Linear and nonlinear approximation results

We state error bounds to linear uniform and nonlinear approximations of X, first with respect to the L₂-norm and, subsequently, we extend our studies and state error bound with respect to Besov norms.

3.1. A class of random functions in Besov spaces 47

1

L₁ p

L_p s

s^∗

ν

−d p

B_τ^s(L_τ), ¹_τ = ^s−ν_d + ¹_p

Figure 3.1: Setting for Corollary 3.12 illustrated in a DeVore-Triebel diagram Error bounds with respect to the L2-norm

Here, we have to consider

α+β >1, or α+β ≥1 and γd <−1, α, γ ∈R, β ∈[0,1], (3.19) in order to ensure X ∈L₂(O) P-a.s., cf. Remark 3.1.

For linear approximation one considers the best approximation from linear subspaces of dimension at mostN, which is given by the orthogonal projection on these subspaces, cf. Section 2.3.2. The corresponding linear approximation error of X with respect to L₂(O) is given by

e^lin_N(X) := inf

E [∥X−X∥ ²_L2(O)]1/2

with the infimum taken over all measurable mappings X : Ω→L₂(O) such that dim(span(X(Ω))) ≤N.

Theorem 3.15. Letα, β, γ in (3.19)be fixed and let X be defined by (3.3). The linear approximation error with respect to L₂(O) satisfies

e^lin_N(X)≍(log₂N)^γd2 N^{−α+β−12} .

Proof. To prove the upper bound, we truncate the decomposition (3.3) of X at some level j₁ ≥j₀ and we obtain a uniform linear approximation

X_j1 :=

j1



j=j0



k∈∇_j

σ_jY_j,kZ_j,kψ_j,k, (3.20) which satisfies

E

∥X−X_j1∥²_L2(O)

≍

∞



j=j1+1

#∇_jσ_j²ρ_j ≍

∞



j=j1+1

j^γd2^{−(α+β−1)jd} ≍j₁^γd2^{−(α+β−1)j1d}. (3.21)

48 Chapter 3. A class of random functions

Since dim(span(X_j1(Ω)))≍j1

j=j0#∇_j ≍2^j1d, we get the upper bound as claimed.

To prove the lower bound we use the fact that ψ_j,k = Φ_blbe_j,k for an orthonormal basis (e_j,k)_j,k in L₂(O) and a bounded linear bijection Φ_blb : L₂(O) → L₂(O), cf.

Remark 3.2. This implies

e^lin_N(X)≍e^lin_N(Φ⁻¹_blbX).

Furthermore, e^lin_N(Φ⁻¹_blbX) depends on Φ⁻¹_blbX only via its covariance operator Q which, as we also know from Remark 3.2, is given by

Qξ =

∞



j=j0

σ_j²ρj



k∈∇_j

⟨ξ, ej,k⟩_L

2(O)ej,k. (3.22)

Consequently, the functions e_j,k form an orthonormal basis of eigenfunctions of Q with associated eigenvalues σ²_jρ_j. Due to a theorem by Micchelli and Wahba, see, e.g., Ritter [139, Proposition III.24], we can conclude

e^lin_N(Φ⁻¹_blbX) =

 _∞



j=j1+1

#∇_jσ_j²ρ_j

1/2

with N =

j1



j=j0

#∇_j. (3.23) Inserting (W3), i.e., #∇_j ≍ 2^jd, and (3.1) into (3.23) implies the asserted lower

bound. _□

The best N-term wavelet approximation imposes a restriction only on η(g) := #



λ : λ ∈ ∇, g=

λ∈∇

c_λψ_λ, c_λ ̸= 0



, (3.24)

the number of non-zero wavelet coefficients of g. Therefore, the correspondingerror of best N-term wavelet approximation for X with respect toL₂(O) is given by

e^best_N (X) := inf

E [∥X−X∥ ²_L2(O)]1/2

with the infimum taken over all measurable mappings X : Ω→L2(O) such that η(X(ω)) ≤N P-a.s.

For deterministic functions x on O the error of best N-term wavelet approximation with respect to the L₂-norm is defined by

e^det_N (x) := inf

∥x−x∥ _L2(O) : x∈L₂(O), η(x) ≤N

, (3.25)

cf. Section 2.3.2. Clearly, we have

e^best_N (X) =

E [e^det_N (X)²]1/2

.

Theorem 3.16. Let α, β, γ in (3.19) be fixed and let X be defined by (3.3). For every ε >0, the error of best N-term wavelet approximation with respect to L₂(O) satisfies

e^best_N (X)⪯

N^−1/ε :if β = 1, N⁻

α+β−1 2(1−β)+ε

:otherwise.

3.1. A class of random functions in Besov spaces 49

Proof. The case β = 1 is a direct consequence of the definition of X. For β < 1, let s and τ satisfy (3.15) with ν = 0 and p= 2, i.e., 1/τ = s/d+ 1/2. By Remark 2.26 in Section 2.3.3 we have that x∈B_τ^s(L_τ(O)) impliese^det_N (x)⪯ ∥x∥_Bτ^s_(Lτ(O))N^−s/d and

therefore it remains to apply Corollary 3.12. _□

For random functions it is also reasonable to impose a constraint on the average number of non-zero wavelet coefficients only, and to study the error of best average N-term wavelet approximation

e^avg_N (X) := inf

E [∥X−X∥ ²_L2(O)]1/2

with the infimum taken over all measurable mappings X : Ω→L₂(O) such that E [η(X)] ≤N.

Theorem 3.17. Let α, β, γ in (3.19) be fixed and let X be defined by (3.3). The error of best average N-term wavelet approximation with respect to L₂(O) satisfies

e^avg_N (X)⪯

N^γd2 2^−αdN2 :if β = 1, (log₂N)^γd2 N⁻

α+β−1

2(1−β) :otherwise.

Proof. LetN_j1 := E[η(X_j1)] forX_j1 as in (3.20). Clearly N_j1 =

j1



j=j0

#∇_jρ_j ≍

j1



j=j0

2^(1−β)jd ≍

j₁ : if β = 1 2^(1−β)j1d : otherwise.

In particular, 2^j1d≍N_j^1/(1−β)₁ if β ∈[0,1). Now, observe the error bound (3.21). _□ The asymptotic behavior of the linear approximation error e^lin_N(X) is determined by the decay of the eigenvalues σ_j²ρ_j of the covariance operator Q, see (3.22), i.e., it is essentially determined by the sum α+β. According to Theorem 3.10, the sum α+β also determines the regularity of X in the scale of Sobolev spaces H^s(O).

For β ∈(0,1] nonlinear approximation issuperior to linear approximation. More precisely, the following holds true. By definition, e^avg_N (X) ≤ e^best_N (X), and for β >0 the convergence of e^best_N (X) to zero is faster than that of e^lin_N(X). For β ∈ (0,1) the upper bounds for e^avg_N (X) ande^best_N (X) slightly differ, and any dependence of e^best_N (X) on the parameter γ is swallowed by the term N^ε in the upper bound. For linear and best average N-term approximation we have

e^avg_N1−β(X)⪯e^lin_N(X) if β ∈(0,1) and e^avg_clog

2N(X)⪯e^lin_N(X) if β = 1 with a suitably chosen constant c >0.

Remark 3.18. We stress that for β ∈ (0,1] the simulation of the approximation X_j1, which achieves the upper bound in Theorem 3.17, is possible at an average computational cost of order N_j1. Let us briefly sketch the method of simulation. Set n_j := #∇_j. For each level j we first simulate a binomial distribution with parameters n_j and ρ_j, which is possible at an average cost of order at most n_jρ_j. Conditional on a

50 Chapter 3. A class of random functions

realization L(ω) of this step, the locations of the non-zero coefficients on level j are uniformly distributed on the set of all subsets of {0, . . . , n_j} of cardinality L(ω). Thus, in the second step, we employ acceptance-rejection to collect the elements of such a random subset sequentially. If L(ω)≤ n_j/2, then all acceptance probabilities are at least 1/2, and otherwise we switch to complements to obtain the same bound for the acceptance probability. In this way, the average cost of the second step is of order n_jρ_j, too. In the last step we simulate the values of the non-zero coefficients. In total, the average computational cost for each level j is of ordern_jρ_j.

Remark 3.19. For Theorems 3.15 and 3.17 we only need the Riesz basis property (W1) and the property (W3), i.e., #∇j ≍2^jd, of the basis Ψ, and (W3) enters only via the asymptotic behavior of the parameters ρ_j and σ_j. After a lexicographic reordering of the indices (j, k) the two assumptions essentially amount to

X=

∞



n=1

σ_nY_nZ_nψ_n

with any Riesz basis {ψ_n}n∈Nfor L₂(O), andσ_n ≍(log₂n)^γd/2n^−α/2 as well as indepen-dent random variables Yn andZn, where Zn is N(0,1)-distributed and Yn is Bernoulli distributed with parameter ρ_n≍n^−β. Therefore, Theorems 3.15 and 3.17 remain valid beyond the wavelet setting. For instance, let ρ_n = 1, which corresponds to β = 0.

Classical examples for Gaussian random functions on O = [0,1]^d are the Brownian sheet, which corresponds to α= 2 and γ = 2(d−1)/d, and L´evy’s Brownian motion, which corresponds to α = (d+ 1)/d and γ = 0. Theorem 3.15 is due to Papageor-giou, Wasilkowski [132], Wo´zniakowski [175] for the Brownian sheet and due to Wasilkowski [173] for L´evy’s Brownian motion. See Ritter [139, Chapter VI]

for further results and references on approximation of Gaussian random functions.

Therefore, for β >0 our stochastic model provides sparse variants of general Gaussian random function.

Error bounds with respect to Besov norms

We extend above findings and state error bounds for linear and nonlinear approximation schemes for the considered random functions with respect to the norms of the Besov spaces B_p^ν(L_p(O)) with ν ∈Rand 1< p <∞.

We define the linear approximation error of X with respect toB_p^ν(L_p(O)) by e^lin_N,p,ν(X) := inf

E[∥X−X∥ ^p_Bν

p(Lp(O))]1/p

with the infimum taken over all measurable mappings X such that dim(span(X(O))) ≤N.

Theorem 3.20. Let β ∈ [0,1), m >0. For a fixed approximation space B_p^ν(L_p(O)), ν ∈R, 1< p <∞, let X be given by (3.3) with

ν+m < d((α−1)/2 +β/p) =: ν+m^∗, (3.26) i.e., X ∈B_p^ν+m(L_p(O)) for all m < m^∗. The linear approximation error with respect to B_p^ν(L_p(O)) satisfies

e^lin_N,p,ν(X)⪯(log₂N)^γd2 N^{−(α−12 +βp−νd)}. (3.27)

3.1. A class of random functions in Besov spaces 51

Proof. Again, as a specific linear approximation, we consider a uniform approximation of the form

X_j1 =

j1



j=j0



k∈∇_j

σ_jY_j,kZ_j,kψ_j,k

for some j₁ ≥j₀, where in particular N ≍2^j1d. With S_j,p as defined in (3.4) and with (3.5), we obtain

E[∥X−X_j1∥^p_Bν

p(Lp(O))]≍E

 _∞



j=j1+1

2^{jp(ν+d(12−p1))}σ^p_jS_j,p



= E

 _∞



j=j1+1

2^jp(^{(ν+m∗)+d(1}₂⁻¹_p⁾)2^−jpm∗σ_j^pS_j,p



=

∞



j=j1+1

2^jpd(^{(ν+m∗)+d(1}₂⁻¹_p⁾)2^−jpm∗σ_j^p#∇_jρ_j.

Inserting (W3), i.e., #∇_j ≍2^jd, (3.1), and (3.26) we get E[∥X−X_j1∥^p_Bν

p(Lp(O))]≍

∞



j=j1+1

2^jpd(^α−12 +^β_p+¹₂−¹_p)2^−jpm∗j^γdp2 2^−αjdp2 2^jd2^−βjd

=

∞



j=j1+1

j^γdp2 2^−jpm∗

≍j

γdp 2

1 2^−j1pm∗

≍(log₂N)^γdp2 N^−p(^α−1₂ ^+β_p^−ν_d),

which yields (3.27). _□

Remark 3.21. In the setting of Theorem 3.20, with a slightly coarser error estimation, for all m < m^∗ we get

E[∥X−X_j1∥^p_Bν

p(Lp(O))]⪯2^−j1pmE

 _∞



j=j0

2^jp(^(ν+m)+d(12−¹_p))σ_j^pS_j,p



⪯N^−pm/dE

∥X∥^p

Bp^ν+m(Lp(O))

 . Since we have E[∥X∥^p

B^ν+mp (Lp(O))] <∞, m < m^∗, by (3.10), we derive that the linear approximation error satisfies

e^lin_N,p,ν(X)⪯N^−m/d



E[∥X∥^p_Bν+m

p (Lp(O))]

1/p

. (3.28)

From (3.28), we observe that, similar to the well-known deterministic setting, see Section 2.3.2, the approximation order which can be achieved by uniform linear schemes depends on the regularity of the object under consideration in the same scale of smoothness spaces.

52 Chapter 3. A class of random functions

The following theorem is a generalization of Theorem 3.15. It states the error bounds for linear wavelet approximation with respect to H^ν.

Theorem 3.22. Let β ∈ [0,1) and m > 0. For a fixed approximation space H^ν(O), ν ∈ R, let X be given by (3.3) with ν +m < d(α−1 + β)/2 =: ν +m^∗, that is, X ∈H^ν+m(O) for all m < m^∗. The linear approximation error with respect to H^ν(O) satisfies

e^lin_N,2,ν(X)≍(log₂N)^γd2 N⁻(^α−1+β₂ ^−ν_d).

Proof. The upper bound is proven in Theorem 3.20, where p= 2. The lower bound is analogously shown as in the proof of Theorem 3.15. Given {e_j,k}_j,k and Φ_blb as in Remark 3.2, we know that

e^lin_N,2,ν(X)≍e^lin_N,2,ν(Φ⁻¹_blbX).

We also know from Remark 3.2 that the covariance operator Q of Φ⁻¹_blbX is given by

Qξ =

∞



j=j0

2^2jνσ²_jρ_j 

k∈∇_j

⟨ξ, e_j,k⟩_Hν(O)e_j,k,

which means, that the functions e_j,k form an orthonormal basis of eigenfunctions of Q with associated eigenvalues 2^2jνσ_j²ρ_j. Using methods, e.g. shown in Ritter [139, Chapter III], we get

e^lin_N,2,ν(Φ⁻¹_blbX) =

 _∞



j=j1+1

#∇_j2^2jνσ²_jρ_j

1/2

with N =

j1



j=j0

#∇_j. (3.29) Inserting (W3), i.e., #∇j ≍2^jd, and (3.1) into (3.29) yields the claim. _□ We define the average nonlinear approximation error of X : Ω→B_τ^s(L_τ(O)) with respect to B_p^ν(Lp(O)) in the scale (3.15), i.e.,

1

τ = s−ν d +1

p, 1< p <∞, and ν < s, cf. Corollary 3.12, by

e^avg_N,p,ν(X) := inf

E[∥X−X∥ ^p_Bν

p(Lp(O))]1/p

with the infimum taken over all measurable mappingsX such that E[η(X)] ≤N. Again, η(g) denotes the number of nonzero wavelet coefficients of g, see (3.24).

Theorem 3.23. Let β ∈ [0,1). For a fixed approximation space B_p^ν(L_p(O)), ν ∈ R, 1 < p < ∞, let X be given by (3.3) with −d/p ≤ ν < d((α−1)/2 +β/p), that is, X ∈B_τ^s(L_τ(O)) in the scale (3.15) for all s < s^∗, where s^∗ is given by (3.16). Then the average nonlinear approximation error with respect to B_p^ν(L_p(O)) satisfies

e^avg_N,p,ν(X)⪯(log₂N)^γd2 N^−1−β1 (^α−12 +^β_p−^ν_d). (3.30)

3.1. A class of random functions in Besov spaces 53

Proof. As a specific nonlinear approximation of X we consider

X_j1,N :=

j1



j=j0



k∈∇j

σ_jY_j,kZ_j,kψ_j,k

for some j₁ ≥j₀, where only the non-zero coefficients N := E[η(X_j1,N)] are retained.

We have

N =

j1



j=j0

#∇_jρ_j ≍2^(1−β)j1d.

WithS_j,p being defined in (3.4) and with (3.5), we use (3.15), wheres =s^∗ andτ = τ^∗, to obtain

E[∥X−X_j1,N∥^p_Bν

p(Lp(O))]≍E

 _∞



j=j1+1

2^jp(^ν+d(1₂⁻¹_p⁾)σ_j^pS_j,p



= E

 _∞



j=j1+1

2^jp(^s∗+d(1₂⁻_τ^1∗))σ_j^pSj,p



=

∞



j=j1+1

2^jp(^s∗+d(1₂⁻_τ¹∗))σ^p_j#∇_jρ_j.

Inserting (W3), i.e., #∇_j ≍2^jd, (3.1), (3.18), and (3.15), where s=s^∗ and τ =τ^∗, we get

E[∥X−Xj1,N∥^p_Bν

p(Lp(O))]≍

∞



j=j1+1

2^jpd(^α−1₂ ⁺_τ^β∗+¹₂−¹

τ∗)j^γdp2 2^−αjdp2 2^jd2^−βjd

=

∞



j=j1+1

j^γdp2 2^{−jp(1−β)(s∗−ν)}

≍j

γdp 2

1 2^{−j1p(1−β)(s∗−ν)}

≍(log₂N)^γdp2 N^−1−βp (^α−1₂ ^+β_p^−ν_d),

which yields (3.30). _□

An analogous statement to Remark 3.21 also holds for the average nonlinear approximation error.

Remark 3.24. Letε >0 ands:=s^∗−εwith s^∗ being defined in (3.16). In the setting of Theorem 3.23, with a slightly coarser error estimation, we get

E[∥X−X_j1,N∥^p_Bν

p(Lp(O))]≍E

 _∞



j=j1+1

2^j(p−τ)(^s+d(1₂⁻¹_τ⁾)σ_j^p−τ2^jτ(^s+d(1₂⁻¹_τ⁾)σ_j^τS_j,p



≍E

 _∞



j=j1+1

2^j(p−τ)(^s+d(1₂⁻¹_τ^−α₂⁾)j^γd2(p−τ)2^jτ(^s+d(1₂⁻_τ¹⁾)σ_j^τS_j,p



54 Chapter 3. A class of random functions

⪯E

 _∞



j=j1+1

2^j(p−τ)(^s+d(12−¹_τ−^α₂))^+δj2^jτ(^s+d(12−_τ¹))σ^τ_jS_j,p

 , for any δ > 0. Insertings =s^∗−ε and δ := p(s−ν)(1−β)(ετ)/d, as well as using (3.18), (3.15), and also (3.15) with s=s^∗ and τ =τ^∗, which yields 1/τ^∗ = 1/τ +ε/d, we get

E[∥X−Xj1,N∥^p_Bν

p(Lp(O))]⪯E

 _∞



j=j1+1

2^j(p−τ)(^{s∗−ε+d(1}₂⁻¹_τ^−α₂⁾)^+δj2^jτ(^s+d(1₂⁻¹_τ⁾)σ_j^τSj,p



≍E

 _∞



j=j1+1

2^j((p−τ)d(β−1)(¹_τ^+ε_d)^+δ) 2^jτ(^s+d(1₂⁻¹_τ⁾)σ_j^τS_j,p



≍E

 _∞



j=j1+1

2^j(p(s−ν)τ(β−1)(τ¹+^ε_d)^+δ) 2^jτ(^s+d(12−_τ¹))σ^τ_jS_j,p



≍E

 _∞



j=j1+1

2−jp(1−β)(s−ν)

2^jτ(^s+d(12−¹_τ))σ_j^τS_j,p



≍2^−j1p(1−β)(s−ν)

E

 _∞



j=j+1

2^jτ(^s+d(1₂⁻¹_τ⁾)σ_j^τSj,p



⪯2^−j1p(1−β)(s−ν)

E

 _∞



j=0

2^jτ(^s+d(1₂⁻¹_τ⁾)σ^τ_jS_j,p



⪯N^−ps−νd E

∥X∥^τ_Bs τ(Lτ(O))



with E[S_j,p] = #∇_jρ_jν_p = #∇_jρ_jν_τν^νp

τ = E[S_j,τ]_ν^νp

τ. Since we have E[∥X∥^τ_Bs

τ(Lτ(O))]<∞ for s < s^∗, by (3.10), we see that the average nonlinear approximation error satisfies

e^avg_N,p,ν(X)⪯N^−s−νd 

E[∥X∥^τ_Bs

τ(Lτ(O))]1/p

. (3.31)

From (3.31) we observe that, similar to the deterministic setting, the approximation order which can be achieved by nonlinear approximation does not depend on the regularity in the same scale of smoothness spaces of the object under consideration, but on the regularity in the corresponding scale (3.15) of Besov spaces.

For the case p= 2, i.e., for nonlinear wavelet approximation with respect to H^ν, also a lower bound for the average nonlinear approximation error can be derived.

Theorem 3.25. Let β ∈[0,1). For a fixed approximation space H^ν(O), ν ∈R, let X be given by (3.3) with −d/2≤ν < d(α−1 +β)/2, i.e., X ∈ B_τ^s(L_τ(O)) in the scale (3.15)for all s < s^∗, wheres^∗ is given by (3.16)with p= 2. Then the average nonlinear approximation error in H^ν(O) satisfies

e^avg_N,2,ν(X)⪰(log₂N)^γd2 N^{−1−β1 (α−1+β2 −νd)}. (3.32) Proof. Let X be defined by (3.3). For every level j, we define the number of scaled coefficients of X larger than a thresholdδ_j >0 as

M(j, δ_j) := #

2^jνσ_jY_j,k|Z_j,k| : 2^jνσ_jY_j,k|Z_j,k|> δ_j, k∈ ∇_j

. (3.33)

3.1. A class of random functions in Besov spaces 55

We set Y_j,β := 

k∈∇jY_j,k and obtain Y_j,β ∼ Bin(2^jd,2^−βjd). Since the (Y_j,k)_j,k are discrete and (Z_j,l)_j,l are identically distributed we can compute

E [M(j, δj)] =

2^jd



l=0

E



M(j, δj)









k∈∇_j

Yj,k =l



 P







k∈∇_j

Yj,k =l





=

2^jd



l=0

lP

2^jνσ_j|Z_j,l|> δ_j

P (Y_j,β =l)

= E[Y_j,β] P(2^jνσ_j|Z_j,k|> δ_j)

= 2^jd(1−β)2



1−Φ_cdf

 δj

2^jνσ_j



,

where Φ_cdf denotes the cumulative distribution function of the standard normal distri-bution. Now, we choose

δ_j := 2^jνσ_j (3.34)

and we obtain E[M(j,2^jνσ_j)] =c₁2^jd(1−β) withc₁ := 2(1−Φ_cdf(1)). For a givenN ∈N0

we set j₁ := min{j : N ≤2^jd2 } and determine a levelj₂, such that E

M(j₂,2^j2νσ_j2)

≥c₁N². (3.35)

This holds for

j₂ =

 j1

1−β



. (3.36)

Up to this point we have shown that, for X and any given N ∈N0, we can find a level j₂, which contains on average at least c₁N² coefficients, that are larger thanδ_j2.

Let

X_N :=

∞



j=j0



k∈∇_j

c_j,kψ_j,k :=

λ∈∇

c_λψ_λ

with E[#∇] = E[η( X_N)]≤N be any approximation of X =

∞



j=j0



k∈∇_j

σ_jY_j,kZ_j,kψ_j,k :=

λ∈∇

d_λψ_λ.

We set |λ|:=j. Then, by using the norm equivalence from (W6), we obtain E[∥X−X_N∥²_Hν(O)] = E















λ∈∇

d_λψ_λ−

λ∈∇

c_λψ_λ









2

H^ν(O)





= E















λ∈∇\∇

d_λψ_λ+

λ∈∇

(d_λ −c_λ)ψ_λ









2

H^ν(O)





≍E







λ∈∇\∇

2^2|λ|ν|dλ|² +

λ∈∇

2^2|λ|ν|dλ−cλ|²



.

56 Chapter 3. A class of random functions

If we omit the second sum and by (3.33) and (3.34), we get

E[∥X−X_N∥²_Hν(O)]⪰E







λ∈∇\∇

2^2|λ|ν|d_λ|²





⪰E







λ∈∇j2\∇

2^2|λ|ν|d_λ|²





≥E







k∈∇_j

2\∇

2^2j2ν|d_j2,k|²





≥E

#

k ∈ ∇_j2 \∇ : 2^2j2ν|d_j2,k|² > δ_j²₂

·δ_j²₂

≥E

M(j₂,2^j2νσ_j2)−#∇

·2^2j2νσ_j²₂

=

E[M(j₂,2^j2νσ_j2)]−E[#∇] 

·2^2j2νσ_j²

2,

so that, by inserting (3.1), (3.35), (3.36), E[#∇] ≤N ≤2^j12d, we can conclude E[∥X−XN∥²_Hν(O)]⪰

2^j1d−2^j12d



2^2j2νj₂^γd2^−αj2d

⪰j₁^γd2^{j1d+2j1−β1ν−αj1−β1d}

≍(log₂N)^γdN^{−1−β1 (α−1+β−2νd)},

which yields (3.32). _□

Remark 3.26. For the proof of Theorem 3.25 it is essential to be able to compute the expected value of M(j, δ_j), i.e., the average number of coefficients on levelj which are larger than the threshold δ_j. This random variable can be derived solely due to the structure of X. Since the threshold δ_j = 2^jνj^γd/22^−αjd/2 decays with increasing levelj, cf. (3.34), the growth of E[M(j, δ_j)] is in compliance with Theorem 3.10.

Remark 3.27. Observe that the upper bound in Theorem 3.23 for p= 2 coincides with the lower bound in Theorem 3.25.

Im Dokument Adaptive wavelet methods for a class of stochastic partial differential equations (Seite 56-66)