Download - A reduced fast component-by-component construction of … · 2018-04-25 · A reduced fast component-by-component construction of (polynomial) lattice points Peter Kritzer Johannes

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


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}.



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.



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.



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

}.



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)} .



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.



• 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.


Tractability

Tractability


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}.


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.


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.


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.


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



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 .



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.



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.



• If

B :=∞∑

j=1

γ1

α−2δj bwj <∞,

then we have strong polynomial tractability with ε-exponent atmost 2

α−2δ .


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}.



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.



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



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


Concluding remarks

Concluding remarks


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.


Concluding remarks

Thank you very much for your attention.


Download - A reduced fast component-by-component construction of … · 2018-04-25 · A reduced fast component-by-component construction of (polynomial) lattice points Peter Kritzer Johannes

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