A reduced fast component-by-component construction of (polynomial) lattice points Peter Kritzer Johannes Kepler University Linz Joint work with J. Dick (Sydney), G. Leobacher (Linz), F. Pillichshammer (Linz) Research supported by the Austrian Science Fund, Projects F5506-N26 and P23389-N18 P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 1
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A reduced fast component-by-componentconstruction of (polynomial) lattice points
Peter KritzerJohannes Kepler University Linz
Joint work withJ. Dick (Sydney), G. Leobacher (Linz), F. Pillichshammer (Linz)
Research supported by the Austrian Science Fund, Projects F5506-N26 and P23389-N18
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 1
1 Introduction and Motivation
2 Tractability
3 The reduced CBC construction
4 The reduced fast CBC construction
5 Concluding remarks
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 2
Introduction and Motivation
Introduction and Motivation
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 3
Introduction and Motivation
Consider integration of functions on [0,1]s,
Is(f ) =∫[0,1]s
f (x) dx ,
where f ∈ H, and H is some Banach space.
Approximate Is by a QMC rule,
Is(f ) ≈ QN,s(f ) =1N
N−1∑k=0
f (xk ),
where PN = {x0, . . . ,xN−1}.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 4
Introduction and Motivation
Worst case error in Banach space H with respect toPN = {x0, . . . ,xN−1} :
eN,s(H,PN) := supf∈H,‖f‖≤1
∣∣Is(f )−QN,s(f )∣∣ .
Need PN that makes eN,s(H,PN) small.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 5
Introduction and Motivation
Weighted Korobov space: Hs,α,γ = space of continuous functions fsuch that ‖f‖s,α,γ <∞, where
‖f‖2s,α,γ =∑h∈Zs
ρα,γ(h)−1 |̂f (h)|2,
and where f̂ (h) =∫[0,1]s f (t)exp(−2πih · t) dt is the h-th Fourier
coefficient of f .
Furthermore, ρα,γ(h) =∏s
j=1 ρα,γj (hj), and
ρα,γ(h) ={
1 h = 0,γ|h|−α h 6= 0.
α is the “smoothness parameter”,
1 = γ1 ≥ γ2 ≥ . . . > 0 are the coordinate weights.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 6
Introduction and Motivation
Here: PN = {x0, . . . ,xN−1} is a lattice point set with generating vectorz = (z1, . . . , zs) ∈ {1, . . . ,N − 1}s.
Points of PN :xn = (xn,1, . . . , xn,s)
with
xn,j =
{nzj
N
}.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 7
Introduction and Motivation
For the Korobov space Hs,α,γ , and for a lattice point set PN , we havean explicit formula for e2(Hs,α,γ ,PN).
e2N(Hs,α,γ ,PN) = e2
N,s,α,γ(z) :=∑
h∈D(z)\{0}
ρα,γ(h),
whereD(z) := {h ∈ Zs : h · z ≡ 0 (N)} .
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 8
Introduction and Motivation
Finite formula:
e2N,s,α,γ(z) = −1 +
1N
N−1∑n=0
s∏j=1
(1 + γjϕα
({nzj
N
})),
where ϕα( k
N
)can be precomputed for all values of k = 0, . . . ,N − 1.
If α = 2k , k ∈ N, ϕα is a constant multiple of the Bernoulli polynomialof degree α.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 9
Introduction and Motivation
• All that remains is to find “good” z ∈ {1, . . . ,N − 1}s.• Rather big search space! (e.g., N = 10 000 and s = 20).• Component by component (CBC) construction: construct zj one at
a time.Size of search space is N − 1 per component.
• Can do fast CBC (Cools & Nuyens), computation cost ofO(sN log N).
• Computation cost of O(sN log N) can still be demanding for big N,s
• Might want to have big N, s simultaneously.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 10
Tractability
Tractability
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 11
Tractability
Let e(N, s) be the Nth minimal (QMC) worst-case error,
e(N, s) = infP
eN(Hs,α,γ ,P),
where the infimum is extended over all N-element point sets P in[0,1]s.
Consider the (QMC) information complexity,
Nmin(ε, s) = min{N ∈ N : e(N, s) ≤ ε}.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 12
Tractability
We say that integration in Hs,α,γ is• weakly QMC tractable, if
lims+ε−1→∞
log Nmin(ε, s)s + ε−1 = 0;
• polynomially QMC-tractable, if there exist c,p,q ≥ 0 such that
Nmin(ε, s) ≤ csqε−p. (1)
Infima over all q and p such that (1) holds: s- and ε-exponent ofpolynomial tractability, respectively;
• strongly polynomially QMC-tractable, if (1) holds with q = 0.Infimum over all p such that (1) holds: ε-exponent of strongpolynomial tractability.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 13
Tractability
For the Korobov space Hs,α,γ it is known that:•
∞∑j=1
γj <∞
is equivalent to strong polynomial tractability.• If
∞∑j=1
γ1/τj <∞
for some τ ∈ [1, α), then one can set the ε-exponent to 2/τ .• The ε-exponent of 2/α is optimal.• Use CBC-constructed lattice point sets to obtain optimal results.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 14
Tractability
Suppose now that∞∑
j=1
γ1/τj <∞
for some τ > α.
So far: CBC construction of lattice point sets that yield optimalε-exponent, but cost of CBC-construction is independent of theweights.
Our new result: CBC construction of lattice point sets that yieldoptimal ε-exponent, but cost of CBC-construction may decrease withthe weights.Exploit situations where weights decrease sufficiently fast.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 15
The reduced CBC construction
The reduced CBC construction
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 16
The reduced CBC construction
Idea: make search space smaller for later components.• Let N be a prime power, N = bm, b prime, m ∈ N• Let w1, . . . ,ws ∈ N0 with 0 = w1 ≤ . . . ≤ ws
• Consider the sequence of reduced search spaces
ZN,wj :=
{{1 ≤ z < bm−wj : gcd(z,N) = 1} if wj < m{1} if wj ≥ m
• Note that
|ZN,wj | :={
bm−wj−1(b − 1) if wj < m1 if wj ≥ m
• write Yj := bwj
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 17
The reduced CBC construction
Algorithm (Reduced CBC construction)
Let N, w1, . . . ,ws, and Y1, . . . ,Ys be as above. Constructz = (Y1z1, . . . ,Yszs) as follows.• Set z1 = 1.• For d ≤ s assume that z1, . . . , zd−1 have already been found. Now
choose zd ∈ ZN,wd such that
e2N,d ,α,γ((Y1z1, . . . ,Ydzd ,Ydzd))
is minimized as a function of zd .• Increase d and repeat the second step until (Y1z1, . . . ,Yszs) is
found.
Usual CBC construction: wj = 0 and Yj = 1 for all j .
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 18
The reduced CBC construction
TheoremLet z = (Y1z1, . . . ,Yszs) ∈ Zs be constructed according to the reducedCBC algorithm. Then for every d ≤ s it is true that
eN,d ,α,γ((Y1z1, . . . ,Ydzd)) ≤
2d∏
j=1
(1 + γ
1α−2δj 2ζ
(α
α−2δ
)bwj
)α/2−δ
N−α/2+δ
for all δ ∈(0, α−1
2
], where ζ is the Riemann zeta function.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 19
The reduced CBC construction
Let δ ∈ (0, α−12 ] and let z be constructed according to the reduced
CBC algorithm.• If
lims→∞
1s
s∑j=1
γjbwj = 0,
then we have weak tractability.• If
A := lim sups→∞
∑sj=1 γ
1α−2δj bwj
log s<∞,
then we have polynomial tractability with ε-exponent at most 2α−2δ
and s-exponent at most 2ζ( αα−2δ )A.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 20
The reduced CBC construction
• If
B :=∞∑
j=1
γ1
α−2δj bwj <∞,
then we have strong polynomial tractability with ε-exponent atmost 2
α−2δ .
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 21
The reduced fast CBC construction
The reduced fast CBC construction
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 22
The reduced fast CBC construction
• The fast CBC construction (Nuyens/Cools) for the non-reducedcase (wj = 0) has a computation cost of O(sN log N).
• The idea also works for the reduced case and yields reduced costby exploiting additional structure of the case wj > 0.
• Bonus: once wj ≥ m the search space contains only one element.Thus the construction of additional components incurs no extracost.
• The computational cost of the reduced fast CBC construction is
O
N log N + min{s, s∗}N +
min{s,s∗}∑j=1
(m − wj)Nb−wj
,
where s∗ := min{j ∈ N : wj ≥ m}.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 23
The reduced fast CBC construction
Example:• Suppose weights γj are γj = j−3.• Fast CBC construction needs O(smbm) operations to compute a
generating vector for which the worst-case error is boundedindependently of the dimension.
• We need O(mbm + min{s, s∗}mbm) operations to compute agenerating vector for which the worst-case error is still boundedindependently of the dimension, as∑
j
γjbwj < ζ(3/2) <∞.
• Reduced fast CBC construction significantly reduces computationcost.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 24
The reduced fast CBC construction
Computation times and log10 worst case error for b = 2, α = 2,γj = j−3:
s = 10 s = 20 s = 50
m = 100.384-1.90
0.724-1.88
1.80-1.88
m = 121.32-2.40
2.62-2.37
6.55-2.37
m = 145.22-2.90
10.4-2.87
26.0-2.86
m = 1621.7-3.40
43.4-3.36
109.-3.35
s = 10 s = 20 s = 50
m = 100.104-1.89
0.120-1.85
0.144-1.79
m = 120.356-2.39
0.400-2.35
0.472-2.31
m = 141.29-2.88
1.45-2.84
1.67-2.79
m = 165.13-3.39
5.68-3.34
6.47-3.30
wj = 0 wj = b 32 logb jc
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 25
The reduced fast CBC construction
s = 10 s = 20 s = 50 s = 100 s = 200 s = 500 s = 1000
m = 100.104-1.89
0.120-1.85
0.144-1.79
0.148-1.74
0.156-1.67
0.164-1.65
0.176-1.65
m = 120.356-2.39
0.400-2.35
0.472-2.31
0.524-2.27
0.564-2.19
0.588-2.10
0.608-2.08
m = 141.29-2.88
1.45-2.84
1.67-2.79
1.88-2.76
2.03-2.72
2.35-2.62
2.50-2.53
m = 165.13-3.39
5.68-3.34
6.47-3.30
7.16-3.28
7.78-3.24
9.27-3.17
11.2-3.10
m = 1822.3-3.89
24.4-3.84
27.2-3.81
29.4-3.79
32.1-3.76
38.2-3.71
47.2-3.65
m = 20118.-4.41
126.-4.35
137.-4.33
145.-4.31
157.-4.30
182.-4.26
223.-4.21
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 26
Concluding remarks
Concluding remarks
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 27
Concluding remarks
• Reduced CBC constructions also works for general weights.• Fast reduced CBC construction so far only for product weights.• Everything (including fast construction for product weights) can be
done analogously for a Walsh space with polynomial lattice pointsinstead of lattice points.
• Instead of setting zj = 1 if wj ≥ m, we can choose these zj atrandom. Error bound essentially stays the same.
• If wj ≥ m, we can even replace the components of the lattice pointset by uniformly distributed random points. We then have a hybridpoint set in the sense of Spanier, the error bound stays the same.
• Error in Korobov space can be related to error of suitablytransformed lattice points in Sobolev spaces.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 28
Concluding remarks
Thank you very much for your attention.
P. Kritzer (JKU Linz) A reduced fast CBC construction ICERM IBC 2014 29