5.3 Spatial approximation by wavelet methods
5.3.1 Complexity estimates using adaptive wavelet solvers
We derive estimates on the number of degrees of freedom within the wavelet setting, which are needed to guarantee that the inexact scheme (5.13) converges with the same order as the exact scheme (5.6). As it turns out, among other things, regularity estimates for the exact solution in specific scales of Besov spaces are essential.
To keep the technicalities at a reasonable level, we focus our analysis on parabolic evolution equations of the form
u′(t) =A(t)u(t) +f(t, u(t)), u(0) =u0, t ∈(0, T], (5.65) in a Gel’fand triple setting (V, U, V∗) with V =H0s(O), U = L2(O), and V∗ =H−s(O) for some s >0, i.e., A: (0, T]×V →V∗ and f : (0, T]×U →U. This way, we are in the setting of Section 5.1 with H=Hν(O) for some smoothness parameter 0≤ν ≤s and G ⊇ H−s(O). Recall that we assume 5.8 and 5.15, i.e., the initial value is given exactly and (5.65) has a unique solution. Furthermore, we assume that an exact scheme (5.6) is given which satisfies Assumption 5.10 on the global Lipschitz continuity of its
operators, as well as Assumption 5.17, i.e., it exhibits convergence order δ.
We split our analysis into two parts. In the first part, we concentrate on the (rather theoretical) case, where the solutions of the stage equations are approximated by using best N-term wavelet approximation; and the complexity estimate is given in Theorem 5.71. Unfortunately, best N-term wavelet approximation is not implementable in our case, since the solutions to the subproblems are not known explicitly, so the N largest wavelet coefficients cannot be extracted directly. Therefore, in the second part, we turn our attention to the case where the stage equations are solved numerically by using an implementable wavelet solver which is asymptotically optimal. In Theorem 5.73 we show that the complexity estimate, derived in Theorem 5.71, immediately extends to this case.
Now to the first part. We consider the inexact scheme (5.13) and apply best N-term wavelet approximation in each stage as an approximation scheme in place of Assumption 5.12.
Remark 5.66. In the case that Ψ is an orthonormal wavelet basis, a best N-term wavelet approximation to a function v can be derived by thresholding, i.e., selecting N wavelet coefficients that are largest in absolute value in the wavelet decomposition of v.
In the biorthogonal case, thresholding yields a bestN-term wavelet approximation up to a constant, cf. Section 2.3.2. Therefore, in this sense bestN-term wavelet approximation is an approximation scheme that fulfills Assumption 5.12.
The error of best N-term wavelet approximation in Hν(O) is defined as edetN,ν(v) := inf
v −
µ∈Λ
cµψµ
Hν(O) : cµ∈R, Λ⊂ ∇, #Λ =N
,
116 Chapter 5. Convergence of the inexact linearly implicit Euler scheme
cf. Section 2.3.2. Furthermore, recall that for v ∈Bqs(Lq(O)), where 1
q = s−ν d + 1
2, 0≤ν < s < s1, (5.66) and under the assumptions (W1)–(W6), we have the estimate
edetN,Hν(O)(v)≤Cnlin∥v∥Bqs(Lq(O))N−s−νd , (5.67) with a constant Cnlin >0, which does not depend on v orN, see Remark 2.26.
We apply Theorem 5.26 and derive an estimate for the number of degrees of freedom needed to compute a solution up to a tolerance (Cexact+T)τδ.
Lemma 5.67. Suppose that (W1)–(W6) and Assumptions 5.8, 5.10, and 5.15 hold.
Let Assumption 5.17 hold for some δ >0 and let the inexact scheme (5.13) be based on best N-term wavelet approximation with the tolerances given by
εk,i:=
S Cτ,k′′ Cτ,k,(i)′ −1
τ1+δ (5.68)
with Cτ,k,(i)′ as in (5.11) and Cτ,k′′ as in (5.22). Let the exact solutions wk,i of the stage equations in (5.13), be given by (5.26), and assume that all wk,i are contained in the same Besov space Bqs(Lq(O)) with (5.66). Then we have (5.24), i.e.,
u(T)−uK
Hν(O) ≤(Cexact+T)τδ,
and the number of the degrees of freedom Mτ,T(δ), given by (5.25), that are needed for the computation of {uk}Kk=0 is bounded from above by
Mτ,T(δ)≤
K−1
k=0 S
i=1
C
d s−ν
nlin ∥wk,i∥
d s−ν
Bqs(Lq(O))
S Cτ,k′′ Cτ,k,(i)′ −1
τ1+δ−s−νd , with Cnlin as in (5.67), and where ⌈·⌉ denotes the upper Gauss-bracket.
Proof. We are in the setting of Theorem 5.26. By (5.67) we may, for each stage equation, choose N ∈N0 as the smallest possible integer, such that
edetN,Hν(O)(wk,i)≤Cnlin∥wk,i∥Bqs(Lq(O))N−s−νd ≤εk,i, holds, that is
N =
Cnlin∥wk,i∥Bqs(Lq(O))
s−νd ε−
d s−ν
k,i
.
Using (5.68) and summing over k and i completes the proof. □ Lemma 5.67 shows that we need estimates for the Besov norms of the exact solutions wk,iof the stage equations in (5.13). We can provide an estimate in the following setting.
Lemma 5.68. Suppose L−1τ,i ∈ L(L2(O), Bqs(Lq(O))) with (5.66), i = 1, . . . , S, and assume that the operatorsRτ,k,i :L2(O)×· · ·×L2(O)→L2(O)are Lipschitz continuous with Lipschitz constants Cτ,k,(i)Lip,R for all k = 0, . . . , K−1, i= 1, . . . S. With Cτ,j,(i)′ as in (5.11), we define
Ck,iBes:=
i−1
l=1
1 + max{Cτ,k,(l)Lip ,∥L−1τ,lRτ,k,l(0, . . . ,0)∥L2(O)}
1 +∥uk∥L2(O)
5.3. Spatial approximation by wavelet methods 117
+
i−1
l=1
1 +Cτ,k,(l)Lip
k−1
j=0
k−1
n=j+1
Cτ,n,(0)′ −1
S
r=1
Cτ,j,(r)′ εj,r
+
i−1
j=1
εk,j i−1
l=j+1
1 +Cτ,k,(l)Lip
. (5.69)
Then all wk,i, as defined in (5.26), are contained in the same Besov space Bqs(Lq(O)) with (5.66), and their norms can be estimated by
∥wk,i∥Bqs(Lq(O))≤ ∥L−1τ,i∥L(L2(O), Bqs(Lq(O)))
×max
Cτ,k,(i)Lip,R,∥Rτ,k,i(0, . . . ,0)∥L2(O)
Ck,iBes. (5.70) Proof. The proof is similar to the one of Theorem 5.24. It is given in Appendix B.7. □ Remark 5.69. In Lemma 5.68, the assumption L−1τ,i ∈ L(L2(O), Bqs(Lq(O))) with (5.66), and the Lipschitz continuity of Rτ,k,i imply Assumption 5.10 withH =Hν(O).
However, this Lipschitz constant may not be optimal.
Remark 5.70. IfRτ,k,i is bounded, then we can prove a similar result as in Lemma 5.68.
The combination of Lemma 5.67 and 5.68 yields the main result of the first part, i.e., the complexity estimate for the case that best N-term wavelet approximations are used for the solution of the stage equations.
Theorem 5.71. Let the assumptions of the Lemmas 5.67 and 5.68 be satisfied. With Cτ,k,(i)′ as in (5.11) and Cτ,k′′ as in (5.22), we have
Mτ,T(δ)
≤
K−1
k=0 S
i=1
C
d s−ν
nlin
max
Cτ,k,(i)Lip,R,∥Rτ,k,i(0, . . . ,0)∥L2(O)
Ck,iBess−νd
×
∥L−1τ,i∥L(L2(O), Bqs(Lq(O)))
s−νd
S Cτ,k′′ Cτ,k,(i)′ −1
τ1+δ−s−νd .
(5.71)
As outlined above, the next step is to discuss the complexity of Rothe’s method in the case that implementable numerical wavelet schemes instead of the bestN-term wavelet approximation are employed for the stage equations. We make the following assumptions, cf. Assumption 5.12.
Assumption 5.72. (i)There exists an implementable asymptotically optimal numeri-cal wavelet scheme for the stage equations arising in (5.13). That is, if the best N-term wavelet approximation in Hν(O) converges with rate
N−s−νd , for some s > ν >0,
then the scheme computes finite index sets Λl ⊂ ∇ and coefficients (cµ)µ∈Λl with
L−1τ,iv−
µ∈Λl
cµψµ
Hν(O)≤Cτ,i,s,νasym(L−1τ,iv) (#Λl)−s−νd (5.72)
118 Chapter 5. Convergence of the inexact linearly implicit Euler scheme
for some constant Cτ,i,s,νasym(L−1τ,iv). Further, for all ε >0 there exists an l(ε) such that
L−1τ,iv−
µ∈Λl
cµψµ
Hν(O)≤ε, l ≥l(ε), and such that
#Λl(ε)≤Cτ,i,s,νasym(L−1τ,iv)ε−s−νd . (ii) The operators Rτ,k,i can be evaluated exactly.
In Section 5.3.3 below, we discuss a prototype of an adaptive wavelet method, fulfilling Assumption 5.72(i), which has been derived in Cohen et al.[29]. It satisfies an optimality estimate of the form (5.72) for the energy norm (2.31), Section 2.4.1.
However, since the energy norm is equivalent to some Sobolev norm∥ · ∥Hν(O), cf. (2.32), the estimate (5.72) also holds for this case. Moreover, it has been shown in Cohen et al. [29] that the constant is of a specific form, which is similar to (5.67). Therefore, we specify Assumption 5.72(i) in the following way.
Assumption 5.72. (iii) The constant Cτ,i,s,νasym(L−1τ,iv) in (5.72) is of the form Cτ,i,s,νasym(L−1τ,iv) =Cτ,iasym∥L−1τ,iv∥Bqs(Lq(O)), 1
q = s−ν d +1
2, with a constant Cτ,iasym independent of L−1τ,iv.
In this setting we are immediately able to state our main result.
Theorem 5.73. Let the assumptions of the Lemmas 5.67 and 5.68 be satisfied. If an optimal numerical wavelet scheme, that satisfies Assumption 5.72, is used for the numerical solution of the stage equations, then the necessary number of degrees of freedom can be estimated as in Theorem 5.71 with Cτ,iasym instead of Cnlin, i.e.,
Mτ,T(δ)
≤
K−1
k=0 S
i=1
Cτ,iasyms−νd max
Cτ,k,(i)Lip,R,∥Rτ,k,i(0, . . . ,0)∥L2(O)
Ck,iBes
s−νd
×
∥L−1τ,i∥L(L2(O), Bqs(Lq(O))s−νd
S Cτ,k′′ Cτ,k,(i)′ −1
τ1+δ−s−νd
. (5.73)
Remark 5.74. The constant Cτ,iasym depends on the concrete design of the adaptive method at hand. As an example this constant may depend on the design of the routines APPLY, RHS, and COARSE, which are further discussed in Section 5.3.3 below.
Moreover, the value of Cτ,iasym depends on the equivalence constants of the energy norm and the Sobolev norm in (2.32). Therefore this constant may grow as τ gets small.
However, this is an intrinsic problem and not caused by our approach.
At this point the question remains if and how the Besov norms of the exact solutions of the stage equations wk,i, cf. (5.70) can be specified, and moreover how all the constants involved in (5.71) and (5.73) can be estimated. Therefore, in the next section we present a detailed study for the most important model problem, that is the linearly-implicit Euler scheme applied to the heat equation.
5.3. Spatial approximation by wavelet methods 119