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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
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1 Introduction and Motivation
2 Tractability
3 The reduced CBC construction
4 The reduced fast CBC construction
5 Concluding remarks
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Introduction and Motivation
Introduction and Motivation
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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}.
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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.
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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.
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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
}.
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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)} .
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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.
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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.
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Tractability
Tractability
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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}.
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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.
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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.
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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.
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The reduced CBC construction
The reduced CBC construction
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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
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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 .
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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.
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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.
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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 .
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The reduced fast CBC construction
The reduced fast CBC construction
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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}.
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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.
• Reduced fast CBC construction: choose, e.g., wj = b32 logb jc.
• 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.
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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
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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
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Concluding remarks
Concluding remarks
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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.
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Concluding remarks
Thank you very much for your attention.
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