Bounds for the Weighted Lp Discrepancy and Tractability of
Integration
Gunther Leobacher and Friedrich Pillichshammer
Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria
E-mail: gunther.leobacher@jku.at, friedrich.pillichshammer@jku.at
Quite recently Sloan and Woźniakowski [4] introduced a new notion of discrepancy, the so called weighted Lp discrepancy of points in the d-dimensional
unit cube for a sequence γ = (γ1 , γ2 , . . .) of weights. In this paper we prove a
nice formula for the weighted Lp discrepancy for even p. We use this formula to
derive an upper bound for the average weighted Lp discrepancy. This bound
enables us to give conditions on the sequence of weights γ such that there
exists N points in [0, 1)d for which the weighted Lp discrepancy is uniformly
bounded in d and goes to zero polynomially in N −1 .
Finally we use these facts to generalize some results from Sloan and Woźniakowski [4] on (strong) QMC-tractability of integration in weighted Sobolev
spaces.
Key Words: Weighted discrepancy, QMC-tractability of integration
1. INTRODUCTION AND BASIC NOTATIONS
Quite recently Sloan and Woźniakowski [4] introduced a new notion of
discrepancy, the so called weighted Lp (star) discrepancy. To introduce
this notion we need a sequence γ = (γ1 , γ2 , . . .) of weights with 1 = γ1 ≥
γ2 ≥ . . . ≥ γd ≥ . . . > 0.
At first we need some
Q notation: let D denote the index set D = {1, 2, . . . , d}.
For u ⊆ D let γu = j∈u γj , γ∅ = 1, |u| the cardinality of u and for a vector
x in I d = [0, 1)d let xu denote the vector from I |u| = [0, 1)|u| containing
Q
the components of x whose indices are in u. Further let dxu = j∈u dxj
and let (xu , 1) be the vector x from I d , with all components whose indices
are not in u replaced by 1.
Now we can give the definition of weighted Lp discrepancy.
1
2
G. LEOBACHER, F. PILLICHSHAMMER
Definition 1.1. For any N points t1 , . . . , tN in the d-dimensional unit
cube I d = [0, 1)d and any x = (x1 , . . . , xd ) in I d let
disc(x) = x1 . . . xd −
|{i : ti ∈ [0, x)}|
.
N
Then the weighted Lp discrepancy LpN,γ for 1 ≤ p < ∞ is defined as
LpN,γ

Z
 X p/2
p
= LN,γ ({ti }) = 
γu
I¯|u|
u⊆D
u6=∅
1/p

|disc((xu , 1))|p dxu 
.
For p = ∞ the weighted star discrepancy is given by
1/2
∗
DN,γ
= sup max γu
u⊆D
x∈I d u6=∅
|disc((xu , 1))|.
In [3] S. Joe gave an explicit formula for the weighted L2 discrepancy.
He proved
Proposition 1.1. For any N points t1 , . . . , tN in I d and any sequence
γ = (γ1 , γ2 , . . .) of weights we have
(L2N,γ )2 =
d Y
j=1
+
1+
N
d
2 XY
γj
γj −
1+
(1 − t2n,j )
3
N n=1 j=1
2
N
N
d
1 X XY
(1 + γj (1 − max(tm,j , tn,j ))),
N 2 m=1 n=1 j=1
where tn,j is the j-th component of the point tn .
In Section 2 we generalize this formula to a formula for the weighted Lp
discrepancy for even p (Theorem 2.1).
In [4] Sloan and Woźniakowski showed that one can find points t1 , . . . , tN
in the d-dimensional unit cube for which the weighted L2 discrepancy L2N,γ
is uniformly bounded
in d and goes to zero polynomially in N −1 as long
P∞
as the series j=1 γj < ∞ is convergent and they anticipated that the
same holds for the weighted Lp discrepancy LpN,γ as long as the series
P∞ p/2
< ∞ is convergent.
j=1 γj
In Section 3 we prove a formula (Theorem 3.1) and an upper bound
(Corollary 3.1) for the average weighted Lp discrepancy LpN,γ for even p.
WEIGHTED DISCREPANCY AND TRACTABILITY
3
Then we use this upper bound to prove the above conjecture of Sloan and
Woźniakowski (Corollary 3.2).
Let us turn now to an application of the previous results, the QMCtractability of integration in weighted Sobolev spaces. First we repeat
some basic facts and definitions. We deal with multivariate integration
Z
Id (f ) := f (t) dt
I¯d
for functions defined over the d-dimensional unit cube I¯d = [0, 1]d which
belong to some normed space Fd . We denote the norm in Fd by k.kd.
A quasi-Monte Carlo (QMC) algorithm QN,d is of the form
QN,d(f ) :=
N
1 X
f (tn ),
N n=1
where tn , 1 ≤ n ≤ N , are points in I d .
We define the worst case error of the QMC algorithm QN,d by its worst
performance over the unit ball of Fd ,
e(QN,d ) :=
sup
f ∈Fd , kf kd ≤1
|Id (f ) − QN,d(f )|.
For N = 0 we set Q0,d (f ) := 0 such that
e(Q0,d ) =
sup
f ∈Fd ,kf kd ≤1
|Id (f )| = kId k
is the initial error, i.e., the a priori error in multivariate integration without
sampling the function.
We would like to reduce the initial error by a factor of ε, where ε ∈ (0, 1).
Let
Nmin (ε, d) = min{N : ∃QN,d such that e(QN,d) ≤ ε e(Q0,d )}.
Definition 1.2.
1. We say that multivariate integration in the space Fd is QMC-tractable
if there exist nonnegative C, α and β such that
Nmin (ε, d) ≤ C dα ε−β
holds for all dimensions d = 1, 2, . . . and for all ε ∈ (0, 1).
2. We say that multivariate integration in the space Fd is strongly QMCtractable if the inequality above holds with α = 0.
4
G. LEOBACHER, F. PILLICHSHAMMER
3. The minimal (infimum) α and β are called the d-exponent and the
ε-exponent of (strong) QMC-tractability.
In this paper we consider the space
Fd = Wq(1,1,...,1) ([0, 1]d ) = Wq1 ([0, 1]) ⊗ . . . ⊗ Wq1 ([0, 1])
(d times),
where Wq1 ([0, 1]) is the Sobolev space of all absolutely continuous functions
with first derivative in Lq ([0, 1]), q ≥ 1. For details on this space see the
appendix.
Now let f ∈ Fd and t1 , . . . , tN ∈ I d . Then Hlawka’s identity [2] states
that
Z
I¯d
f (x)dx −
Z
N
X
∂ |u|
1 X
disc(xu , 1)
(−1)|u|
f (ti ) =
f (xu , 1) dxu .
N i=1
∂xu
∅6=u⊆D
I¯|u|
For a proof of Hlawkas’s identity see also [5].
Let γ = (γ1 , . . . , γd ) be a sequence of weights with 1 = γ1 ≥ γ2 ≥ . . . ≥
1/2
γd > 0. By multiplying and dividing by γu we obtain from Hlawka’s
identity
Z
I¯d
f (x)dx −
X
∅6=u⊆D
N
1 X
f (ti ) =
N i=1
(−1)|u|
Z
−1/2
1/2
γu disc(xu , 1)γu
I¯|u|
∂ |u|
f (xu , 1) dxu .
∂xu
Finally an application of Hölder’s inequality yields
Z
N
X
1
f (x)dx −
f (ti ) ≤ LpN,γ · kf kd,γ,q ,
N i=1
¯d
(1)
I
where for 1 ≤ q < ∞ p is such that p1 + 1q = 1, and LpN,γ is the weighted
Lp discrepancy of the point set {t1 , . . . , tN } with respect to the sequence
γ of weights and

kf kd,γ,q := 
X
u⊆D
γu−1
1/q
q
Z |u|
∂
 .
∂xu f (xu , 1) dxu
I¯|u|
5
WEIGHTED DISCREPANCY AND TRACTABILITY
Remark 1. 1. It is well known that k · kd,γ,q is a norm on the linear space
(1,...,1)
Wq
([0, 1]d ) (see the appendix).
In [4] Sloan and Woźniakowski showed that integration in the P
weighted
d
(1,...,1)
γj
j=1
([0, 1]d ) is QMC-tractable iff lim sup log
<
d
Pd
∞ and strongly QMC-tractable iff j=1 γj < ∞.
In Section 4 we generalize some of their results to the non-Hilbertian case,
i.e., q 6= 2. We show that integration in the weighted tensor product space
tensor product space W2
Pd
(1,...,1)
([0, 1]d ) is QMC-tractable if lim sup
P∞ p/2
QMC-tractable if j=1 γj < ∞, where p1 +
(Theorem 4.1).
Wq
p/2
γj
log d
1
q = 1
j=1
< ∞ and strongly
and where p is even
2. A FORMULA FOR THE WEIGHTED Lp DISCREPANCY
In this section we prove a generalization of Joe’s formula for the weighted
Lp discrepancy for even p.
Theorem 2.1. Let p be an even, positive integer. For any N points
t1 , . . . , tN in the d-dimensional unit cube I d and for any sequence of weights
γ = (γ1 , γ2 , . . .) we have
(LpN,γ )p
l
p X
1
p
−
=
l
N
l=0
X
(u1 ,...,ul )
∈{1,...,N }l
d
Y
j=1

1 +
p/2
γj
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1


where ti,j is the j-th component of the point ti .
Proof.
we have
For even p, for x = (x1 , . . . , xd ) ∈ [0, 1)d and for u ⊆ D, u 6= ∅,
disc((xu , 1))p
p
N Y
X
1
1[0,xj ) (ti,j )
xj −
= 
N
i=1 j∈u
j∈u

=
Y
l=0
=
l
N Y
X
1
−
xp−l
1[0,xj ) (ti,j )
j
N
j∈u
i=1 j∈u
p Y
X
p
l
p Y
X
p
l=0
l

l
1
−
xp−l
j
N
j∈u
X
(u1 ,...,ul )
∈{1,...,N }l
l Y
Y
k=1 j∈u
1[0,xj ) (tuk ,j )
6
G. LEOBACHER, F. PILLICHSHAMMER
l
p X
p
1
=
−
N
l
l
p X
1
p
−
=
l
N
l=0
l=0
l
Y
xp−l
j
(u1 ,...,ul )
∈{1,...,N }l
Y
j∈u
X
Y
xp−l
1[0,xj ) ( max tuk ,j ).
j
X
1[0,xj ) (tuk ,j )
k=1
1≤k≤l
j∈u
(u1 ,...,ul )
∈{1,...,N }l
Therefore we get
Z
disc((xu , 1))p dxu =
I¯|u|
=
p X
p
l
l=0
1
−
N
l
Z Y
X
(u1 ,...,ul )
I¯|u|
∈{1,...,N }l
xp−l
1[0,xj ) ( max tuk ,j ) dxu .
j
1≤k≤l
j∈u
Now we evaluate the integral in the above expression. We have
Z
Y
I¯|u| j∈u
xp−l
1[0,xj ) ( max tuk ,j ) dxu =
j
1≤k≤l
j∈u
=
xp−l
1[0,xj ) ( max tuk ,j ) dxj
j
1≤k≤l
0
1
YZ
j∈u
=
1
YZ
max1≤k≤l tuk ,j
xp−l
dxj
j
max tp−l+1
uk ,j
Y 1 − 1≤k≤l
p−l+1
j∈u
.
Thus we get
Z
p
I¯|u|
disc((xu , 1)) dxu =
p X
p
l=0
l
1
−
N
l
X
(u1 ,...,ul )
∈{1,...,N }l
max tp−l+1
uk ,j
Y 1 − 1≤k≤l
j∈u
p−l+1
Inserting this in the definition of weighted Lp discrepancy we obtain
(LpN,γ )p
=
X
u⊆D
u6=∅
p/2
γu
l
p X
1
p
−
l
N
l=0
l
p X
1
p
=
−
N
l
l=0
X
X
(u1 ,...,ul )
∈{1,...,N }l
XY
(u1 ,...,ul ) u⊆D
∈{1,...,N }l u6=∅
j∈u
max tp−l+1
uk ,j
Y 1 − 1≤k≤l
j∈u
p/2
γj
p−l+1
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1
.
.
7
WEIGHTED DISCREPANCY AND TRACTABILITY
Now we use the fact that
XY
u⊆D
u6=∅
p/2
γj
1 − max tp−l+1
uk ,j
j∈u
1≤k≤l
p−l+1
= −1 +
d
Y
j=1

1 +
p/2
γj
1 − max tp−l+1
uk ,j
1≤k≤l


p−l+1
(this can be proved by induction on d, or see [3]) and that
l
p X
1
p
N l = 0.
−
N
l
l=0
Hence we obtain
(LpN,γ )p
=
p X
p
l=0
l
1
−
N
l
X
(u1 ,...,ul )
∈{1,...,N }l
d
Y
j=1

1 +
p/2
γj
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1

,
and we are done.
3. AVERAGING THE WEIGHTED Lp DISCREPANCY
In [1] Heinrich et al. analyzed the Lp -star discrepancy for uniformly
distributed points in [0, 1)d by computing bounds on the average Lp -star
discrepancy. In the sequel we follow their calculations to derive bounds
on the average weighted Lp discrepancy. For even p we define the average
weighted Lp discrepancy as
avγp (N, d)
=
Z
[0,1]N d
p
LpN,γ (t1 , . . . , tN )
!1/p
dt
,
where t = (t1 , . . . , tN ) ∈ I N d .
With Theorem 2.1 we have
avγp (N, d)p =
l
p X
1
p
=
−
N
l
l=0
X
(u1 ,...,ul )
∈{1,...,N }l
Z
[0,1]N d
d
Y
j=1

1 + γjp/2
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1

 dt.
For given (u1 , . . . , ul ) ∈ {1, . . . , N }l let κ(u1 , . . . , ul ) be the number of
different ui ’s and denote by v1 , . . . , vκ the different ui ’s (κ = κ(u1 , . . . , ul )).
8
G. LEOBACHER, F. PILLICHSHAMMER
Then we have

d
Y
Z
[0,1]N d j=1

Z
d
Y

=
=
j=1
1
...
0
j=1
d
Y
1 + γjp/2
"
Z
1
0
p/2
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1

1 + γjp/2
γj
1+
p−l+1
Z
1
...
0

 dt =
1 − max tp−l+1
vk ,j
1≤k≤κ
p−l+1
Z
1


 dtv1 ,j . . . dtvκ ,j 
#
min (1 − tp−l+1
vk ,j ) dtv1 ,j . . . dtvκ ,j .
0 1≤k≤κ
In [1] it is shown, that
Z
0
1
...
Z
1
min (1 − tp−l+1
vk ,j ) dtv1 ,j . . . dtvκ ,j =
0 1≤k≤κ
p−l+1
.
κ(u1 , . . . , ul ) + p − l + 1
Thus we obtain
d
Y
Z
[0,1]N d j=1
=

1 + γjp/2
d Y
p/2
1 + γj
j=1
1 − max tp−l+1
uk ,j
1≤k≤l
p−l+1

 dt =
1
κ(u1 , . . . , ul ) + p − l + 1
.
Inserting this back into the above expression we have
avγp (N, d)p =
l
p X
1
p
=
−
l
N
l=0
X
(u1 ,...,ul )
∈{1,...,N }l
d Y
p/2
1 + γj
j=1
1
κ(u1 , . . . , ul ) + p − l + 1
.
Now let #(l, k, N ) be the number of tuples (u1 , . . . , ul ) ∈ {1, . . . , N }l
such that k different elements occur (this is the number of mappings from
{1, . . . , l} to {1, . . . , N } such that the range has cardinality k). With this
notation we obtain
avγp (N, d)p =


l X
p l
d X
Y
p
1
1
p/2
 · #(l, k, N ).

=
−
1 + γj
l
N
k
+
p
−
l
+
1
j=1
l=0
k=0
WEIGHTED DISCREPANCY AND TRACTABILITY
9
Fortunately the numbers #(l, k, N ) are well known in combinatorics and
can be expressed by Stirling numbers s(k, i) of the first kind and Stirling
numbers S(l, k) of the second kind. We have
#(l, k, N ) = k!
k
X
N
s(k, i) S(l, k) N i
S(l, k) =
k
i=0
(this is easy calculation using the fact that the number of surjective mappings from {1, . . . , l} to {1, . . . , k} is given by k! S(l, k)). Here we use
#(0, 0, N ) = 1 and #(l, 0, N ) = 0 for l > 0. This is consistent with the
above formula if we use the usual definitions s(0, 0) = S(0, 0) = 1 and
S(l, 0) = 0 for l > 0. Therefore we get
avγp (N, d)p =


l X
p l
d X
Y
1
p
1
p/2
×

−
=
1 + γj
N
k
+
p
−
l
+
1
l
j=1
l=0
k=0
×
k
X
s(k, i) S(l, k) N i
i=0


Y
d 1
p
p/2

 s(k, i) S(l, k)N −l+i
1 + γj
(−1)l
=
l
k
+
p
−
l
+
1
j=1
l=0 k=0 i=0


p
l X
l
d XX
Y
1
p
p/2

 s(k, i) S(l, k)N −l+i .
1 + γj
(−1)l
=
l
k
+
p
−
l
+
1
i=0
j=1
p X
l X
k
X
l=0
k=i
Letting
A(i, l) :=
l
X
k=i


Y
d 1
p
p/2

 s(k, i) S(l, k)
1 + γj
(−1)l
l
k
+
p
−
l
+
1
j=1
we have
avγp (N, d)p =
p X
l
X
l=0 i=0
=
p
X
r=0
A(i, l) · N −l+i =
C(r, p, d) · N −r ,
p−r
p X
X
r=0 i=0
A(i, i + r) · N −r
10
G. LEOBACHER, F. PILLICHSHAMMER
where
C(r, p, d) =
p−r
X
A(i, i + r)
i=0
=
p−r X
i+r
X
(−1)i+r
i=0 k=i


Y
d 1
p
p/2

×
1 + γj
i+r
k
+
p
−
i−r+1
j=1
×s(k, i) S(i + r, k).
Now we consider C(r, p, d). First note that C(p, p, d) = 0. Next we write
C(r, p, d) in the form
r Y
d X
p/2
1 + γj
C(r, p, d) = (−1)
r
l=0 j=1
1
p+1−r+l
· β(r, p, l)
with
β(r, p, l) =
p−r+l
X k=l
p
(−1)k−l s(k, k − l) S(k − l + r, k).
r+k−l
In [1, Lemma 1] it is shown that β(r, p, l) = 0 for r = 0, . . . , p/2 − 1
and l = 0, . . . , r for all even integers p. Thus we have C(r, p, d) = 0 for
r = 0, . . . , p/2 − 1.
We summarize:
Theorem 3.1. Let p be an even integer and let d ∈ N. Then we have
avγp (N, d)p =
p−1
X
r=p/2
C(r, p, d) · N −r ,
where C(r, p, d) is given by
C(r, p, d) =
p−r X
i+r
X
i=0 k=i
(−1)i+r


Y
d 1
p
p/2
×

1 + γj
k
+
p
−
i−r+1
i+r
j=1
×s(k, i) S(i + r, k).
Remark 3. 1. It is easy to prove that for the special case p = 2 we have
1/2

d d Y
Y
γ
1
γ
j
j
γ
 .
−
1+
av2 (N, d) = 1/2 · 
1+
2
3
N
j=1
j=1
WEIGHTED DISCREPANCY AND TRACTABILITY
11
Compare this formula to the subsequent Corollary 3.1.
So to obtain upper bounds on avγp (N, d) one needs to have upper bounds
for |C(r, p, d)|. We continue in the following way:
|C(r, p, d)| ≤
≤


X
i+r Y
d p
1
p/2

×
1 + γj
i+r
k
+
p
−
i−r+1
j=1
p−r
X
i=0
d
Y
j=1
k=i
p/2
1 + γj
1
p−r+1
×|s(k, i) S(i + r, k)|
X
i+r
p
|s(k, i) S(i + r, k)|.
i+r
X
p−r i=0
k=i
Now following the proof of [1, Theorem 5] we get
|C(r, p, d)| ≤
d Y
p/2
1 + γj
j=1
1
p−r+1
· (r + 1) (4 p)p .
We use this bound to get an upper estimate on avγp (N, d):
p−1
X
r=p/2
|C(r, p, d)| · N −r ≤
≤
=
≤
Noting that
Theorem 3.1:


1
p/2

 (r + 1) (4 p)p 1
1 + γj
p−r+1
Nr
r=p/2 j=1


X
p−1
d p
1
p (4 p)p  Y
p/2

1
+
γ
N −(r− 2 )
j
p/2
2
N
j=1
r=p/2
 p

d 2 −1
X
p (4 p)p  Y
p/2 1 
1
+
γ
N −r
j
2
N p/2 j=1
r=0


d
p (4 p)p  Y
p/2 1 
1 + γj
.
2
p/2
2
N
j=1
p−1
X
d Y
√
√
p p ≤
2 for even p we get the following corollary from
Corollary 3.1. Let p be an even integer and let d ∈ N. Then we have
avγp (N, d) ≤ 8 p
1
N 1/2

1/p
d Y
1
p/2
 .
·
1 + γj
2
j=1

12
G. LEOBACHER, F. PILLICHSHAMMER
From this corollary we immediately get the following result, which was
already conjectured by Sloan and Woźniakowski in [4].
Corollary 3.2. Let p be an even integer and let d ∈ N.
1.If the series
∞
X
p/2
γj
j=1
< ∞,
then one can find points t1 , . . . , tN in I d such that we have
LpN,γ ({ti }) < C ·
1
where a := e 2 p
2.If
P∞
j=1
p/2
γj
a
N 1/2
,
and where C := 8 p.
lim sup
d−→∞
p/2
Pd
j=1
γj
< ∞,
log d
then one can find points t1 , . . . , tN in I d such that we have
dα
LpN,γ ({ti }) < C · √ ,
N
where α :=
1
2p
· lim supd−→∞
p/2
Pd
γj
log d
j=1
and where C := 8 p.
Proof. From Corollary 3.1 we know that there exist points t1 , . . . , tN
in I d for which


1/p
d Y
1
1
p/2

1 + γj
LpN,γ ({ti }) ≤ 8 p 1/2 · 
2
N
j=1
holds. Further we have
d P∞
Pd
Y
p/2
p/2 1
1
p/2 1
1 + γj
= e j=1 log(1+γj 2 ) ≤ e 2 j=1 γj
2
j=1
and part 1 follows. Since
e
Pd
j=1
p/2 1
2)
log(1+γj
1
≤ e2
Pd
j=1
p/2
γj
1
= d2
p/2
γ
j
j=1 log d
Pd
13
WEIGHTED DISCREPANCY AND TRACTABILITY
we have also part 2 and we are done.
4. CONDITIONS FOR TRACTABILITY IN q-SPACES
In this section we shall give sufficient conditions on the sequence of
weights γ = (γ1 , γ2 , . . .) for (strong) QMC-tractability of integration in
(1,...,1)
the space Fd,γ = Wq
([0, 1]d ), q ≥ 1, endowed with the norm k · kd,γ,q .
For this purpose we prove upper bounds on N = Nmin (ε, d) such that
e(QN,d ) ≤ ε e(Q0,d)
holds. As in [4] we use reproducing kernels to derive the initial error e(Q0,d ).
Define
Kd,γ (x, t) :=
d
Y
j=1
(1 + γj min(1 − xj , 1 − tj )).
Then Kd,γ is a reproducing kernel for the space Fd,γ , i.e.,
f (t) = hf (.), K(., t)i,
where for every element f ∈ Fd,γ and every element f ′ in the Banach dual
space of Fd,γ , hf, f ′ i := f ′ (f ) denotes the action of f ′ on f .
Here K(., t) clearly is in the dual space (see appendix) for every t and
hf (.), K(., t)i =
If we define hd (x) :=
on Fd,γ , i.e.,
hf, hd i = hf (.),
R
X
γu−1
u⊆D
Z
f (x)K(x, t)dx,
I¯|u|
(f ∈ Fd,γ ).
K(x, t)dt then hd is the representer of integration
I¯d
Z
K(., t)dti =
Z
I¯d
I¯d
hf (.), K(., t)idt =
Z
f (t)dt,
I¯d
for all f ∈ Fd,γ . Thus we have
Z
e(Q0,d ) = kId k = sup f (t)dt = khd kd,γ,p,
f ∈Fd
kf k
≤1 ¯d
d,γ,q
I
14
G. LEOBACHER, F. PILLICHSHAMMER
′
such that the initial error is the norm of hd in Fd,γ
(here p is such that
1
1
+
=
1).
Further
we
have
p
q
hd (x) =
Z
Kd (x, t)dt
I¯d
=
Z
[0,1]d
d
Y
j=1
(1 + γj min(1 − xj , 1 − tj )) dt
1
=
d Z
Y
j=1 0
=
d
Y
j=1
1 + γj min(1 − xj , 1 − tj )dtj

1 + γj
Z1
0

min(1 − xj , 1 − tj )dtj 
and
Z1
0
min(1 − xj , 1 − tj )dtj =
1
(1 − x2j ).
2
Putting this together we get
hd (x) =
d Y
j=1
1
1 + γj (1 − x2j )
2
and
khd kpd,γ,p
p
= hd (1) +
X
∅6=u⊆D
−p/2
γu
p
Z |u|
∂
∂xu hd (xu , 1) dxu ,
I¯|u|
where 1 is the vector in Rd with all components equal to 1. We evaluate
the integral:
Y
Y
1
∂ |u|
2
1 + γj (1 − xj ) ,
(−γj )(1 − xj )
hd (x) =
∂xu
2
j∈u
j ∈u
/
Y
∂ |u|
γj (xj − 1),
hd (xu , 1) =
∂xu
j∈u
15
WEIGHTED DISCREPANCY AND TRACTABILITY
and hence
p
Z Y
Z |u|
∂
|γj |p |xj − 1|p dxu
∂xu hd (xu , 1) dxu =
j∈u
I¯|u|
I¯|u|
=
Y
j∈u
γjp
Z1
0
|xj − 1|p dxj
Y γjp
=
.
p+1
j∈u
Since hd (1) = 1 we have
khd kpd,γ,p
= 1+
−p/2
γu
X
∅6=u⊆D
p/2
d
Y γjp
Y
γj
1+
=
p + 1 j=1
p+1
j∈u
!
,
such that
p
e(Q0,d ) =
d
Y
j=1
p/2
γj
1+
p+1
!
.
We summarize:
Proposition 4.1. The initial error for integration in the weighted space
(1,...,1)
Fd,γ = Wq
([0, 1]d ) with weight sequence γ = (γ1 , γ2 , . . .) and for q ≥ 1
is given by
e(Q0,d ) =
d
Y
j=1
where p is such that
1
p
+
1
q
p/2
γj
1+
p+1
!1/p
,
= 1.
Remark 4. 1. Observe that the initial error e(Q0,d ) is bounded as a
P∞ p/2
function of d iff j=1 γj is finite.
Now we are ready to state our tractability results for integration in the
space Fd,γ .
Theorem 4.1. Let q =
Then we have
p
p−1 ,
where p is an even integer, and let d ≥ 1.
16
G. LEOBACHER, F. PILLICHSHAMMER
1.If
∞
X
p/2
γj
j=1
< ∞,
then integration in the weighted space Fd,γ is strongly QMC-tractable. More
detailed for ε ∈ (0, 1) we have Nmin ≤ C · ε−2 where the constant C is given
p−1
by C = (8pa)2 and where a = e 2p(p+1)
2.If
b := lim sup
d−→∞
P∞
j=1
p/2
γj
.
p/2
d
X
γj
< ∞,
log d
j=1
then integration in the weighted space Fd,γ is QMC-tractable. More detailed
for ε ∈ (0, 1) we have Nmin ≤ C · dα · ε−2 , where C = (8p)2 and where
p−1
α = p(p+1)
· b.
Remark 4. 2. Note that in the case of QMC-tractability the d-exponent
α tends to zero as q tends to one. But unfortunately C depends on p and
therefore tends to infinity in this case.
Proof. From the definition of the worst case error e(QN,d) and from the
generalized Koksma-Hlawka bound (1) for the integration error in Fd,γ we
get
e(QN,d) ≤ LpN,γ =
LpN,γ
j=1 1 +
Qd
p/2
γj
p+1
1/p · e(Q0,d ).
Due to Corollary 3.1 we can find N points t1 , . . . , tN in the d-dimensional
unit cube I d with weighted Lp discrepancy LpN,γ such that
LpN,γ
Qd
j=1 1 +
1/p ≤
p/2
γj
p+1

d
Y
8p 
·
N 1/2 j=1
1+
1+
p/2
γj
2
p/2
γj
p+1
1/p



1+
d
1 X
8p

log =
· exp 
p j=1
N 1/2
1+

p/2
γj
2
p/2
γj
p+1




17
WEIGHTED DISCREPANCY AND TRACTABILITY


d
X
1
8p
p
−
1
p/2 
· exp 
≤
log 1 +
γ
p j=1
2(p + 1) j
N 1/2


d
X
p
−
1
8p
p/2
γ .
· exp 
≤
2p(p + 1) j=1 j
N 1/2
Hence there is a point set {t1 , . . . , tN } in I d such that


d
X
8p
p
−
1
p/2
e(QN,d ) ≤ 1/2 · exp 
γ  · e(Q0,d ).
2p(p + 1) j=1 j
N
Let ε ∈ (0, 1).
P∞ p/2
1. If j=1 γj < ∞, then we have


d
X
8p
8p
p
−
1
p/2
γ  ≤ 1/2 · a,
· exp 
2p(p + 1) j=1 j
N 1/2
N
where a is as in the statement of Theorem 4.1. Now the right hand side of
this inequality is smaller than ε iff (8pa)2 · ε−2 < N and the result follows.
Pd γjp/2
2. It b := lim supd−→∞ j=1 log
d < ∞, then we have

8p
p−1
· exp 
2p(p + 1)
N 1/2
d
X
j=1

p/2
γj  =
≤
Pd
p−1
8p
· d 2p(p+1) j=1
1/2
N
p−1
8p
8p
· d 2p(p+1) ·b = 1/2 · dα/2 ,
N 1/2
N
where α is as in the statement of Theorem 4.1. Now
(8p)2 · dα · ε−2 < N and the result follows.
APPENDIX:
p/2
γ
j
log d
8p
N 1/2
· dα/2 < ε iff
THE SPACE Wq(1,...,1) ([0, 1]d )
For the whole section let 1 ≤ q ≤ ∞. D = {1, . . . , d}.
(1,...,1)
Definition A.1. Wq
([0, 1]d ) is the space of functions
f : [0, 1]d −→ R
18
G. LEOBACHER, F. PILLICHSHAMMER
|u|
∂
which are absolutely continuous on [0, 1]d , for which ∂x
f (xu , 1) exists for
u
|u|
almost every xu ∈ [0, 1] , u ⊆ D := {1, . . . , d}, and for which
kf kd,q
Example A.1.
Wq1 ([0, 1])
1/q
q
X Z ∂ |u|

< ∞.
:= 
∂xu f (xu , 1) dxu

u∈D ¯|u|
I
For the case d = 1 we have
:= {f : [0, 1] −→ R : f absolutely continuous,
Z1
0
|f ′ (x)|q < ∞}
Definition A.2. Let X and Y be linear spaces. The algebraic tensor
product X ⊙ Y is the linear space of linear combinations of elementary
tensors x ⊗ y, x ∈ X and y ∈ Y ,
X ⊙ Y := span{x ⊗ y : x ∈ X, y ∈ Y }
with the relations (“simplification rules”)
1. x1 ⊗ y + x2 ⊗ y = (x1 + x2 ) ⊗ y,
2. x ⊗ y1 + x ⊗ y2 = x ⊗ (y1 + y2 ),
3. λ(x ⊗ y) = (λx) ⊗ y = x ⊗ (λy).
Each element z ∈ X ⊙ Y has a representation as a sum of elementary
tensors, i.e.,
z=
m
X
k=1
xk ⊗ yk
x1 , . . . , xm ∈ X, y1 . . . , ym ∈ Y . Note that we do not need scalar factors
due to simplification rule 3.
Example A.2. Let X be a space of functions on a set A and Y be a
space of functions on a set B. Then we can identify elementary tensors
with the functions
f ⊗ g : A × B −→ R
(a, b) 7−→ f (a)g(b)
(a ∈ A, b ∈ B),
WEIGHTED DISCREPANCY AND TRACTABILITY
19
for f ∈ X, g ∈ Y .
Every element in X ⊙ Y is of the form
m
X
k=1
fk ⊗ gk
for f1 , . . . , fm ∈ X, g1 , . . . , gm ∈ Y , and can be identified with the function
(a, b) 7−→
m
X
k=1
fk ⊗ gk (a, b) =
m
X
fk (a)gk (b)
k=1
for a ∈ A, b ∈ B.
Let X1 , . . . , Xn be linear spaces. Then the tensor product of these spaces
is defined by induction
X1 ⊙ . . . ⊙ Xn := (X1 ⊙ . . . ⊙ Xn−1 ) ⊙ Xn .
Definition A.3. Let X1 , . . . , Xn be normed spaces. Then we define
their q-direct product as the Cartesian product
n
Y
k=1
Xk := {(x1 , . . . , xn ) : xk ∈ Xk }
equipped with the norm
k(x1 , . . . , xn )k⊕q :=
n
X
k=1
kxk k
q
!1/q
,
Ln
Ln
and we denote it by
k=1 Xk . Note that
k=1 Xk is complete iff each
Xk , 1 ≤ k ≤ n is complete.
(1,...,1)
Proposition A.1. The spaces Wq
are isometrically isomorphic.
([0, 1]d ) and
L
Proof. Define a linear mapping
I:
M
u⊆D
Lq ([0, 1]|u| ) −→ Wq(1,...,1) ([0, 1]d )
u⊆D
Lq ([0, 1]|u| )
20
G. LEOBACHER, F. PILLICHSHAMMER
by
X
I ((fu )u⊆D ) :=
|u|
(−1)
u={k1 ,...,k|u| }⊆D
Z1
...
xk1
Z1
fu (ξ1 , . . . , ξ|u| )dξ1 . . . dξ|u| .
xk|u|
I is a linear operator and I ((fu )u⊆D ) is absolutely continuous as a finite
sum of primitive functions.
L
I is isometric: Let (fu )u⊆D ∈ u⊆D Lq ([0, 1]|u|).
kI ((fu )u⊆D ) kqd,q =
q
1
1
Z
Z
Z
|u|
X
X
∂
|v|
...
fv (ξv )dξv dxu
(−1)
=
∂xu
u⊆D ¯|u| v={k1 ,...,k|v| }⊆D
xk|v|
xk1
I
Z
X
X
q
q
=
|fu (xu )| dxu =
kfu kqq = (k(fu )u⊆D k⊕q ) .
u⊆D ¯|u|
I
u⊆D
(1,...,1)
It remains to show that I is onto Wq
Define
fu (xu ) :=
(1,...,1)
([0, 1]d ). Let g ∈ Wq
([0, 1]d ).
∂ |u|
g(xu , 1).
∂xu
Then it is easy to show that I ((fu )u⊆D ) = g.
Note that for f1 , . . . , fd ∈ Wq1 the function
[0, 1]d −→ R
(x1 , . . . , xd ) 7−→ f1 (x1 ) . . . fd (xd )
(1,...,1)
is in Wq
([0, 1]d ). Thus Wq1 ([0, 1]) ⊙ . . . ⊙ Wq1 ([0, 1]) can be viewed as
(1,...,1)
a subspace of Wq
([0, 1]d ), as in Example A.2.
Proposition A.2. The d-fold algebraic tensor product Wq1 ([0, 1])⊙ . . .⊙
(1,...,1)
Wq1 ([0, 1]) is dense in Wq
([0, 1]d ).
(1,...,1)
Proof. Let f ∈ Wq
([0, 1]d ). For all u ⊆ D there is a function
gu ∈ Lq [0, 1] ⊙ . . . ⊙ Lq [0, 1] (|u|-times), such that
q
Z |u|
q
gu (xu ) − ∂ f (xu , 1) dxu < ε ,
∂xu
2d
I¯|u|
WEIGHTED DISCREPANCY AND TRACTABILITY
21
since Lq [0, 1] ⊙ . . . ⊙ Lq [0, 1] (|u|-times) is dense in Lq ([0, 1]|u| ). Define
g := I ((gu )u⊆D ) ∈ Wq(1,...,1) ([0, 1]d ),
where I is the isomorphism from Proposition A.1. It is straightforward
to show that g is a linear combination of simple tensors, i.e., g lies in
Wq1 ([0, 1]) ⊙ . . . ⊙ Wq1 ([0, 1]). Now
q
|u|
X Z gu (xu ) − ∂ f (xu , 1) dxu
∂xu
kg − f kqd,q =
u⊆D ¯|u|
I
X q
<
u⊆D
ε
= εq ,
2d
and we are done.
Definition A.4. Let X1 , . . . , Xd be normed spaces with norms p1 , . . . , pd
respectively. A norm p on the algebraic tensor product X1 ⊙ . . . ⊙ Xd is
called a tensor product norm, if
p(x1 ⊗ . . . ⊗ xd ) = p1 (x1 ) . . . pd (xd ),
for every elementary tensor x1 ⊗ . . . ⊗ xd , xk ∈ Xk , 1 ≤ k ≤ d.
So what would a tensor product norm p on Wq1 ([0, 1]) ⊙ . . . ⊙ Wq1 ([0, 1])
look like?
Let f1 , . . . , fd ∈ Wq1 ([0, 1]). We write f = f1 ⊗ . . . ⊗ fd
p(f ) = kf1 k1,q . . . kfd k1,q
Z
d Y
q
=
|fk (1)| +
0
k=1

= 
XY
u⊆D k∈u
/

= 
= kf kd,q .
|fk (1)|q
q
!1/q
∂
∂xk fk (xk ) dxk
YZ
k∈u
1
0
1/q
q
∂

∂xk fk (xk ) dxk
1/q
q
|u|
∂
f (xu , 1) dxu 
I¯|u| ∂xu
XZ
u⊆D
1
So our norm k.kd,q is a tensor product norm and it indeed looks like the
most natural one. (But be careful: There still may be tensor product norms
22
G. LEOBACHER, F. PILLICHSHAMMER
p on Wq1 ([0, 1]) ⊙ . . . ⊙ Wq1 ([0, 1]) with p(f1 ⊗ . . . ⊗ fd ) = kf1 ⊗ . . . ⊗ fd kd,q
but p 6= k.kd,q .)
Definition A.5. We call the completion of Wq1 ([0, 1]) ⊙ . . . ⊙ Wq1 ([0, 1])
(d-times) with respect to the norm k.kd,q the tensor product of the spaces
Wq1 ([0, 1]), . . . . . . , Wq1 ([0, 1]) and we simply denote it by Wq1 ([0, 1]) ⊗ . . . ⊗
Wq1 ([0, 1]).
Now we introduce the weights γ1 ≥ γ2 ≥ . . . 0. Obviously the function
pk : Wq1 ([0, 1]) −→ [0, ∞)
1/q
R1
f 7−→ |f (1)|q + γk−1 0 |f ′ (x)|q dx
,
1 ≤ k ≤ d, defines a norm on Wq1 ([0, 1]) which is equivalent to k.k1,q .
As in the unweighted case this norm can be extended in a natural way to
(1,...,1)
the d-fold tensor product Wq1 ([0, 1]) ⊗ . . . ⊗ Wq1 ([0, 1]) = Wq
([0, 1]d ).
REFERENCES
1. Heinrich, S., Novak, E., Wasilkowski, G., Woźniakowski, H., The inverse of the stardiscrepancy depends linearly on the dimension. Acta Arith. 96: 279-302, 2001.
2. Hlawka, E., Über die Diskrepanz mehrdimensionaler Folgen mod. 1. Math. Zeitschr.
77: 273-284, 1961.
3. Joe, S., Formulas for the Computation of the Weighted L2 Discrepancy. Department
of Mathematics Research Report 55, 1997, The University of Waikato, Hamilton,
New Zealand.
4. Sloan, I., Woźniakowski, H., When are Quasi-Monte Carlo Algorithms Efficient for
High Dimensional Integrals? J. Complexity 14: 1-33, 1998.
5. Zaremba, S. K., Some applications of multidimensional integration by parts. Ann.
Poln. Math. 21:85-96, 1968.