 0 < p ≤ ∞ Asymptotic behaviour of the constants as d → ∞ Explicit two-sided estimates of an for large n / small n Similar results for – approximation in the sup-norm – spaces of dominating mixed smoothness − → talk by Winfried Sickel

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 7 / 18

Reduction to sequence spaces

Commutative diagram Hs,p(Td) L2(Td) ℓ2(Zd) ℓ2(Zd) Id A D B with Af := (ws,p(k) ck(f ))k∈Zd , Bξ :=

k∈Zd ξk eikx

and a diagonal operator D(ξk) := (ξk/ws,p(k)) A and B are unitary operators = ⇒ an(Id) = an(D) = sn(D)

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 8 / 18

Diagonal operators and combinatorics

Let (σn)n∈N be the non-increasing rearrangement of (1/ws,p(k))k∈Zd. With this piecewise constant sequence we have an(Id : Hs,p(Td) → L2(Td)) = sn(D : ℓ2(Zd) → ℓ2(Zd)) = σn . The ”sequence” (ws,p(k))k∈Zd attains all values (1 + rp)s/p, r ∈ N, in fact each of them at least 2d times, for k = ±re1, ±re2 . . . , ±red.

Lemma

Let r ∈ N and n = #{k ∈ Zd : d

j=1 |kj|p ≤ rp} . Then

an(Id : Hs,p(Td) → L2(Td)) = σn = (1 + rp)−s/p . In principle, this gives an(Id) for sufficiently many n′s, but to compute these cardinalities exactly is impossible. However, good estimates will be enough.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 9 / 18

Grid, covering and entropy numbers

(Quasi-)norms on Rd xp :=    d

j=1 |xj|p1/p

, 0 < p < ∞ max1≤j≤d |xj| , p = ∞ with (closed) unit balls Bd

p := {x ∈ Rd : xp ≤ 1}

Let A ⊆ Rd. Grid number G(A) := #(A ∩ Zd) Covering numbers Nε(A) := minimal n ∈ N such that there are x1, . . . , xn ∈ Rd with A ⊆ n

i=1(xi + εBd ∞)

Entropy numbers εn(A) := {inf ε > 0 : Nε(A) ≤ n} Here, covering and entropy numbers are always w.r.t. the sup-norm.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 10 / 18

Grid numbers vs. covering numbers

A subset A ⊂ Rd is called solid, if (xj) ∈ A and |yj| ≤ |xj| implies (yj) ∈ A. Examples: rBd

p for all r > 0 and 0 < p ≤ ∞

Lemma

Let A ⊆ Rd be a solid subset and 0 < ε < 1/2. Then N1(A) ≤ G(A) ≤ Nε(A) .

• Proof. Given x ∈ Rd, define k(x) = (kj) ∈ Zd by kj = signxj · [|xj|].

Then the set {k(x) : x ∈ A} is a 1-net for A and, since A is solid, it is equal to A ∩ Zd. This proves N1(A) ≤ G(A). The inequality G(A) ≤ Nε(A) follows from the fact that each ball of radius ε < 1/2 in ℓd

∞ is a cube of side length 2ε < 1, whence it

contains at most one element of Zd. Therefore every covering of A by ε-balls in ℓd

∞ must have at leastG(A) elements.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 11 / 18

Covering numbers are homogeneous, in the sense of Nε(A) = Nλε(λA) for all λ, ε > 0 . This is an advantage over grid numbers! For large ℓp-balls the previous lemma can be improved.

Lemma

Let 0 < p < ∞, d ∈ N and r > d1/p/2. Set ˜ p = min(1, p) and ℓ = ℓ(r, p, d) = (r ˜

p − d ˜ p/p/2˜ p)1/˜ p

L = L(r, p, d) = (r ˜

p + d ˜ p/p/2˜ p)1/˜ p

Then N1/2(ℓ Bd

p ) ≤ G(r Bd p ) ≤ N1/2(L Bd p )

Prooof: By triangle inequality and volume arguments. Note that ℓ(r, p, d) ≍ r ≍ L(r, p, d) as r → ∞.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 12 / 18

Appoximation in Hs,p(Td) via entropy of ℓp-balls

Recall that we have already shown an(Id : Hs,p(Td) → L2(Td)) = (1 + rp)−s/p for r ∈ N and n = #{k ∈ Zd : kp ≤ r} = G(rBd

p ).

Together with the two lemmata this implies the following result.

Theorem

Let s > 0, 0 < p ≤ ∞ and d ∈ N. Then, for all n ∈ N, one has

• 2−1−1/pεn(Bd

p )

s ≤ an(Id : Hs,p(Td) → L2(Td)) ≤

• 4εn(Bd

p )

s For p = ∞ we have n−1/d ≤ εn(Bd

∞) ≤ 4n−1/d,

for all n ∈ N, For 0 < p < ∞ the entropy numbers εn(Bd

p ) = εn(id : ℓd p → ℓd ∞) are

also completely understood.

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 13 / 18

Entropy numbers of ℓp-balls

Lemma (Sch¨ utt 1984, Edmunds/Triebel 1996, K. 2001)

Let 0 < p < ∞ and d ∈ N. Then εn(id : ℓd

p → ℓd ∞) ∼

       1 , 1 ≤ n ≤ d

• log(1+d/ log n)

log n

1/p , d ≤ n ≤ 2d d−1/pn−1/d , n ≥ 2d We have explicit expressions for the constants hidden in ∼ . In particular, for fixed d, we have lim

n→∞ n1/dεn(id : ℓd p → ℓd ∞) = vol(Bd p )1/d

2 .

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 14 / 18

Asymptotic constants

Putting everything together, we can show the existence of asymptotically (in n) optimal constants.

Theorem (K./Sickel/Ullrich, J.Complexity 2014)

Let 0 < s, p < ∞ and d ∈ N. Then lim

n→∞ ns/d an(Id : Hs,p(Td) → L2(Td)) = vol(Bd p )s/d ∼ d−s/p

Optimal constant is of order d−s/2 for the natural norm (p = 2) d−1/2 for the modified natural norm (p = 2s) We got the correct order n−s/d of the an in n and the exact decay rate d−s/p of the constants in d. Polynomial decay in d of the constants helps in error estimates!

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 15 / 18

Estimates for large n

Theorem (KSU 2014, case p = 1)

Let s > 0 and n ≥ 6d/3. Then d−sn−s/d ≤ an(Id : Hs,1(Td) → L2(Td)) ≤ (4e)sd−sn−s/d . We have similar estimates for all other 0 < p < ∞ , but for p = 1 the constants are nicer. Note the correct d-dependence d−s of the constants!

Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 16 / 18

Estimates for small n

Theorem (KSU 2014)

Let p = 1 and 2 ≤ n ≤ 2d. Then

• 1

2 + log2 n s ≤ an(Id : Hs,1(Td) → L2(Td)) ≤ log2(2d + 1) log2 n s . This estimate was shown by combinatorial arguments, which only worked for p = 1. Using the relation to entropy, we could close the gap between lower and upper bounds and treat arbitrary p’s.

Theorem (K/Mayer/Ullrich, preprint 2014)

Let s > 0, 0 < p < ∞ and 2 ≤ n ≤ 2d. Then an(Id : Hs,p(Td) → L2(Td)) ∼ log2(1 + d/ log2 n)) log2 n s/p . (We have explicit expressions for the hidden constants.)

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Thank you for your attention!

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