an algorithm for the construction of weighted degree lattice rules

Introduction Degree of exactness Worst case error CBC construction Numerical results Conclusion

An algorithm for the construction ofweighted degree lattice rules

Dirk Nuyensa

Joint work with R. Coolsa and F. Y. Kuob

aDepartment of Computer Science, K.U.Leuven, BelgiumbSchool of Mathematics & Statistics, University of NSW, Australia

MCQMC2010Warsaw, Poland

August 15–20, 2010

Weighted degree lattice rules Dirk Nuyens


Outline

1 Introduction

2 Degree of exactness

3 Worst case error

4 CBC construction

5 Numerical results

6 Conclusion



Multivariate integration

Multivariate integration by lattice rules

Approximate the d-dimensional integral

I (f ) :=∫

[0,1)df (x) dx

by an n-point (rank-1) lattice rule

Q(f ; z,n) :=1n

n−1∑k=0

f({

zkn

})

with “good” generating vector z ∈ (Z×n )d .

→What kind of goodness?



Multivariate integration

Multivariate integration by lattice rules

Approximate the d-dimensional integral

I (f ) :=∫

[0,1)df (x) dx

by an n-point (rank-1) lattice rule

Q(f ; z,n) :=1n

n−1∑k=0

f({

zkn

})

with “good” generating vector z ∈ (Z×n )d .

→What kind of goodness?→ Combine degree of exactness and worst case error.



Preliminaries

Imagery of good lattice rules and sequences

(a) rank-1 rule (b) Fibonacci lattice (c) rank-2 copy rule

(d) 33 seq points (e) 64 seq points (f) 34 seq pointsWeighted degree lattice rules Dirk Nuyens

latti

cese

quence

inbas

e3

fixed

latti

ceru

les


Preliminaries

Error of integration

For f with Fourier representation

f (x) =∑

h∈Zd

f (h) exp(2πi h · x)

we have

Q(f ; z,n)− I (f ) =∑

h∈Zd

f (h)1n

n−1∑k=0

exp(2πi h · zk/n)− f (0)



Preliminaries

Error of integration

For f with Fourier representation

f (x) =∑

h∈Zd

f (h) exp(2πi h · x)

we have

Q(f ; z,n)− I (f ) =∑

0 6=h∈Zd

h·z≡0 (mod n)

f (h).

The error is given as a sum over the dual lattice:

{h ∈ Zd : h · z ≡ 0 (mod n)} =: Λ⊥.

→Denotes Fourier coefficients which contribute to error.Weighted degree lattice rules Dirk Nuyens


Degree of exactness

Degree of exactness

Mark region of interestAd(m) in Fourier domain of“degree” m (in d dimensions).

Ask to integrate those Fourier terms exactly. I.e.

Λ⊥ ∩ Ad(m) = {0}.

⇒ Rule of degree (at least) m.Different regionsAd(m) possible:

Trigonometric degree.Zaremba cross degree.Product trigonometric degree.. . .

Aim: create construction algorithm for relatively high d.



Preview

The result

We show:

Need extra weights to control exponential growth innumber of dimensions. No surprise.→ “Weighted degree of exactness”

Standard CBC construction delivers rules with“reasonable” degree of exactness.

New algorithm delivers quite good degrees of exactness.(Comparison for classical trigonometric degree.)

Results published in R. Cools, F. Y. Kuo and D. Nuyens,Constructing lattice rules based on weighted degree of exactnessand worst case error, Computing (2010) 87:63–89.



Classical degrees

Degrees of exactness

Different regions lead to different degrees of exactness:Trigonometric degree: crosspolytope (diamond shape)

Ad(m) =

{h ∈ Zd :

d∑j=1

|hj | ≤ m

}.

Zaremba degree: hyperbolic cross

Ad(m) =

{h ∈ Zd :

d∏j=1

max(1, |hj |) ≤ m

}.

Note: classical Zaremba index ρ = m + 1.Product degree: hypercube

Ad(m) ={

h ∈ Zd : max1≤j≤d

|hj | ≤ m}.



Classical degrees

And their Fourier spectra in 2D

Take m = 5 (and d = 2):

Trigonometric degree Zaremba degree Product degree

For d →∞ these shapes grow exponentially. . .



Classical degrees

Exponential growth

In fact:

Trigonometric degree (Cools & Sloan, 1996)

|Ad(m)| =min(d,m)∑`=0

(d`

)(m`

)2` = O(md).

Zaremba degree

3d−1(2m+1) ≤ |Ad(m)| ≤ (1+2ζ(τ))d mτ for all τ > 1.

Product degree

|Ad(m)| = (2m + 1)d .



Classical degrees

And need exponential amount of nodes

When the setAd(m) is convex and centrally symmetric thenminimal number of points

nmin ≥∣∣∣Ad

(⌊m2

⌋)∣∣∣ = O(md),

to achieve degree m with a rule of the form

n∑k=1

wk f (t k).

(Cools & Reztsov, 1997)



Weighted degree of exactness

Assume some variables more important than others

Inspired by classical theory of product rules and weightedspaces from QMC (Sloan & Wozniakwoski, 1998), introduce a“weighted degree of exactness”:

Importance of dimension j modelled by weight βj .

Decaying sequence of positive weights

1 = β1 ≥ β2 ≥ · · · ≥ 0.

The setAd(m) intersects the jth axis at bβjmc.

Note: For rank-1 lattice rules with n prime theone-dimensional (projected) degrees are always n − 1.




Weighted degrees

Weighted trigonometric degree:

Ad(m) =

{h ∈ Zd :

d∑j=1

|hj |βj≤ m

}.

Weighted Zaremba degree:

Ad(m) =

{h ∈ Zd :

d∏j=1

max(

1,|hj |βj

)≤ m

}.

Weighted product degree:

Ad(m) ={

h ∈ Zd : max1≤j≤d

|hj |βj≤ m

}.




And their controlled sizes

Weighted Zaremba degree:

2m+1 ≤ |Ad(m)| ≤ mτd∏

j=1

(1 + 2ζ(τ)βτj

)for all τ > 1.

Weighted product degree:

|Ad(m)| =d∏

j=1

(1 + 2bβjmc

)≤ exp

(2m

d∑j=1

βj

).

(Similar proof as in (Kuo, Sloan & Wozniakowksi, 2006).)Weighted trigonometric degree (see product degree).

→ These sets are bounded in size independently of d if∞∑

j=1

βj <∞.




And their 2D Fourier spectra

Take m = 5, β1 = 1, β2 = 0.6 (and d = 2):

Trigonometric degree Zaremba degree Product degree



Classical Korobov space

Worst case error in Korobov space Eα

Traditional setting for lattice rules:Absolutely convergent Fourier series representation

f (x) =∑

h∈Zd

f (h) exp(2πi h · x).

Then f ∈ Eα, for an α > 1, if

‖f ‖2Eα :=

∑h∈Zd

rα(h,γ) |f (h)|2 <∞

with

rα(h,γ) :=d∏

j=1

rα(hj , γj), rα(hj , γj) :=

1, if hj = 0,

|hj |α

γj, otherwise.




Worst case error in Eα

Worst case error:

e(Q; H) := supf ∈H‖f ‖H≤1

|I (f )−Q(f )|.

For rank-1 lattice rule in Eα:

e2(z,n; Eα) = −1 +1n

n−1∑k=0

d∏j=1

1 + γj

∑0 6=h∈Z

exp(2πi hzjk/n)|h|α

=

∑0 6=h∈Zd

h·z≡0 (mod n)

1rα(h,γ)

.




Interpretation

We had

Q(f ; z,n)− I (f ) =∑

0 6=h∈Zd

h·z≡0 (mod n)

f (h).

So, the squared worst case error

e2(z,n; Eα) =∑

06=h∈Zd

h·z≡0 (mod n)

1rα(h,γ)

is the error of a “worst case function” with Fourier expansion

ξ(x) :=∑

h∈Zd

exp(2πi h · x)rα(h,γ)

.



A new setting: modified Korobov space

A new worst case setting

Amend the Korobov space Eα to make new space H :

‖f ‖H :=

( ∑h∈Ad(m)

|f (h)|2 +∑

h/∈Ad(m)

|f (h)|2 rα(γ,h)

)1/2

,

with1 ≥ γ1 ≥ γ2 ≥ · · · > 0.

Reproducing kernel of H is then

K (x, y) =∑

h∈Ad(m)

exp(2πi h·(x−y))+∑

h/∈Ad(m)

exp 2πi h · (x − y)rα(γ,h)

.



A new setting: modified Korobov space

Worst case error

The squared worst case error of a rank-1 lattice rule is now

e2n,d(z) =

∑06=h∈Ad(m)

h·z≡0 (mod n)

1 +∑

h/∈Ad(m)h·z≡0 (mod n)

1rα(γ,h)

.

When the rule has degree m this worst case error is the sameas in the Korobov space Eα.



Aims

Towards an algorithm

We now want an algorithm which:

Is component-by-component (extensibility in d),(Sloan, Joe, Kuo, Dick, . . . ).

Achieves a prescribed (weighted) degree m.

Has near optimal worst case error in our function space.

Is strongly tractable with respect to increasingdimensions (given conditions on the weights).

Has acceptable construction cost.→ Fast CBC concstruction (N. & Cools, 2006).



Inductive steps

Component-by-component construction

Theorem (Inductive step for degree of exactness: s → s + 1)

Let c > 1 be fixed, m be given, and suppose that n > m is aprime number satisfying

n > 1 +c

c − 1|As+1(m)| − |As(m)| − 2bβs+1mc

2.

If z ∈ Z∗ns satisfies

h · z 6≡ 0 (mod n) ∀ h ∈ As(m) \ {0},

then there are “more than 1c (n − 1)” choices of zs+1 such that

(h,hs+1) · (z, zs+1) 6≡ 0 (mod n) ∀ (h,hs+1) ∈ As+1(m)\{0}.



Inductive steps

On the number of points. . .

The required n is quite large:

n > 1 +c

c − 1|As+1(m)| − |As(m)| − 2bβs+1mc

2.

For unweighted product degree the lower bound is

nmin & (m + 1)s+1

while the condition above (for c = 2) says

n > 1 + 2m((2m + 1)s − 1).

This is due to a very conservative estimate in the proof wherewe exclude “bad choices” for zs+1: every bad choice isexcluded at least twice by point-symmetry,

h · z ≡ −hs+1zs+1 (mod n) ≡ −h · z ≡ hs+1zs+1 (mod n),

but the same zs+1 may be excluded many times more.Weighted degree lattice rules Dirk Nuyens


Inductive steps

Add in result on the worst case error

Theorem (Combined inductive step: s → s + 1)

Let c > 1 and n > m be a prime number large enough (previouscondition). Suppose that z ∈ Z∗ns has weighted degree m, and

e2n,s(z) ≤

(c

n − 1

s∏j=1

(1 + 2ζ(αλ)γλj

))1/λ

for all λ ∈ (1/α, 1].

Then there is “at least one” zs+1 such that we have weighteddegree m in s + 1 dimensions, and

e2n,s+1(z, zs+1) ≤

(c

n − 1

s+1∏j=1

(1 + 2ζ(αλ)γλj

))1/λ

.

Proof uses idea from (Dick, Pillichshammer & Waterhouse, 2008).Weighted degree lattice rules Dirk Nuyens


The basic algorithm

A CBC algorithm

Let c > 1 and n > m be a prime number satisfying

n > 1 +c

c − 1max

1≤s≤d−1

|As+1(m)| − |As(m)| − 2bβs+1mc2

.

1 Set z1 = 1 and form the setA1(m).2 For each s = 1, . . . ,d − 1, with z = (z1, . . . , zs) fixed andAs(m) already formed, do the following:(a) Form the setAs+1(m) fromAs(m).(b) Exclude zs+1 ∈ Z∗

n if hs+1zs+1 ≡ −h · z (mod n) for any(h,hs+1) ∈ As+1(m) \ {0}.

(c) Find the zs+1 from the remaining choices in Z∗n that

minimizes e2n,s+1(z, zs+1).



The basic algorithm

Construction cost

If the conditions

n > 1 +c

c − 1|As+1(m)| − |As(m)| − 2bβs+1mc

2.

and∞∑

j=1

βj <∞

are fulfilled then the construction cost is

O(|Ad(m)|+ dn log n) = O(dn log n),

using a modified fast component-by-component algorithmbased on (N. & Cools, 2006).



Tractability

Tractability conditions

Supposen > 1 + |Ad(m)|

then to have en,d(z) ≤ ε it is sufficient to take

n ≥ 1 +cε2λ

d∏j=1

(1 + 2ζ(αλ)γλj

)for all λ ∈ (1/α, 1].

From which

n(ε,d) ≤ max

(1 +

2ε2λ

d∏j=1

(1 + 2ζ(αλ)γλj

), 1 + |Ad(m)|

)We have strong tractability when

∞∑j=1

γ1/αj <∞ and

∞∑j=1

βj <∞.



Basic algorithm

Numerical results

Basic example for product degree m = 3 and n = 4001 where

α = 2, γj = 0.75j−1, βj = γ1/αj = (0.866025 . . .)j−1.

s zs |As(3)| en,s(z) drop1 1 7 4.53e-042 1478 35 3.30e-03 10 (12)3 563 175 1.66e-02 60 (68)4 1844 525 4.88e-02 174 (174)5 827 1575 1.07e-01 524 (524)6 1318 4725 1.79e-01 1574 (1574)7 586 14175 2.63e-01 3324 (4724)8 121 42525 3.53e-01 3982 (14174)9 1339 42525 4.26e-01 0 (0)

→ Almost always same choice as classical CBC.



Modified algorithm

A modified algorithm

The basic algorithm fixes the degree m throughout the entireCBC construction. Therefore:

We might not get the highest degree possible.

The claimed degree might be much lower than the actualdegree.

Modification: dynamically change degree m in each step,decreasing the degree when the number of bad choices isabove a preset threshold κ.



Modified algorithm

Results for modified algorithm

Example for trigonometric degree for n = 4001 andκ = 4000− 2 where

α = 2, γj = 0.75j−1, βj = 1.

s zs ms |As(ms)| en,s(z) drop1 1 4000 8001 4.53e-042 899 88 15665 3.77e-03 3990 (7656)3 525 25 22151 1.98e-02 3996 (10400)4 538 14 29961 7.45e-02 3998 (12922)5 1717 9 22363 1.18e-01 3862 (8352)6 919 8 40081 1.94e-01 3996 (13496)7 1610 6 19825 2.72e-01 3438 (5412)8 635 6 40081 3.54e-01 3910 (10122)9 64 6 75517 4.28e-01 3996 (17712)

10 1518 5 36365 4.94e-01 3716 (6996)11 24 5 56695 5.50e-01 3894 (10160)



Modified algorithm

Comparison for trigonometric degree

This new algorithm can quickly construct rules in tens ofdimensions for the unweighted trigonometric degree.

How does it compare?

Comparing to (Cools & Lyness, 2001) in three and fourdimensions and (Cools & Govaert, 2003) in five and sixdimensions we do slightly worse.

Comparing to higher dimensional results as in (Cools,

Novak & Ritter, 1999) we do much better.



Conclusion

Introduced a concept of weighted degree.Derived component-by-component algorithm whichachieves

a prescribed weighted degree; andoptimal worst case error.

Adjustable to other degrees of exactness.

R. Cools, F. Y. Kuo and D. Nuyens, Constructing lattice rules based onweighted degree of exactness and worst case error, Computing (2010)87:63–89.


an algorithm for the construction of weighted degree lattice rules

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