IMA Journal of Numerical Analysis (2023) 43, 2934–2964
https://doi.org/10.1093/imanum/drac059
Advance Access publication on 15 October 2022
Tino Franz and Holger Wendland∗
Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany
∗ Corresponding author: holger.wendland@uni-bayreuth.de
[Received on 28 March 2022; revised on 23 August 2022]
Quasi-interpolation methods are well-established tools in multivariate approximation. They are efficient
as they do not require, in contrast to interpolation, the solution of a linear system. Quasi-interpolations
are often first studied on infinite grids. Here, it is usually required that the quasi-interpolation operator
reproduces polynomials of a certain degree exactly. This degree corresponds to the approximation order
of the quasi-interpolation process. Unfortunately, if a radial, compactly supported kernel is employed for
building the quasi-interpolation operator, it is well known that polynomial reproduction is impossible
in two or more dimensions. As such operators are numerically appealing and are frequently used in
particle methods, we will, in this paper, look at such quasi-interpolation operators that do not reproduce
polynomials and show that they lead, when employed in a multilevel scheme, to an efficient and converging
approximation method.
Keywords: multivariate approximation; quasi-interpolation; multilevel methods.
1. Introduction
The efficient approximation of an unknown multivariate function f : Rd → R using only information
of that function at a discrete point set, i.e., fj = f (xj ), j ∈ J, is one of the most fundamental problems in
approximation theory and numerical analysis. While interpolation and approximation with radial basis
functions (see for example Buhmann, 2003; Wendland, 2005; Fasshauer, 2007) and moving-least squares
(see for example Levin, 1998; Wendland, 2001) are by now well-established tools, they often require the
solution of a linear system. A related, but easier-to-implement, concept is that of quasi-interpolation;
see for example Buhmann & Jäger (2022), de Boor (1990). Here, the approximation to f : Rd → R is
formed using a fixed kernel Φ : Rd → R and certain functionals λj , j ∈ J, by setting
Qf :=
λj (f )Φε (· − xj ),
j∈J
where Φε := ε−d Φ(·/ε) is a scaled version of the original kernel. Crucial for an efficient evaluation is,
of course, that the functionals λj can be computed in constant time. Quasi-interpolation processes have
also been thoroughly investigated in the context of shift-invariant spaces; see for example de Boor et al.
(1994a,b, 1998), Johnson (1997).
Besides their role in classical multivariate approximation theory, quasi-interpolation operators are
extremely popular in particle methods such as the smoothed-particle hydrodynamics method (see for
example Di Lisio et al., 1998; Vila, 1999; Monaghan, 2005, 2012; Violeau, 2012) for discretising partial
© The Author(s) 2022. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Multilevel quasi-interpolation
MULTILEVEL QUASI-INTERPOLATION
2935
Qp = p,
p ∈ πm (Rd ).
(1.1)
This can either be achieved by imposing certain conditions on the kernel Φ or on the functionals λj or
both. The best investigated situation is when the data sites xj form a regular grid of size h > 0, i.e.,
xj = hj, j ∈ Zd , the scaling parameter ε is proportional to h, i.e., ε = ch with c > 0 and the functionals
are essentially point evaluation functionals, i.e., the operator takes the form
Qh f = hd
f (hj)Φch (· − hj).
(1.2)
j∈Zd
In this situation, it is well known that polynomial reproduction is equivalent to the so-called Strang–Fix
conditions; see for example Strang et al. (1973). Moreover, these conditions are equivalent to the fact
that the space spanned by the translates of the scaled kernel provides good approximations. For example,
for any Sobolev function f ∈ W2m+1 (Rd ), we have the optimal approximation result
inf f − sL2 (Rd ) ≤ Chm+1 f W m+1 (Rd ) ,
s∈Sh
2
Sh := span{Φch (· − hj) : j ∈ Zd }.
As mentioned above, particularly in the area of particle methods, compactly supported and radial
kernels Φ = φ( · 2 ) are used. Here, however, we face the problem that such kernels do not satisfy the
Strang–Fix conditions in any space dimension d ≥ 2; see Wendland (2017), Wu (1997). Hence, such
kernels, when employed in quasi-interpolation operators (1.2), will not lead to a convergent result. As a
matter of fact, it is known that the error behaves like (h/ε)m+1 ; see for example Raviart (1983).
As interpolation with compactly supported radial basis functions faces the same problem, in that
context, the idea of multilevel approximation schemes has been introduced (see Floater & Iske, 1996)
and shown to lead to converging schemes (see for example Le Gia et al. 2010; Wendland, 2010, 2017).
It is the goal of this paper to show that such multilevel techniques also convert nonconverging multilevel
quasi-interpolation schemes into converging ones. The only additional assumption on the kernel Φ is that
it satisfies a so-called moment condition of a certain order m; see Definition 2.2 below. This condition
is essentially a continuous version of the polynomial reproduction property (1.1) above. Its advantage
is that it is possible to have compactly supported radial kernels that satisfy the moment condition. As
a matter of fact, any non-negative, radial kernel with compact support satisfies the moment condition
of order m = 2. The moment condition can also be seen as a generalisation of the generalised StrangFix condition introduced in Wu & Liu (2005). In that paper, similar techniques to this paper are used
to analyse the approximation power of quasi-interpolation operators. However, these operators are a
significant modification of operator (1.2). In particular, the authors do not study the stationary setting
with ε = ch but a nonstationary setting, which is numerically less attractive.
Unfortunately, the moment condition itself cannot lead to a converging quasi-interpolation operator
(1.2), as the moment condition is not equivalent to the Strang–Fix conditions. Hence, we need to combine
the moment condition with a multilevel approximation scheme.
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differential equations in fluid dynamics. Here, Φ : Rd → R is usually a compactly supported radial
function.
In general, quasi-interpolation operators are required to reproduce polynomials of a certain degree
m ∈ N0 , i.e., to satisfy
2936
T. FRANZ AND H. WENDLAND
Mn f = Mn−1 f + Qhn En−1 f ,
En f = En−1 f − Qhn En−1 f .
This means that the global, multilevel approximation Mn f is computed by correcting the previous
approximation Mn−1 f by a local approximation Qhn En−1 f of the previous error.
The following result is a specific, simplified version of the general convergence results, which we
will prove in this paper.
Theorem 1.1 Let 1 ≤ p < ∞ and σ ∈ N with σ > d/p. Let Φ ∈ Wp2σ (Rd ) ∩ L1 (Rd ) satisfy the
moment condition of order m < σ . Let hn = μn with μ ∈ (0, 1). Then there is a constant C > 0 and a
constant q = q(μ, m) such that q ∈ (0, 1) for sufficiently small μ and sufficiently large m and such that
f − Mn f Lp (Rd ) ≤ Cqn f Wpσ (Rd ) ,
f ∈ Wpσ (Rd ).
Hence, the multilevel quasi-interpolation approximation scheme converges linearly in the number of
levels.
So far, results of this form seem to have limited use, as the terms sufficiently small and sufficiently
large cannot yet be quantified. However, the numerical results at the end of this paper clearly indicate
that convergence can be achieved with reasonable values. This means in particular that this paper is just
a first step in understanding the convergence of multilevel quasi-interpolants.
The paper is organised as follows. In the rest of this section, we introduce the necessary notation. In
the next section, we analyse one-level approximations by splitting the error f − Qh f into a convolution
and a discretisation error. In the third section, we analyse the multilevel quasi-interpolation method and
prove our main results. In the fourth section, we review techniques for constructing kernels satisfying
the moment condition of an arbitrary order. In the final section, we give some numerical examples.
1.1 Notation
As usual, Wpσ (Rd ), σ ∈ N0 denotes the Sobolev space of all σ -times weakly differentiable functions
in Lp (Rd ) also having weak derivatives in Lp (Rd ). Throughout this paper, we will restrict ourselves to
integer σ , though the results also hold for fractional-order Sobolev spaces by interpolation. We will
use the standard Sobolev norm denoted by · Wpσ (Rd ) , and the standard Sobolev seminorm denoted by
| · |Wpσ (Rd ) .
In the following, we will use a weighted variation of the standard Sobolev norm. For f ∈ Wpσ (Rd )
and ε > 0, the weighted Sobolev norm is defined by
f Wp,ε
σ (Rd ) =
⎧
σ
⎪
p
⎪
⎨
εkp |f | k
k=0
Wp (Rd )
1/p
⎪
⎪
⎩ max εk |f |W∞
k (Rd )
0≤k≤σ
for 1 ≤ p < ∞,
for p = ∞.
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The idea behind the multilevel scheme is a simple residual correction method. Instead of one data
set hZd , we will use a sequence of data sets hj Zd with h1 ≥ h2 ≥ h3 ≥ · · · . Then, we start the scheme
by setting M0 f = 0 and E0 f = f and compute for n = 1, 2, . . . , recursively,
2937
MULTILEVEL QUASI-INTERPOLATION
p,1
|f |Wp,ε
σ (Rd ) =
p
It is easy to see that f W σ (Rd ) =
p,ε
εσ |f |Wpσ (Rd )
for 1 ≤ p < ∞,
εσ |f |
for p = ∞.
σ (Rd )
W∞
σ
p
k (Rd ) . Next, for 0
k=0 |f |Wp,ε
≤ τ1 ≤ τ2 ≤ σ , we will define
the following (generalised) weighted Sobolev seminorms as the sum over the above-defined weighted
seminorms. To be more precise, we set
τ1 ,τ2
|f |Wp,ε
(Rd ) =
⎧
1/p
τ1
⎪
p
⎪
kp
⎨
ε |f |W k (Rd )
for 1 ≤ p < ∞,
⎪
⎪
⎩ max
for p = ∞.
p
k=τ1
τ1 ≤k≤τ2
εk |f |W∞
k (Rd )
σ ,σ
Note that we simply have |f |Wp,ε
= f Wp,ε
σ (Rd ) and |f | 0,σ
σ (Rd ) .
(Rd ) = |f |Wp,ε
Wp,ε (Rd )
We will also use the following notation for p norms. Let J be an index set and (fj )j∈J be a sequence.
Then
⎧
1/p
⎪
⎪
p
⎪
⎨
|fj |
for 1 ≤ p < ∞,
j∈J
(fj )j p (J) :=
⎪
⎪
⎪sup |fj |
for p = ∞.
⎩
j∈J
If the set J is given by {τ1 , τ1 + 1, . . . , τ2 } with two integers τ1 ≤ τ2 , we simply write (fj ) p (τ1 ,τ2 )
instead of (fj )j p ({τ1 ,...,τ2 }) .
2. One-level approximation
In this section we will investigate the one-level quasi-interpolation operator Qh as defined in (1.2) but
will allow for a general scaling ε > 0 instead of only looking at ε = ch. As we intend to interpret the
quasi-interpolation operator as the discretisation of a convolution, we will first discuss these processes
separately.
After that, we will combine both concepts to analyse the approximation scheme. We will first give
all estimates in the generalised Sobolev seminorms and then simplify the results to the standard norms
and seminorms.
2.1 Convolution
The convolution f ∗ g : Rd → R of two functions f , g : Rd → R is given by the parameter-dependent
integral
f ∗g=
Rd
f (y)g(· − y) dy,
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Obviously, we have that, for ε = 1, the weighted Sobolev norm and the standard norm coincide, i.e.,
· W σ (Rd ) = · Wpσ (Rd ) . The weighted Sobolev seminorm is given by
2938
T. FRANZ AND H. WENDLAND
Lemma 2.1
1. Young’s inequality: let 1 ≤ p, q, r ≤ ∞ be such that 1p + 1q = 1 + 1r . If f ∈ Lp (Rd ) and g ∈ Lq (Rd )
then f ∗ g ∈ Lr (Rd ) and
f ∗ gLr (Rd ) ≤ f Lp (Rd ) gLq (Rd ) .
2. Generalised Minkowski inequality: let (S1 , μ1 ) and (S2 , μ2 ) be σ -finite measure spaces and let
1 ≤ p < ∞. If F : S1 × S2 → R is measurable, then
1/p
1/p
p
F(x, y) dμ (y) p dμ (x)
≤
|F(x,
y)|
dμ
(x)
dμ2 (y).
2
1
1
S1
S2
S2
S1
Young’s inequality immediately carries over to our generalised weighted Sobolev spaces. For
example, with r = p, we have q = 1 and find for 0 ≤ τ1 ≤ τ2 ≤ σ , f ∈ Wpσ (Rd ), g ∈ L1 (Rd )
and δ > 0, by straightforward calculations,
|f ∗ g|W τ1 ,τ2 (Rd ) ≤ |f |W τ1 ,τ2 (Rd ) gL1 (Rd ) .
p,δ
(2.1)
p,δ
In contrast to classical quasi-interpolation schemes that satisfy the discrete requirement of reproducing polynomials in the sense of (1.1), we will only require a corresponding continuous condition, which
leads to a good approximation by convolution.
Definition 2.2 A kernel Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N if
i)
Rd
ii)
Rd
iii)
CΦ,m :=
Rd
Φ(x)dx = 1,
xα Φ(x)dx = 0,
α ∈ Nd0 ,
1 ≤ |α| < m,
xm
2 |Φ(x)|dx < ∞.
When working with a continuous compactly supported function Φ : Rd → R, all integrals are
obviously well defined. Moreover, the third condition is automatically satisfied. However, the constant
CΦ,m will play a crucial role in what follows.
It is easy to verify that for an ε > 0, the scaled kernel Φε := ε−d Φ(·/ε) satisfies the moment
condition of order m if and only if Φ satisfies the moment condition of order m; see for example Franz
& Wendland (2018).
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whenever the integral on the right-hand side is well defined. Throughout this paper, we will use standard
estimates for convolution and other integral operators. In particular, we will employ Young’s inequality
and a generalised version of Minkowski’s inequality, which we state in the following lemma for the
convenience of the reader. A proof of Young’s inequality can be found in any book on higher analysis
or functional analysis. Minkowski’s generalised inequality can, for example, be found in Stein (1971).
2939
MULTILEVEL QUASI-INTERPOLATION
δ0 := 1 + ΦL1 (Rd ) ,
δm := CΦ,m
dm
,
m!
m ≥ 1,
(2.2)
where CΦ,m is the constant from Definition 2.2.
In our first result, we will keep the same weights in the seminorms on the left-hand and right-hand
sides of the inequality.
Theorem 2.3 Assume Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N. Let 0 ≤ τ1 ≤
σ (Rd ) ∩ Cσ (Rd ),
τ2 ≤ σ with σ ≥ m + τ2 and ε, δ > 0. If f ∈ Wpσ (Rd ) for 1 ≤ p < ∞ or f ∈ W∞
then
|f − f ∗ Φε |W τ1 ,τ2 (Rd ) ≤ δm
ε m
p,δ
δ
|f |W m+τ1 ,m+τ2 (Rd ) ,
p,δ
where δm , m ∈ N0 is given by (2.2).
Proof. We begin by discussing the case σ = τ2 , i.e., m = 0. In this situation, we can use Young’s
inequality in the form (2.1) to derive
|f − f ∗ Φε |W τ1 ,τ2 (Rd ) ≤ |f |W τ1 ,τ2 (Rd ) + |f ∗ Φε |W τ1 ,τ2 (Rd )
p,δ
p,δ
p,δ
≤ 1 + ΦL1 (Rd ) |f |W τ1 ,τ2 (Rd ) ,
p,δ
using also Φε L1 (Rd ) = ΦL1 (Rd ) .
Next we consider the case σ > τ2 . For 1 ≤ p < ∞, we first assume that f ∈ C∞ (Rd ) ∩ Wpm (Rd ).
Hence, in both cases, 1 ≤ p < ∞ and p = ∞, we can take the Taylor expansion of f up to order m with
integral remainder
f (x − y) = f (x) +
1≤|α|<m
+m
Dα f (x)
(−y)α
α!
1 0 |α|=m
(1 − s)m−1 (−y)α α
D f (x − sy) ds.
α!
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The convergence order in terms of ε of the approximation by convolution with a kernel satisfying
the moment condition is well known for Sobolev functions; see, for example, Raviart (1983). We will
generalise these results using weighted Sobolev seminorms. Unfortunately, in such a situation, we might
lose convergence in ε if the norms are weighted proportionally. To have convergence in our multilevel
scheme, we thus need to calculate all constants involved as accurately as possible. In particular, we need
to track how the constants depend on the space dimension d and, more importantly, on the order m of
the moment condition. In the rest of the paper, we therefore need the constants
2940
T. FRANZ AND H. WENDLAND
f (x) − f ∗ Φε (x) =
Rd
[f (x) − f (x − y)] Φε (y) dy
= −
Rd
m
1 0 |α|=m
(1 − s)m−1 (−y)α α
D f (x − sy)Φε (y) ds dy.
α!
With this equation and the triangle inequality, we have
1 m−1
α
(1 − s)
(−y) α
D f (· − sy)Φε (y) ds dy W τ1 ,τ2 (Rd )
|f − f ∗ Φε |W τ1 ,τ2 (Rd ) = m
p,δ
α!
0 |α|=m
Rd
p,δ
1 F (·, z)dz τ1 ,τ2 d ,
≤m
α
Wp,δ (R )
α! S2
|α|=m
using S2 := Rd × [0, 1], z = (y, s) ∈ S2 and Fα (·, z) = (1 − s)m−1 (−y)α Dα f (· − sy)Φε (y).
From now on, we will restrict ourselves to the case 1 ≤ p < ∞. The case p = ∞ is handled in a
similar way. Thus, the seminorm on the right-hand side becomes
p
τ2
kp
β
F (·, z) dz p τ ,τ
δ
α
D Fα (·, z) dz
Wp,δ1 2 (Rd ) =
S2
S2
|β|=k
k=τ1
(2.3)
Lp (Rd )
and the generalised Minkowski inequality from Lemma 2.1 together with the change of variables from
x − sy to x yields for the Lp -norm on the right-hand side,
Dβ F (·, z) dz
α
S2
1 ≤
≤
Rd
Rd
0
Rd
Rd
S2
Lp (Rd )
=
≤
1/p
β
dz
D Fα (x, z) p dx
1/p
ds dy
(1 − s)m−1 yα Φε (y) p |Dβ+α f (x − sy)|p dx
ym
2 |Φε (y)| dy
1
(1 − s)m−1 ds
0
Rd
1/p
|Dβ+α f (x)|p dx
1
C εm Dβ+α f Lp (Rd ) ,
m Φ,m
where, in the last step, we have used that the integral over s is bounded by 1/m and that for the integral
over y the moment condition of the function Φ yields
Rd
m
ym
2 |Φε (y)| dy = ε
Rd
m
ym
2 |Φ(y)| dy = CΦ,m ε .
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Multiplying this equation by Φε (x), integrating it over Rd and using the first two properties of the moment
condition for Φε leads to
2941
MULTILEVEL QUASI-INTERPOLATION
⎛
⎞
1/p
τ2
1
F (·, z) dz τ1 ,τ2 d ≤ ⎝
C εm Dβ+α f p⎠
δ kp
α
Lp (Rd ) m Φ,m
Wp,δ (R )
S2
|β|=k
k=τ1
=
1
CΦ,m εm Dα f W τ1 ,τ2 (Rd ) ,
p,δ
m
which then gives the bound
1 |f − f ∗ Φε |W τ1 ,τ2 (Rd ) ≤ m
Fα (·, z) dz W τ1 ,τ2 (Rd )
p,δ
p,δ
α! S2
|α|=m
≤ CΦ,m εm
1
|Dα f |W τ1 ,τ2 (Rd ) .
p,δ
α!
|α|=m
The remaining sum can be bounded by
⎛
⎞
1
1
⎠ max |Dα f | τ1 ,τ2 d
|Dα f |W τ1 ,τ2 (Rd ) ≤ ⎝
Wp,δ (R )
p,δ
α!
α! |α|=m
|α|=m
|α|=m
⎛
⎞1/p
τ2
dm
p
max ⎝
≤
δ kp |Dα f |W k (Rd ) ⎠
p
m! |α|=m
k=τ1
⎛
≤
dm
m!
⎝
τ2
⎞1/p
p
Wp
δ kp |f | k+m
k=τ1
(Rd )
⎠
⎛
⎞1/p
m+τ
dm −m ⎝ 2 kp p
δ
≤
δ |f |W k (Rd ) ⎠
p
m!
k=m+τ1
=
dm
m!
δ −m |f |W m+τ1 ,m+τ2 (Rd ) .
p,δ
The fact that Wpσ (Rd ) ∩ C∞ (Rd ) is dense in Wpσ (Rd ) finishes the proof for this case. As mentioned
above, the case p = ∞ is treated in an analogous way.
Direct consequences of this theorem are the following estimates for weighted Sobolev norms and
seminorms. The first of the next two estimates follows by setting τ1 = 0 and τ2 = τ and the second one
by setting τ1 = τ2 = τ .
Corollary 2.4 Assume Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N. Let 0 ≤ τ ≤ σ
σ (Rd ) ∩ Cσ (Rd ), then we have
with σ ≥ m + τ and ε, δ > 0. If f ∈ Wpσ (Rd ) for 1 ≤ p < ∞ or f ∈ W∞
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Inserting this into (2.3) and taking the pth root shows
2942
T. FRANZ AND H. WENDLAND
f − f ∗ Φε W τ (Rd ) ≤ δm
ε m
|f |W m,m+τ (Rd ) ,
δ
p,δ
p,δ
and on the other hand
|f − f ∗ Φε |W τ (Rd ) ≤ δm
ε m
δ
p,δ
|f |W m+τ (Rd ) .
p,δ
Next we will modify these estimates by allowing different weights in the Sobolev norm. However,
we will do this only in the case where the Sobolev norm on the right-hand side is not weighted by δ > 0
but instead by ε, i.e., the scaling parameter of our kernel Φε .
Theorem 2.5 Assume Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N. Let 0 ≤ τ1 ≤
τ2 ≤ σ and ε, δ > 0. Set k1 := min{σ − m − 1, τ2 } and k2 := max{τ1 , σ − m}. Let f ∈ Wpσ (Rd ) for
σ (Rd ) ∩ Cσ (Rd ). Then
1 ≤ p < ∞ or f ∈ W∞
δ k
|f − f ∗ Φε |W τ1 ,τ2 (Rd ) ≤ δm |f |Wp,ε
m+k d
(R )
p,δ
ε
δ
+
ε
k
δσ −k k
k
p (τ1 ,k1 )
(2.4)
|f |Wp,ε
σ (Rd ) .
p (k2 ,τ2 )
Proof. We only deal with the case 1 ≤ p < ∞. The case p = ∞ is treated similarly. We first note that
we have with Fε := f − f ∗ Φε the identity
τ2
p
=
Wp,δ (Rd )
|Fε | k
k=τ1
k1
k=τ1
p
+
Wp,δ (Rd )
|Fε | k
τ2
p
.
Wp,δ (Rd )
|Fε | k
k=k2
(2.5)
To see this, we notice that in the case of σ − m ≤ τ1 we have k2 = τ1 , i.e., the second sum in (2.5) is the
full sum, and because of σ − m − 1 ≤ τ1 − 1 ≤ τ2 − 1, also k1 = σ − m − 1 ≤ τ1 − 1, i.e., the first sum
is empty.
If, however, σ − m > τ1 then we have k2 = σ − m. If σ − m > τ2 then the second sum in (2.5) is
empty and k1 = τ2 , i.e., the first sum is the full sum. If σ − m ≤ τ2 , then we have k1 = σ − m − 1, i.e.,
k2 = k1 + 1, meaning that both sums on the right-hand side of (2.5) yield the sum on the left-hand side.
Thus, using the second part of Corollary 2.4 with δ = ε, we see for 1 ≤ p < ∞,
p
Wp,δ
|f − f ∗ Φε | τ1 ,τ2
(Rd )
≤
k1 δ
k=τ1
p
≤ δm
kp
ε
p,ε
k1 δ
k=τ1
p
|f − f ∗ Φε |W k (Rd ) +
ε
kp
p
+
Wp,ε (Rd )
|f | m+k
τ2 δ
k=k2
ε
kp
p
|f − f ∗ Φε |W k (Rd )
ε
k=k2
τ2 δ
kp
p,ε
p
p
δσ −k |f |W σ (Rd ) ,
p,ε
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on the one hand
2943
MULTILEVEL QUASI-INTERPOLATION
To simplify the notation, we will, from now on, assume that the smoothness σ of the target function
and the order of the moment condition m satisfy σ ≥ m.
We can easily specialize the last theorem to the case of weighted Sobolev norms and to standard
Sobolev norms.
Corollary 2.6 Assume Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N. Let σ ≥ m and
σ (Rd ) ∩ Cσ (Rd ). If m = σ then
ε, δ > 0. Let f ∈ Wpσ (Rd ) for 1 ≤ p < ∞ or f ∈ W∞
δ
f − f ∗ Φε W σ (Rd ) ≤ p,δ
ε
k
δσ −k k
|f |Wp,ε
σ (Rd ) .
(2.6)
p (0,σ )
If m < σ and δ ≤ ε, then
δ
f − f ∗ Φε W σ (Rd ) ≤ δm f Wp,ε
σ (Rd ) + p,δ
ε
k
δσ −k k
|f |Wp,ε
σ (Rd ) .
(2.7)
p (σ −m,σ )
Finally, if m ≤ σ , then
|f − f ∗ Φε |W σ (Rd ) ≤ δ0
p,δ
δ
ε
σ
|f |Wp,ε
σ (Rd ) .
(2.8)
Proof. With τ1 = 0, τ2 = σ and σ ≥ m, we have k1 = σ − m − 1 and k2 = σ − m. Hence, (2.4)
becomes
δ k
f − f ∗ Φε W σ (Rd ) ≤ δm |f |Wp,ε
m+k d
(R )
p,δ
ε
δ
+
ε
k
δσ −k k
k
p (0,σ −m−1)
|f |Wp,ε
σ (Rd ) .
p (σ −m,σ )
In the case σ = m, the first term comprises an empty sum and thus this inequality reduces to the
stated bound (2.6). If, however, m < σ then, because of δ ≤ ε, the first norm can be bounded by
δ
ε
k
|f |Wp,ε
m+k d
(R )
k
≤ |f |Wp,ε
m+k d
(R ) p (0,σ −m−1)
k
= f Wp,ε
σ −1 d ≤ f W σ (Rd ) ,
(R )
p,ε
p (0,σ −m−1)
yielding (2.7). Finally, for (2.8), we have to set τ1 = τ2 = σ , which yields with σ ≥ m this time
k1 = σ − m − 1 and k2 = σ . Thus, again the first norm in (2.4) vanishes and the second norm reduces
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where in the application of Corollary 2.4 to the second sum we notice that the necessary condition σ −
k2 ≤ m is automatically satisfied.
Taking the pth root and using the triangle inequality for p gives the stated bound (2.4).
2944
T. FRANZ AND H. WENDLAND
δ
|f − f ∗ Φε |W σ (Rd ) ≤ p,δ
ε
k
δσ −k k
|f |Wp,ε
σ (Rd ) = δ0
p (σ ,σ )
δ
ε
σ
|f |Wp,ε
σ (Rd ) ,
which is the stated inequality.
The next result concerns intermediate Sobolev smoothness.
Corollary 2.7 Assume Φ ∈ L1 (Rd ) satisfies the moment condition of order m ∈ N. Let 0 ≤ τ ≤ σ
with σ ≥ m and ε ∈ (0, 1]. If f ∈ Wpσ (Rd ), then
τ
− min{σ −m−1,τ }
−k
f − f ∗ Φε Wpτ (Rd ) ≤ δm ε
f Wp,ε
ε δσ −k |f |Wp,ε
σ (Rd ) +
σ (Rd ) .
k=σ −m
Proof. Here, we have τ1 = 0 and τ2 = τ giving k1 = min{σ − m − 1, τ } and k2 = σ − m. With δ = 1,
this gives, as in the last corollary,
1
f − f ∗ Φε Wpτ (Rd ) ≤ δm ε
1
+
ε
≤ δm ε
using the fact that every
k
|f |Wp,ε
m+k d
(R )
k
δσ −k − min{σ −m−1,τ }
k
k
p (0,min{σ −m−1,τ })
|f |Wp,ε
σ (Rd )
p (σ −m,τ )
1
f Wp,ε
σ (Rd ) + ε
p norm can be bounded by the 1 norm.
k
δσ −k k
|f |Wp,ε
σ (Rd ) ,
1 (σ −m,τ )
2.2 Quadrature
We will investigate the approximation of an integral over Rn by a simple quadrature formula. To this
end, we will divide the space Rd into infinitely many cubes of length h > 0 with midpoints xj = hj for
j ∈ Zd . For this midpoint rule, the following result is well known; see for example Raviart (1983).
Theorem 2.8 Let f ∈ W1σ (Rd ), σ > d and xj = hj, j ∈ Zd , h > 0. Then there exists a constant Cd,σ > 0
such that
d
f (xj ) ≤ Cd,σ hσ |f |W σ (Rd ) .
d f (x) dx − h
1
R
j∈Zd
The proof of this theorem employs the Bramble–Hilbert lemma. This, however, means that the
constant Cd,σ given in this result is not explicitly known. There exist formulations of this result for
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to one term with index k = σ , i.e.,
2945
MULTILEVEL QUASI-INTERPOLATION
Lemma 2.9 Let 0 ≤ τ1 ≤ τ2 ≤ σ and 1 ≤ p, q, r ≤ ∞ such that 1 + 1p = 1q + 1r . Then there exists a
constant C > 0 such that
f ∗ g − hd
f (hj)g(· − hj) W τ1 ,τ2 (Rd )
p,δ
j∈Zd
α α−β ≤ Cd,σ hσ
f
D
Lq (Rd )
β
|α|=σ β≤α
δ |γ | Dβ+γ g
τ1 ≤|γ |≤τ2
Lr (Rd )
holds for all f ∈ Wqσ (Rd ) ∩ C(Rd ) and g ∈ Wrσ +τ2 (Rd ) ∩ C(Rd ).
Proof. We start by looking at the specific case τ1 = τ2 = 0. Let x ∈ Rd . A direct application of
Theorem 2.8, which will be justified by the following calculations, shows
≤ C hσ |f (·)g(x − ·)| σ d .
f ∗ g(x) − hd
f
(hj)g(x
−
hj)
d,σ
W1 (R )
j∈Zd
Using the multivariate Leibniz rule, the seminorm on the right-hand side can then be bounded by a sum
of convolutions as
|f (·)g(x − ·)|W σ (Rd ) =
1
|α|=σ
Rd
α
D (f (y)g(x − y)) dy
α
β α−β
β
=
f (y)D g(x − y) dy
(−1) D
d
β
|α|=σ R β≤α
α ≤
Dα−β f (y)Dβ g(x − y) dy
β Rd
|α|=σ β≤α
=
α |Dα−β f | ∗ |Dβ g| (x).
β
|α|=σ β≤α
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sub-classes of functions, from which the constant can be estimated. However, for functions in W1σ (Rd ),
no such results seem to be known.
In a first step, we will apply the above result to analyse the quadrature of a general convolution
integral. After that, we will apply this result to the specific situation, where the convolution takes place
with kernels from Section 2.1.
2946
T. FRANZ AND H. WENDLAND
f ∗ g − hd
f (hj)g(· − hj)
j∈Zd
α |Dα−β f | ∗ |Dβ g| ≤ Cd,σ hσ |α|=σ β≤α β
d
Lp (R )
≤ Cd,σ hσ
|α|=σ β≤α
≤ Cd,σ h
σ
Lp (Rd )
α |Dα−β f | ∗ |Dβ g| Lp (Rd )
β
α α−β β f
g
,
D
D
Lq (Rd )
Lr (Rd )
β
|α|=σ β≤α
which finishes the proof in the situation τ1 = τ2 = 0. For the general situation, we only consider the case
τ1 ,τ2
(Rd ), we can
1 ≤ p < ∞. The case p = ∞ is dealt with similarly. Recall that for an arbitrary f ∈ Wp,δ
p
p
write the generalised weighted seminorm by f τ1 ,τ2 d = τ1 ≤|γ |≤τ2 δ p|γ | Dγ f L (Rd ) . Inserting the
Wp,δ
(R )
p
result above yields
f ∗ g − hd
f
(hj)g(·
−
hj)
τ1 ,τ2
Wp
j∈Zd
⎛
≤⎝
(Rd )
⎛
τ1 ≤|γ |≤τ2
⎞p ⎞1/p
α ⎠ ⎠
δ p|γ | ⎝Cd,σ hσ
Dα−β f Dβ+γ g
Lq (Rd )
Lr (Rd )
β
|α|=σ β≤α
⎛
α α−β ⎝
≤ Cd,σ hσ
f
D
Lq (Rd )
β
|α|=σ β≤α
≤ Cd,σ hσ
|α|=σ β≤α
using again the fact that the
α α−β f
D
Lq (Rd )
β
Lr (Rd )
τ1 ≤|γ |≤τ2
δ |γ | Dβ+γ g
⎞1/p
p
δ p|γ | Dβ+γ g
Lr (Rd )
τ1 ≤|γ |≤τ2
⎠
,
p norm can always be bounded by the 1 norm.
As mentioned above, we will now specify this result by considering the convolution with our scaled
kernels.
Corollary 2.10 Let 0 ≤ τ1 ≤ τ2 ≤ σ . Let ε, δ, h > 0 and Φ ∈ Wpσ +τ2 (Rd ) ∩ C(Rd ). Then
f ∗ Φ − hd
f
(hj)Φ
(·
−
hj)
ε
ε
τ1 ,τ2
d
j∈Z
Wp,δ
≤
Cσ ,τ2 ,d
h
ε
σ
max
δ
ε
τ1
,
δ
ε
τ2 f Wp,ε
σ (Rd )
(Rd )
holds for all f ∈ Wpσ (Rd ) ∩ C(Rd ), where Cσ ,τ2 ,d = Cσ ,τ2 ,d (Φ) = 2σ σ +d−1
d−1 Cd,σ |Φ| W σ +τ2 (Rd ) .
1
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The triangle inequality and Young’s inequality then yield
2947
MULTILEVEL QUASI-INTERPOLATION
Wp,δ
≤ Cd,σ hσ
(R )
α α−β f
D
Lp (Rd )
β
|α|=σ β≤α
δ |γ | Dβ+γ Φε L1 (Rd )
τ1 ≤|γ |≤τ2
.
For the p-norm of f , we see that
α−β f
≤ |f | W |α|−|β| (Rd ) ≤ ε|β|−|α| |f | W |α|−|β| (Rd ) ≤ ε|β|−σ f Wp,ε
D
σ (Rd ) .
d
p
Lp (R )
p,ε
The sum over γ can be estimated by
δ |γ | Dβ+γ Φε ≤
d
L1 (R )
τ1 ≤|γ |≤τ2
τ1 ≤|γ |≤τ2
≤ε
−|β|
β+γ Φ
D
|β|+|γ |
δ |γ |
ε
L1 (Rd )
ΦW σ +τ2 (Rd ) max
1
δ
ε
τ1
δ
,
ε
τ2 .
The remaining sum computes to
α
|α|=σ β≤α
β
= 2σ
= 2σ
|α|=σ
σ +d−1
.
d−1
Again, this immediately leads to the following bounds for the full weighted Sobolev norm and the
weighted Sobolev seminorm by setting τ1 = 0, τ2 = τ and τ1 = τ2 = τ , respectively.
Corollary 2.11 Let 0 ≤ τ ≤ σ , ε, δ, h > 0 and Φ ∈ Wpσ +τ (Rd ) ∩ C(Rd ). Then
σ
τ
h
δ
f ∗ Φ − hd
f Wp,ε
f (hj)Φε (· − hj)
≤ Cσ ,τ ,d
max 1,
σ (Rd ) ,
ε
ε
ε
τ d
j∈Zd
Wp,δ (R )
f ∗ Φ − hd
f (hj)Φε (· − hj)
ε
τ
j∈Zd
Wp,δ
≤ Cσ ,τ ,d
h
ε
σ δ
ε
τ
f Wp,ε
σ (Rd )
(Rd )
for every f ∈ Wpσ (Rd ) ∩ C(Rd ).
Note that the constant Cσ ,τ ,d appears to be quite large. However, it is based on a rather pessimistic
estimate, which seems to be worse than it actually is. We will see this later on when looking at specific
examples.
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Proof. We use Lemma 2.9 with q = p, r = 1 and g = Φε . Hence, we have
f ∗ Φ − hd
f
(hj)Φ
(·
−
hj)
ε
ε
τ1 ,τ2 d
d
j∈Z
2948
T. FRANZ AND H. WENDLAND
We will now combine both approximation techniques to analyse our quasi-interpolation operator. We
have to be aware that we need point evaluations of the function f in our approximation scheme. Hence,
if 1 ≤ p < ∞, we will assume that σ > d/p to ensure that f ∈ Wpσ (Rd ) is continuous. For p = ∞, we
σ +1 (Rd ). Under this condition, the theorem of Morrey states that f
will assume that σ > 0 and f ∈ W∞
σ
,1
d
belongs to the Hölder space C (R ), such that the conditions of Theorem 2.5 are satisfied.
Let Φ ∈ Wpk (Rd ). For 1 ≤ p < ∞, we define the quasi-interpolation operator Q : Wpσ (Rd ) →
k
Wp (Rd ) with σ > d/p as in (1.2), i.e., by
Qf := hd
f (hj)Φε (· − hj).
j∈Zd
For p = ∞ the operator is defined as a mapping Q : Wpσ +1 (Rd ) → Wpk (Rd ).
As mentioned at the beginning of this section, we can split up the error of the approximation operator
into a convolution error and a quadrature error by
|f − Qf |W τ1 ,τ2 (Rd ) ≤ |f − f ∗ Φε |W τ1 ,τ2 (Rd ) + |f ∗ Φε − Qf |W τ1 ,τ2 (Rd ) .
p,δ
p,δ
p,δ
(2.9)
We can use Theorem 2.5, or, more precisely, Corollaries 2.6 and 2.7 to bound the first term. The
quadrature error can be bounded by employing Corollary 2.10.
For the analysis of the multilevel method, we will need two particular versions of (2.9). First, we
need the above estimate for the Wpτ (Rd ) norm.
Corollary 2.12 Let 0 ≤ τ ≤ σ . If 1 ≤ p < ∞, then let σ > d/p and f ∈ Wpσ (Rd ). Otherwise, let
σ +1 (Rd ). Let h > 0 and 0 < ε ≤ 1. Suppose Φ ∈ W σ +τ (Rd ) ∩ L (Rd ) satisfies the
σ > 0 and f ∈ W∞
p
1
moment condition of order m ≤ σ . Then
f − Qf Wpτ (Rd ) ≤ ε−τ
τ
k=σ −m
ετ −k δσ −k |f |Wp,ε
σ (Rd )
h
+ δm εmax{τ −σ +m+1,0} + Cσ ,τ ,d
ε
σ
f Wp,ε
σ (Rd ) .
Proof. As mentioned above, we split the error according to (2.9) with the specific choices δ = 1, τ1 = 0
and τ2 = τ . Then Corollary 2.7 gives for the first term in (2.9) the bound
f − f ∗ Φε Wpτ (Rd ) ≤ δm ε
− min{σ −m−1,τ }
f Wp,ε
σ (Rd ) +
τ
k=σ −m
ε
−k
δσ −k |f |Wp,ε
σ (Rd ) .
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2.3 Analysis of the quasi-interpolation operator
2949
MULTILEVEL QUASI-INTERPOLATION
f ∗ Φε − Qf Wpτ (Rd ) ≤ Cσ ,τ2 ,d
= Cσ ,τ ,d
h
ε
h
ε
σ
max
σ 1
ε
τ1
δ
ε
τ2 δ
,
ε
f Wp,ε
σ (Rd )
τ
f Wp,ε
σ (Rd ) .
Both bounds together yield the stated estimate.
The second result we need prepares for a level-to-level estimate in the multilevel method. Hence, the
order of the Sobolev norm remains the same while the weights are changing.
Corollary 2.13 If 1 ≤ p < ∞, then let σ > d/p and f ∈ Wpσ (Rd ). Otherwise, let σ > 0 and
σ +1 (Rd ). Let ε, δ, h > 0 with δ ≤ ε ≤ 1. Suppose Φ ∈ W 2σ (Rd ) ∩ L (Rd ) satisfies the moment
f ∈ W∞
p
1
condition of order m < σ . Then,
f − Qf W σ (Rd ) ≤
p,δ
σ
δ
ε
δσ −k
k=σ −m
and
|f − Qf |W σ (Rd ) ≤
p,δ
k
δ
ε
h
|f |Wp,ε
σ (Rd ) + δm + Cσ ,σ ,d
ε
σ δ0 |f |Wp,ε
Cσ ,σ ,d
σ (Rd ) + σ
f Wp,ε
σ (Rd )
(2.10)
σ
h
ε
f Wp,ε
σ (Rd ) .
(2.11)
Proof. We start again with the basic error split (2.9). For (2.10), we have τ1 = 0 and τ2 = σ . Thus, the
bound (2.7) from Corollary 2.6 yields for the convolution error
δ
f − f ∗ Φε W σ (Rd ) ≤ δm f Wp,ε
σ (Rd ) + p,δ
ε
p,δ
= Cσ ,σ ,d
h
ε
σ
h
ε
σ
k
k=σ −m
using again that the p -norm can be bounded by the
according to Corollary 2.10, this time be bounded by
δσ −k σ
δ
ε
≤ δm f Wp,ε
σ (Rd ) +
f ∗ Φε − Qf W τ (Rd ) ≤ Cσ ,τ2 ,d
k
k
|f |Wp,ε
σ (Rd )
p (σ −m,σ )
δσ −k |f |Wp,ε
σ (Rd ) ,
1 -norm. The second term in (2.9) can, again
max
δ
ε
τ1
δ
,
ε
τ2 f Wp,ε
σ (Rd )
f Wp,ε
σ (Rd ) ,
using in particular τ1 = 0 and δ ≤ ε. Both bounds together yield the stated estimate (2.10).
The second bound (2.11) is proven in the same way, using again the estimate just derived on the
discretisation error and using (2.8) from Corollary 2.6 for the convolution error.
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The second term in (2.9) can, according to Corollary 2.10, be bounded by
2950
T. FRANZ AND H. WENDLAND
2.4 Discussion of the constants
For our further analysis, it is important to know the size of the constants involved in the convergence
estimates.
Here, we have first of all the constants δm from (2.2) playing a crucial role in the analysis of the
m
convolution given in Theorem 2.3. They are mainly given by δm = CΦ,m dm! . While the quotient tends
to zero for m → ∞, we still need to study the behaviour of the constant CΦ,m , which is defined by the
moment condition. In applications, we will choose kernels Φ having compact support in the closed unit
ball B1 [0]. In this case, the constant CΦ,m can be bounded by
CΦ,m =
Rd
xm
2 |Φ(x)| dx ≤
2π d/2
ΦL∞ (Rd )
Γ (d/2)
1
rm+d−1 dr =
0
2π d/2
1
ΦL∞ (Rd ) ,
m + d Γ (d/2)
which means that CΦ,m also converges to zero for m → ∞ or for d → ∞.
The constants appearing in the analysis of the quadrature process are more complicated. They are
given by
σ σ +d−1
Cd,σ |Φ| W σ +τ2 (Rd ) ,
Cσ ,τ2 ,d = Cσ ,τ2 ,d (Φ) = 2
1
d−1
where Cd,σ is the constant from Theorem 2.8 and mainly determined by the constant of the Bramble–
Hilbert lemma. Even if no analytical estimate exists for this constant, numerical tests seem to indicate
that Cd,σ < 2. The remaining part of the constant, however, will be large and exponentially growing in
d and σ . Nonetheless, as mentioned in Section 2.2, the estimates made to achieve this result are far from
being optimal.
3. The multilevel method
We now want to analyse the multilevel quasi-interpolation operator. To this end, we first rewrite the
previous results in an appropriate form, reflecting the level structure of the method. Then we recall the
definition of the multilevel scheme and summarise some additional assumptions and straightforward
representations. Finally, we derive our error analysis.
3.1 Notation and auxiliary results
For every level n ∈ N, let xnj = hn j, j ∈ Zd be the midpoints of the cubes with edge length hn > 0.
Let Φ : Rn → R be a continuous and integrable kernel. For each level n, let Φn := Φεn be the scaled
kernel with scaling parameter εn ∈ (0, 1] with εn+1 ≤ εn .
On each level, we define the quasi-interpolation operator Qn as in Section 2.3 by
Qn f = hdn
j∈Zd
f (xnj )Φn (· − xnj ).
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As mentioned before, this result means that we do not show convergence in the stationary setting
ε = ch.
2951
MULTILEVEL QUASI-INTERPOLATION
Lemma 3.1 If 1 ≤ p < ∞, then let σ > d/p and f ∈ Wpσ (Rd ). Otherwise, let σ > 0 and
σ +1 (Rd ).
f ∈ W∞
1. If 0 ≤ τ ≤ σ and Φ ∈ Wpσ +τ (Rd ) ∩ L1 (Rd ) satisfies the moment condition of order m < σ , then
f − Qn f Wpτ (Rd ) ≤
εn−τ
τ
k=σ −m
εnτ −k δσ −k |f |Wp,n
σ (Rd )
hn
max{τ −σ +m+1,0}
+ δm ε
+ Cσ ,τ ,d
εn
σ
f Wp,n
σ (Rd ) .
2. If Φ ∈ Wp2σ (Rd ) ∩ L1 (Rd ) satisfies the moment condition of order m < σ , then
σ
εn+1 k
f − Qn f W σ (Rd ) ≤
δσ −k |f |Wp,n
σ (Rd )
p,n+1
εn
k=σ −m
hn σ
f Wp,n
+ δm + Cσ ,σ ,d
σ (Rd ) .
εn
3. If Φ ∈ Wp2σ (Rd ) ∩ L1 (Rd ) satisfies the moment condition of order m < σ , then
εn+1 σ
hn σ
f Wp,n
δ0 |f |Wp,n
|f − Qn f |W σ (Rd ) ≤
σ (Rd ) + Cσ ,σ ,d
σ (Rd ) .
p,n+1
εn
εn
Next we intend to derive a recursion formula, which will be necessary for the analysis of the
multilevel scheme. To this end, we assume that there are constants 0 < μ ≤ 1 and 0 < Cμ ≤ 1
such that
Cμ μhn ≤ hn+1 ≤ μhn ,
n ∈ N0 ,
(3.1)
which means that the individual sets of points become uniformly finer at the same rate. Moreover, we
will assume that the support radius of the kernel on each level depends linearly on the mesh size hn , i.e.,
that there is a constant 0 < ν < 1 such that
hn = νεn ,
n ∈ N0 .
(3.2)
Corollary 3.2 If 1 ≤ p < ∞, then let σ > d/p and f ∈ Wpσ (Rd ). Otherwise, let σ > 0 and
σ +1 (Rd ). Assume Φ ∈ W 2σ (Rd ) ∩ L (Rd ) satisfies the moment condition of order m < σ . For
f ∈ W∞
p
1
(n)
(n)
(n)
(n)
n ∈ N, define numbers αj , βj , 0 ≤ j ≤ n, as follows. The numbers αn , βn are chosen arbitrarily
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For the sake of readability, we will write · Wp,n
σ (Rd ) = · W σ (Rd ) for the weighted Sobolev
p,εn
norm. For the weighted seminorms, we will use analogous notation. For the convenience of the reader,
we summarise the results of Corollaries 2.12 and 2.13 using this notation.
2952
T. FRANZ AND H. WENDLAND
σ
(n)
(n)
(n)
αj := αj+1 μσ δ0 + βj+1
k=σ −m
(n)
μk δσ −k ,
βj
(n)
(n)
:= Cσ ,σ ,d ν σ μσ αj+1 + δm + Cσ ,σ ,d ν σ βj+1 .
Then, for j = n − 1, n − 2, . . . , 1, the error f − Qj f satisfies the recursion
(n)
αj+1 |f − Qj f |W σ
d
p,j+1 (R )
(n)
+ βj+1 f − Qj f W σ
d
p,j+1 (R )
(n)
(n)
(n)
≤ αj |f |W σ (Rd ) + βj f W σ (Rd ) .
p,j
p,j
(n)
Moreover, if for 0 ≤ τ ≤ σ the numbers αn and βn are defined by
τ
αn(n) :=
k=σ −m
εnτ −k δσ −k ,
βn(n) := δm εnmax{τ −σ +m+1,0} + Cσ ,τ ,d ν σ ,
then the error satisfies the bound
(n)
f − Qn f Wpσ (Rd ) ≤ εn−τ αn(n) |f |Wp,n
σ (Rd ) + βn f W σ (Rd ) .
p,n
(n)
(n)
Proof. For simplicity, we will write αj , βj instead of αj and βj .
Using j instead of n and multiplying the second and third bound of Lemma 3.1 by βj+1 and αj+1 ,
respectively, yields, together with (3.1) and (3.2),
αj+1 |f − Qj f |W σ
≤ αj+1 μσ δ0 |f |W σ (Rd ) + Cσ ,σ ,d ν d f W σ (Rd ) ,
p,j
p,j
σ
k
d
μ δσ −k |f |Wp,n
Cσ ,σ ,d ν f W σ (Rd ) .
σ (Rd ) + δm + (Rd ) ≤ βj+1
d
p,j+1 (R )
βj+1 f − Qj f W σ
p,j+1
p,j
k=σ −m
Adding these two equations, reordering the terms and denoting the resulting factor in front of |f |W σ (Rd )
p,j
by αj and the factor in front of f W σ (Rd ) by βj gives the first statement. The second statement
p,j
immediately follows from the first bound in Lemma 3.1.
3.2 The method
Let f ∈ Wpσ (Rd ). As mentioned in the introduction, the multilevel quasi-interpolation approximation
method is defined by two sequences of operators (Mn ) and (En ), where En describes the error at level n
and Mn describes the approximation at level n. These operators are defined by setting M0 f = 0, E0 f = f
and then by computing recursively
Mn f = Mn−1 f + Qn En−1 f ,
En f = En−1 f − Qn En−1 f .
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and then, for j = n − 1, n − 2, . . . , 1, they are given recursively by
2953
MULTILEVEL QUASI-INTERPOLATION
n
I
Qn Qn−1 · · · Qm
Qj :=
j=m
if n < m,
if n ≥ 1,
where I denotes the identity operator. As the operators Qj usually do not commute, the order of the
operators is crucial.
Proposition 3.3 The operators Mn and En of the multilevel quasi-interpolation method satisfy
Mn =
n
Qj
j−1
(I − Q ),
n
En =
(I − Qj ),
=1
j=1
j=1
where I is the identity operator.
Proof. This follows easily by induction. For example, for the error operators, we obviously have
En+1 = (I − Qn+1 )En ,
E1 = I − Q1 ,
which immediately leads to the stated representation. Similarly, for the multilevel operators Mn , we have
M1 = M0 + Q1 E0 = Q1 and
Mn+1 = Mn + Qn+1 En =
n
Qj
j−1
(I − Q ) + Qn+1
=1
j=1
j−1
n
n+1
(I − Qj ) =
Qj
(I − Q ).
j=1
j=1
=1
3.3 Convergence results
In this subsection we will prove our main result on the convergence of the multilevel method.
Theorem 3.4 Let 0 ≤ τ ≤ σ . If 1 ≤ p < ∞, then let σ > d/p and f ∈ Wpσ (Rd ). Otherwise, let σ > 0
σ +1 (Rd ). If Φ ∈ W 2σ (Rd ) ∩ L (Rd ) satisfies the moment condition of order m < σ , then the
and f ∈ W∞
p
1
error f − Mn f satisfies
(n)
(n)
f − Mn f Wpτ (Rd ) ≤ εn−τ α1 ε0σ μσ + β1 f Wpσ (Rd )
(n) f W σ (Rd ) ,
α1(n) ε0σ μσ + β
≤ ε0−τ 1
p
(n)
(n)
(n)
where the coefficients (αj ) and (βj ) are from Corollary 3.2 and the new coefficients (
αj )j and
−τ −τ n+1−j (n)
−τ −τ n+1−j (n)
(n)
(n)
(n)
α
= Cμ μ
α and β
= Cμ μ
β and satisfy for
(β )j are given by j
j
j
j
j
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Given this recursion, it is easy to see that multilevel quasi-interpolation operator Mn and the multilevel
error operator En satisfy the following recursions. To formulate these, we use the notation
2954
T. FRANZ AND H. WENDLAND
αj(n)
=
(n) −τ σ −τ (n) Cμ−τ
Cμ μ
αj+1
δ0 + β
j+1
σ
k−τ
μ
k=σ −m
δσ −k ,
(n)
(n)
(n) = Cσ ,σ ,d Cμ−τ ν σ μσ −τ αj+1 + Cμ−τ μ−τ δm + Cσ ,σ ,d ν σ β
β
j
j+1
with initial values
αn(n) = Cμ−τ μ−τ αn(n) ,
n(n) = Cμ−τ μ−τ βn(n) .
β
Proof. As before, we simplify the notation by omitting the upper index (n) in the definition of the
coefficients. Using the second statement of Corollary 3.2 to start the recursion and then the first statement
multiple times yields
f − Mn f Wpτ (Rd ) = En−1 f − Qn En−1 f Wpτ (Rd )
≤ εn−τ αn |En−1 f |Wp,n
σ (Rd ) + βn En−1 f W σ (Rd )
p,n
≤ εn−τ α1 |E0 f |W σ (Rd ) + β1 E0 f W σ (Rd )
p,1
p,1
−τ
= εn α1 |f |W σ (Rd ) + β1 f W σ (Rd )
p,1
p,1
σ σ
−τ
≤ εn α1 ε0 μ + β1 f Wpσ (Rd ) ,
where, in the last step, we have used that |f |W σ (Rd ) = ε1σ |f |Wpσ (Rd ) ≤ ε0σ μσ f Wpσ (Rd ) and f W σ (Rd ) ≤
p,1
p,1
f Wpσ (Rd ) .
−τ
n
= ε0−τ Cμ−τ μ−τ .
From Cμ μεn−1 ≤ εn , we can conclude εn−τ ≤ (Cμ μεn−1 )−τ ≤ Cμn μn ε0
With this, we can show the stated recursion of the new coefficients, as we have
−τ −τ n+1−j
Cμ μ
αj
σ
−τ −τ n+1−j
σ
k
= Cμ μ
μ δσ −k
αj+1 μ δ0 + βj+1
αj =
k=σ −m
σ
j+1
= Cμ−τ μ−τ αj+1 μσ δ0 + β
μk δσ −k
k=σ −m
σ
j+1 Cμ−τ
= αj+1 Cμ−τ μσ −τ δ0 + β
μk−τ δσ −k
k=σ −m
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j = n − 1, n − 2, . . . , 1 the recursions
2955
MULTILEVEL QUASI-INTERPOLATION
j = Cμ−τ μ−τ n+1−j βj
β
!
n+1−j
= Cμ−τ μ−τ
Cσ ,σ ,d ν σ βj+1
Cσ ,σ ,d ν σ μσ αj+1 + δm + !
j+1
αj+1 + δm + Cσ ,σ ,d ν σ β
= Cμ−τ μ−τ Cσ ,σ ,d ν σ μσ j+1 .
= Cσ ,σ ,d Cμ−τ ν σ μσ −τ αj+1 + Cσ ,σ ,d Cμ−τ ν σ μ−τ β
So far it is not obvious that the above result indeed guarantees convergence under reasonable
assumptions. Typically for multilevel methods, such assumptions usually comprise the refinement ratio μ
and the ratio ν between the grid size and the support radius and convergence is attained if both quantities
are sufficiently small. Here, however, we will also need to consider the order m of the moment condition.
Let us now introduce
η := Cσ ,σ ,d ν σ ,
σ
γ :=
k=σ −m
μk δσ −k .
(3.3)
As both μ and ν can be chosen arbitrarily small, it is possible to make both parameters as small as
required. In particular, we can choose them in such a way that
γ ≤
δm
+ 1,
η
(3.4)
which is, for example, satisfied if we require the stronger condition γ ≤ 1. With this notation, the
recursion of the coefficients αj , βj can be expressed as
(n)
αj
(n)
(n)
= αj+1 μσ δ0 + βj+1 γ ,
(n)
βj
(n)
(n)
= αj+1 μσ η + βj+1 (δm + η).
Hence, both recursions have the same pattern and differ only in some constants. Using the fact that δ0 ≥ 1
and assuming (3.4) gives
(n)
αj
δ (n)
(n)
(n)
(n)
≤ δ0 αj+1 μσ + βj+1 γ = 0 αj+1 μσ η + βj+1 γ η
η
δ
δ
(n)
(n)
(n)
≤ 0 αj+1 μσ η + βj+1 (δm + η) = 0 βj ,
η
η
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and, in the same way,
2956
T. FRANZ AND H. WENDLAND
δ (n) σ
(n) σ
(n) (n) δm + η ≤ 0 βj+1
δm + η
βj(n) = αj+1
μ η + βj+1
μ η + βj+1
η
"
#
(n)
= δ0 μσ + δm + η βj+1
"
#n−j (n)
≤ δ0 μσ + δm + η
βn ,
αj(n) ≤
"
#n−j (n)
δ0 (n) δ0 βj ≤
δ0 μσ + δm + η
βn .
η
η
j )j , this becomes
For the modified sequences (
αj )j and (β
#n−j (n)
δ0 "
n ,
δ0 Cμ−τ μσ −τ + Cμ−τ μ−τ δm + η
β
η
(3.5)
"
#
(n) ≤ δ0 Cμ−τ μσ −τ + Cμ−τ μ−τ δm + η n−j β
n(n) .
β
j
(3.6)
(n)
αj
≤
This leads to the following convergence result.
Corollary 3.5 Suppose that the conditions of Theorem 3.4 are satisfied and let ε0 = 1. Suppose that
there is a constant Cν > 0 such that μ = Cν ν. Let ν > 0 be sufficiently small and m sufficiently large
such that
σ
k=σ −m
μk δσ −k ≤ 1
and
q := Cμ−τ δ0 μσ −τ + μ−τ δm + Cσ ,σ ,d ν σ < 1.
Then there exists a constant C = C(Cν , σ , δ0 ) independent of n such that the quasi-interpolation
multilevel method is converging with
f − Mn f Wpτ (Rd ) ≤ Cqn f Wpσ (Rd ) .
Proof. The definition of η from (3.3) shows that (3.5) and (3.6) yield
α1(n) ≤
δ0 n−1 (n)
n ,
q β
η
n(n) .
(n) ≤ qn−1 β
β
1
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so that we have the recursive inequalities
MULTILEVEL QUASI-INTERPOLATION
2957
(n)
(n) f W σ (Rd )
α1 μσ + β
1
p
δ0
n(n) f W σ (Rd ) .
≤
Cνσ + 1 qn−1 β
p
Cσ ,σ ,d
f − Mn f Wpτ (Rd ) ≤
Finally, from its definition, we can conclude
n(n) = Cμ−τ μ−τ βn(n)
β
= Cμ−τ μ−τ δm εnmax{τ −σ +m+1,0} + Cσ ,τ ,d ν σ
≤ Cμ−τ μ−τ δm + Cσ ,τ ,d ν σ
$
Cσ ,τ ,d
≤ max 1,
q,
Cσ ,σ ,d
which yields the stated result.
The above result shows that we can indeed achieve convergence, if we choose the parameters m, ν
Cσ ,σ ,d ν σ μ−τ become sufficiently small. In
and μ = Cν ν such that the three terms δ0 μσ −τ , δm μ−τ and the case of pure approximation errors, i.e., τ = 0, this simplifies dramatically, as the negative influence
μ−τ disappears. If we additionally choose the constants Cμ = Cν = 1, then we have μ = ν and hn = ν n ,
εn = hn /ν = hn−1 and only need to guarantee that
σ
k=σ −m
ν k δσ −k ≤ 1,
σ
δ0 ν + δm + Cσ ,σ ,d ν σ < 1
is satisfied. Obviously, if we first choose m ∈ N such that δm < 1, then there is a ν0 > 0 such that for all
ν ∈ (0, ν0 ) these inequalities are satisfied.
4. Kernel construction
The goal of this section is to remind the reader of techniques that allow the easy construction of kernels
satisfying a higher-order moment condition. Proofs of the results can be found in Franz & Wendland
(2018), Ramming & Wendland (2018) but go back to ideas from Beale & Majda (1985).
4.1 Construction scheme
We will concentrate on radial basis functions as kernels, as in this situation, the moment condition can
be expressed using only univariate integrals. To be more precise, the following result holds.
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Thus, Theorem 3.4 shows, together with μ = Cν ν and ε0 = 1,
2958
T. FRANZ AND H. WENDLAND
1
sd−1 φ(s) ds =
0
1
Γ ( d2 )
,
2π d/2
s2k+d−1 φ(s) ds = 0,
(4.1)
1 ≤ k ≤ M − 1,
(4.2)
0
1
s2M+d−1 φ(s) ds < ∞,
(4.3)
0
where Γ denotes the Gamma function; see Gradshteyn & Ryzhik (2000, 8.310). Then Φ = φ( · 2 )
satisfies the moment condition of order m = 2M.
We now want to give a simple way to construct radial kernel functions satisfying the moment
condition of arbitrary order. For that, we follow the ideas of Beale & Majda (1985), see also Majda
& Bertozzi (2002) and Ramming & Wendland (2018), by constructing a linear combination of scaled
kernel functions. Suppose that M ∈ N. Consider the kernel as a linear combination of scaled functions,
i.e., consider
φ(s) =
M
s
aj
λj ψ
j=1
,
s ∈ [0, ∞),
(4.4)
with λj ∈ R for 1 ≤ j ≤ M, 0 < a1 < a2 < · · · < aM = 1 ∈ R and ψ : [0, ∞) → R being a continuous
function with compact support. With this kind of kernel, we can change the conditions imposed on the
kernel to conditions imposed on the parameters λj and aj . Inserting (4.4) into (4.1), we find the new
condition
M
j=1
⎞−1
d ⎛∞
Γ
2 ⎝
λj adj =
sd−1 ψ(s) ds⎠ =: c0 .
2π d/2
(4.5)
0
Furthermore, condition (4.2) will be satisfied if
M
λj a2k+d
= 0,
j
1 ≤ k ≤ M − 1.
(4.6)
j=1
With predefined aj , this leads to a linear system of M equations for the M coefficients λj , which can be
shown to have a unique solution. As a matter of fact, it is possible to write down this solution explicitly
as follows.
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Lemma 4.1 Let φ ∈ C[0, ∞) have compact support and let M ∈ N. Suppose φ satisfies
2959
MULTILEVEL QUASI-INTERPOLATION
λj =
M
c0 a2k
adj k=1 a2k − a2j
k =j
a21 · · · a2j−1 a2j+1 · · · a2M
c
= (−1)j−1 0d 2
aj (aj − a21 ) · · · (a2j − a2j−1 )(a2j+1 − a2j ) · · · (a2M − a2j )
for 1 ≤ j ≤ M.
This method of construction of the kernel gives us a very simple way to satisfy the moment condition
of any desired order. Moreover, this construction is, except for the constant c0 , independent of the choice
for the function ψ. However, the smoothness of the function φ is determined by the smoothness of the
function ψ.
4.2 Examples of kernel functions
For the sake of simplicity, we now choose a ∈ RM to be equidistant; more precisely we assume that
λj := adj λj via
aj = bj for b > 0 and for 1 ≤ j ≤ M. Then we can calculate λj = c0
M
k=1
k =j
k2
k
k
(bk)2
=
c
=
c
0
0
k−jk+j
(bk)2 − (bj)2
k 2 − j2
m
M
k=1
k =j
k=1
k =j
⎛
⎞⎛
⎞
j−1
M
M
(M! )2 ⎝ 1 ⎠ ⎝ 1 ⎠
1
= c0 2
2j
k−j
k−j
k+j
j
k=1
= (−1)j−1 2c0
= (−1)j−1 2c0
= (−1)j−1 2c0
(M! )2
j
⎛
⎝
k=j+1
j−1
1
k=1
k
⎞⎛
⎠⎝
M−j
k=1
⎞⎛
1⎠ ⎝
k
k=1
M+j
k=1+j
⎞
1⎠
k
M!
M!
(M + j)! (M − j)!
j
M+1−k
k=1
M+k
.
With this representation for λj , which is independent of the choice of b, we can easily verify that
M+1−j
λj = − M+j λj−1 . However, the usual choice of is b = 1/M to ensure that φ has the same compact
support as ψ. Examples of typical values for λj ∈ R for 1 ≤ j ≤ M and 0 < a1 < a2 < · · · < aM = 1 ∈ R
can be found in Table 1.
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Proposition 4.2 Let M ∈ N, aj > 0 be pairwise distinct for 1 ≤ j ≤ M and c0 ∈ R. Then the
coefficients λj are given by
2960
T. FRANZ AND H. WENDLAND
Parameters for d = 2 and c0 = 1 on the left and d = 3 and c0 = 1 on the right
M
a
1
2
a1
a1
a2
a1
a2
a3
a1
a2
a3
a4
3
4
λ
1
1/2
1
1/3
2/3
1
1/4
1/2
3/4
1
λ1
λ1
λ2
λ1
λ2
λ3
λ1
λ2
λ3
λ4
1
16/3
−1/3
27/2
−27/20
1/10
128/5
−16/5
8/315
−1/35
M
a
1
2
a1
a1
a2
a1
a2
a3
a1
a2
a3
a4
3
4
λ
1
1/2
1
1/3
2/3
1
1/4
1/2
3/4
1
λ1
λ1
λ2
λ1
λ2
λ3
λ1
λ2
λ3
λ4
1
32/3
−1/3
81/2
−81/10
1/10
512/5
−32/5
32/945
−1/35
Table 2 Errors for the first test example with d = 2 and c0 = 1; ψ is the C6 Wendland function given
by r → (1 − r)+ (32r3 + 25r2 + 8r + 1)
Level
hn
εn
1
1.25e−1
1.
2
6.25e−2
5e−1
3
3.13e−2 2.5e−1
4
1.56e−2 1.25e−1
5
7.81e−3 6.25e−2
6
3.91e−3 3.13e−2
7
1.95e−3 1.56e−2
m=2
Error
q
m=4
Error
q
m=6
Error
q
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
1.54e−2
4.35e−2
3.62e−3
1.64e−2
8.04e−4
4.40e−3
1.26e−4
7.84e−4
1.29e−5
8.46e−5
7.94e−7
7.25-6
2.76e−8
3.04e−7
—
—
0.24
0.38
0.22
0.27
0.16
0.18
0.10
0.11
0.06
0.09
0.03
0.04
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
2.73e−3
1.35e−2
4.57e−4
2.65e−3
5.47e−5
3.57e−4
3.67e−6
3.21e−5
1.30e−7
1.59e−6
5.45e−9
8.58e−8
3.85e−10
7.06e−9
—
—
0.17
0.02
0.11
0.14
0.07
0.09
0.04
0.05
0.04
0.05
0.07
0.08
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
2
∞
1.59e−3
8.53e−3
2.37e−4
1.44e−3
2.18e−5
1.56e−4
9.69e−7
1.03e−5
2.70e−8
4.43e−7
1.61e−9
2.68e−8
6.89e−11
1.51e−9
—
—
0.15
0.17
0.09
0.11
0.04
0.07
0.03
0.04
0.06
0.06
0.04
0.06
5. Numerical tests
In this section we want to verify our analytical error bounds numerically. Since our analysis is given for
functions defined on all of Rd , we can simplify the error computation by looking either at compactly
supported or periodic target functions. We will use the former. We will look at two test examples in R2 .
The first one consists of an infinitely smooth target function, while the second example uses a target
function of limited smoothness.
In both cases, we use point sets with grid length hn = 2−2−n and grid points (xnj )k = 2−2−n (2−1 +jk )
for j ∈ Zd , 1 ≤ k ≤ d and 1 ≤ n ≤ 7. In this case, we have μ = 1/2 and Cμ = 1. We will choose
d = 2, εn = 8hn , such that ν = 1/8 and ε1 = 1. As kernels, we use the normalised versions of the C4
(k = 2) and C6 (k = 3) Wendland functions. To obtain a higher-order moment condition, we use the
construction from Section 4.
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Table 1
MULTILEVEL QUASI-INTERPOLATION
2961
Fig. 2. Example 1: ∞ error of each level on the left-hand side and values for q on the right-hand side.
5.1 Test example 1
A first test example approximates the infinitely smooth blob function f : R2 → R given by
e−1/(1−x2 )
0
2
x = (x1 , x2 ) →
if x2 < 1,
else.
The results of this test example can be found in Table 2 and in Figs 1 and 2. We also give estimates on
the constant q, which is required to be in (0, 1) for convergence. The estimates are simply computed by
dividing two successive errors.
As one can see, the error is smaller for kernels satisfying a higher-order moment condition. When
comparing the order 6 kernel with the order 2 kernel, we see that we need about two levels less to achieve
a comparable error. Moreover, the value for q, the improvement of the error per level, is smaller for the
higher-order kernels.
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Fig. 1. Example 1: 2 error of each level on the left-hand side and values for q on the right-hand side.
2962
T. FRANZ AND H. WENDLAND
Fig. 4. Example 2: ∞ error of each level on the left-hand side and values for q on the right-hand side.
Nonetheless, though our theoretical result requires m to be sufficiently large, we even have good
convergence for m = 2, which describes the kernels mostly used in particle methods.
In the last two levels, we see that the value for q is smaller for the kernels with m = 2 than for
higher-order kernels. Nevertheless, we gain convergence for all kernel functions.
5.2 Test example 2
For our second example, we choose the C1 -function f : R2 → R given by
x = (x1 , x2 ) →
(1 − x2 )2
0
if x2 < 1,
else.
In this case, the test function is not particularly smooth. This means that it does not satisfy the required
conditions for our error analysis for larger m.
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Fig. 3. Example 2: 2 error of each level on the left-hand side and values for q on the right-hand side.
MULTILEVEL QUASI-INTERPOLATION
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