4.3 The composite trapezoidal rule
Throughout this thesis we approximate the integral Qd[f] :=
Z
Rd
f(x)dx, (4.10)
by the weighted sum
Qdh[f] := hdX
j∈Zd
f(hj), (4.11)
where h >0denotes the step size, i.e. we use the composite trapezoidal rule.
In this section we prove that the composite trapezoidal rule, which coincides with the midpoint rule or rectangular rule on Rd, is convergent of high order for certain classes of weighted continuously differentiable functions. Our presentation follows in large parts [40], [42] and [30] extending their results to the case of arbitrary space dimensions.
For the forthcoming analysis we introduce the following spaces of differentiable functions, all of which are Banach spaces equipped with their respective norms.
Forn, d ∈N and p∈R we define BCpn(Rd) :=
ψ ∈BCn(Rd) :kψkBCn
p(Rd) <∞ , with
kψkBCpn(Rd) := max
|α|≤nkwp∂αψkBC(Rd), where
wp(x) := (1 +|x|)p, x∈Rd.
In other words this means that for f ∈ BCpn(Rd), there exists a constant c > 0 so that
|∂αf(x)| ≤ c
(1 +|x|)p, x∈Rd and α∈Nd with |α| ≤n.
Remark 4.6. We note the trivial identities
wpwq =wp+q, p, q ∈R (4.12)
and
w−p = 1 wp
, p∈R. (4.13)
Lemma 4.7. Let f ∈ BCpn(Rd) with p > d. Then ∂αf ∈ L1(Rd) for α ∈ Nd with
|α| ≤n.
Proof. whereωd denotes the surface area of the unit sphere in Rd.
To handle kernel functions on Rd×Rd we define in an analogue way the space BCpn(Rd×Rd) :=
The following multi-dimensional variant of the Leibniz rule orgeneralised product rule will be used in the sequel.
Lemma 4.8 (Leibniz rule). For f, g ∈Cn(Rd) it holds that Nd are given through
α
Proof. See e.g. [28, p. 247, Theorem 1].
Lemma 4.9. For α, β ∈Nd with β ≤α we have the identity
4.3 The composite trapezoidal rule
Proof. We first note the one-dimensional version of this rule 2n= (1 + 1)n=
which is a simple consequence of the binomial theorem. The general rule follows by successive summation and the use of (4.15)
X
The first building block of the error estimate for the composite trapezoidal rule is the next Lemma that examines relations of weighted and non weighted spaces of differentiable functions. Proof. (i) Using the Leibniz rule and the identity (4.14) we calculate
|∂α(f g)(x)|=
for allα ∈Ndwith|α| ≤n. We use thatwpwq=wp+q forp, q ∈R, cf.Remark4.6, multiply this inequality with wp+q(x) and take the supremum over x ∈ Rd. This yields the estimate (4.16).
(ii) This is a direct consequence of (i), if we first note that for a∈BCpn(Rd×Rd)it holds that a(x,·)∈BCpn(Rd) for all x∈Rd.
The second building block it the following Lemma, which is a consequence of a generalisation of the Euler-Maclaurin expansion for the multi-dimensional case. Let
C0n([0,1]d) :=
ψ ∈Cn([0,1]d) :∂αψ|∂([0,1]d) = 0 for |α| ≤n ,
the space of n-times continuously differentiable functions on[0,1]d that vanish with all their derivatives up to ordern on the boundary.
Lemma 4.11. Let m, n ∈ N, g ∈ C0m([0,1]d) and define h := 1/n and N :=
(n, . . . , n)∈Nd. Then Z
[0,1]d
g(x)dx−hd
N−1
X XX XXXXXX
j=1
g(hj)
≤CkgkCm([0,1]d)hm, where the constant C > 0 depends only on m.
Proof. For details of a proof see e.g. the comments in [51].
The following theorem is an extension of [42, Lemma 3.10] to the multi-dimensional case and the proof presented given here closely follows the one in [42].
Lemma 4.12. If, for some p > d and m∈N, f ∈BCpm(Rd), then
|Qdf−Qdhf| ≤CkfkBCm
p (Rd)hm, h >0, (4.18) where the constant C > 0 depends only on m and p.
Proof. Letφ∈C∞(R)be such thatφ(s) = −12 fors≤0andφ(s) = 12 fors≥1. Let ψ0(s) := φ(s)−φ(s−1) and letψj(s) :=ψ0(s−j) for j ∈Z. Thenψj ∈Ccomp∞ (R), with suppψj = [j,2 +j] and
X
j∈Z
ψj(s) = 1, s∈R,
so that we have a partition of unity on R. The family of functions Ψj(x) :=ψj1(x1)·. . .·ψjd(xd), j ∈Zd, x∈Rd.
4.3 The composite trapezoidal rule and by Lemma 4.11 and Lemma 4.10,
|ej(h)| ≤CkΨjfkBCm(supp Ψj) hm
where C > 0 depends only on m and p. The convergence of the multi-dimensional series can be concluded from a generalisation of the integral test. We first note that the function wp :Rd>0 →R, t7→(1 +|t|)−p is an integrable and non-negative mono-tone decreasing function, which means that the maximum of wp over the interval [j, j + 2] is attained at the left end point, i.e.
t∈[j,2+j]max (1 +|t|)−p = (1 +|j|)−p. Using symmetry properties we can write the series in the form
X
The convergence of the series now follows from the estimate wp(j)≤
Z
[j,j+1]
wp(t)dt, j ∈Zd>0,
which is a direct consequence of the monotonicity and which implies that X
Chapter 5
Operator approximations
In this chapter we analyse operator approximations that can be used either
• in the context of Nyström methods together with the finite section method as a truncation scheme
or
• in the context of the recently proposed multi-section method, where, in con-trast to the classical Nyström method, a linear least squares problem has to be solved.
We are interested in approximating integral operators (Aψ)(x) :=
Z
R2
K(x,y)J(y)ψ(y)dy, x∈R2, (5.1) whereJ(y) =p
1 +|∇f(y)|2denotes as usual the surface area element, by operators of the form
(Ahψ)(x) := X
j∈Z2
Kh,j(x)J(hj)ψ(hj), x∈R2, h >0, (5.2) where Kh,j,j ∈Z2,is a family of regular functions that have to be determined.
For a reasonable large class of kernel functions and densities one can get an ap-proximation of the form (5.2) by replacing the integral by the composite trapezoidal rule. Thus we get
(Ahψ)(x) = Q2h[K(x,·)J ψ] = X
j∈Z2
Kh,j(x)J(hj)ψ(hj), x∈R2 where
Kh,j(x) :=h2K(x, hj), j ∈Z2. (5.3) These kind of operator approximations are used in Nyström methods and it is known that the operatorAh cannot converge in norm to the original operator Afor h→0. The best one can hope for instead is pointwise convergence, i.e.
kAψ−Ahψk →0, h→0, (5.4)
for a reasonable class of functions ψ and an appropriate norm k · k.
5.1 Approximations for weighted differentiable kernels
In the case that the kernel and density exhibits a certain rate of decay, we are able to prove convergence and quantify a convergence rate.
Lemma 5.1. Let A be given through (5.1) with K ∈BCpm(Rd×Rd), ψ ∈BCqm(Rd) with p+q ≥ d and f ∈ BCm+1(Rd) for some m ∈ N. Then, the operator Ah for h >0 given through (5.2), with Kh,j given through (5.3), converges pointwise to A and the error can be estimated through
kAψ−AhψkBC(Rd) ≤CkK(x,·)kBCm
p (Rd)kJkBCm(Rd)kψkBCm
q (Rd)hm, (5.5) for some constant C > 0 dependent only on m and p.
Proof. For f ∈ BCm+1(Rd) we see that J ∈ BCm(Rd) so that Lemma 4.10 yields K(x,·)J ψ ∈ BCp+qm (Rd). Lemma 4.7 ensures that K(x,·)J ψ ∈ L1(Rd) for all x∈Rd, as p+q > d. HenceLemma 4.12 yields
kAψ−AhψkBC(Rd) = sup
x∈Rd
Qd
K(x,·)J ψ
−Qdh
K(x,·)J ψ
≤ckK(x,·)J ψkBCm
p+q(Rd)hm so that the bound (5.5) follows from Lemma 4.10.
Remark 5.2. The reason for our interest in this particular case is that the global part of the single-layer and double-layer operator satisfy these conditions for d= 2 and p = 2, as we have shown in the estimates (1.22) and (1.28). To retain the convergence order, it suffices that the density satisfies the mild decay property ψ ∈ BCqm(R2) for q >0.