Quasi-Monte Carlo integration on manifolds with mapped low-discrepancy points and greedy minimal Riesz s-energy points

(1)

Quasi-Monte

Carlo

integration

on

manifolds

with

mapped

low-discrepancy

points

and

greedy

minimal

Riesz

s-energy

points

Stefano De

Marchi

a

,

∗

,

Giacomo Elefante

b

a_University_of_Padova,_Department_of_Mathematics,_Via_Trieste,_63,_I-35121_Padova,_Italy

b_University_of_Fribourg,_Department_of_Mathematics,_Chemin_du_Musée,_23,_CH-1700_Fribourg,_Switzerland

Keywords:

Cubatureonmanifolds Quasi-MonteCarlomethod Measurepreservingmaps Low-discrepancysequences GreedyminimalRieszs energypoints

Inthispaperweconsider twosets of pointsforQuasi-Monte Carlointegrationon two-dimensional manifolds. The ﬁrst is the set of mapped low-discrepancy sequence by a measurepreservingmap, from a rectangle U ⊂ R2 _to _the _manifold. _The _second _is _the greedyminimalRieszs-energypoints extractedfromasuitablediscretizationofthemanifold. ThankstothePoppy-seedBagelTheorem weknowthattheclasses ofpointswithminimal Rieszs-energy,undersuitableassumptions,areasymptoticallyuniformlydistributedwith respecttothenormalizedHausdorffmeasure.Theycanthenbeconsideredasquadrature points onmanifolds via the Quasi-Monte Carlo(QMC) method. On the otherhand, we donot knowifthegreedyminimalRieszs-energypointsareagoodchoicetointegrate functions with the QMC method on manifolds. Through theoretical considerations, by showingsomeproperties ofthesepointsand by numericalexperiments,weattempt to answertothesequestions.

1. Introduction

MonteCarlo(MC) andQuasi-MonteCarlo(QMC)methods are well-known techniquesin numericalanalysis, statistics,

ineconomy, inﬁnancialengineeringandinmanyﬁeldswhereitisrequiredtonumericallycomputefastlyandaccurately,

theintegralofamultivariatefunction f .BothMCandQMCmethodsapproximatetheintegral

_X f

(x)

d

μ

(x)

,withX

⊂ R

d_,

bytheaverage ofthefunctionvaluesatasetof N pointsof X uniformlydistributedwithrespecttoa givenmeasure

μ

.

MonteCarlouses randompoints whereas the Quasi-MonteCarlomethodconsiders deterministic pointsets, inparticular

low-discrepancysequences.

Letusconsidertheintegral

1

H

d

(

M

)

M

f

(

x

)

d

H

d

(

x

)

(1)

where

M

is a d-dimensional manifold and

H

d is the Hausdorffmeasure (for thedeﬁnition of thismeasure we refer to

[8,§11.2]). Inthiscase,theQMCmethodis preferabletoother cubaturetechniques,since itrequiresonlythe knowledge

*

Correspondingauthor.

E-mailaddresses:demarchi@math.unipd.it(S. DeMarchi),giacomo.elefante@unifr.ch(G. Elefante).

http://doc.rero.ch

Published in "Applied Numerical Mathematics 127: 110–124, 2018"

which should be cited to refer to this work.

(2)

of f ona well-distributedpoints setofthemanifold.It isworthmentioning,that othercubaturetechniques mayrequire

more information onthe approximation space, like forexample inthe nonnegative leastsquares or those basedon the

ApproximateFeketePoints(cf.[2])whereitisrequiredtheknowledgeofasuitablepolynomialbasisofthemanifold.There

existalsoChebyshev-typequadratureformulasonmultidimensionaldomains,asthosestudiedforinstancein[11].

ConvergenceresultsanderrorboundsforMCandQMCmethods,areusuallystudiedon X

= [

0

,

1

)

d.Inparticularthe

er-rorboundin

[

0

,

1

)

d_for_the_QMC_method_is_given_by_the_well-known_{Koksma–Hlawka}_{inequality (see}_{Theorem 2}_of_Section₂_).

ForothercloseddomainsormanifoldsthereexistsimilarinequalitiesasrecalledinTheorems 3 and4(cf.[1,4,18]).

In ordertoprove convergenceto theintegral ona manifold, we canchoose alow-discrepancy sequence,whichturns

out to be uniformlydistributedwithrespect to theHausdorff measure ofthemanifold. Due to Poppy-seedBagelTheorem

(seetheweightedversion,Theorem 5ofSection4)weknowthatminimalRieszs-energypoints,undersomeassumptions,

areuniformlydistributedwithrespecttotheHausdorffmeasure

H

d (see[10,9]).Thereforethesepointsrepresentpotential

candidates forintegrating functions via the QMC method on manifolds. As observed in [3], it is also possible to use a

continuousandpositiveonthediagonal(CPD)weightfunction,sayw,todistributethesepointsuniformlywithrespectto

agivendensity.

To compute the minimal Riesz s-energy points we make use of a greedy technique, obtaining the so-called greedy

ks-energypoints (or Léja–Gorskipoints) and thegreedy

(

w

,

s

)

-energy points in the weighted case (see Section 4, below).

So far,we do not knowiftheseapproximatepoints of theminimal Rieszs-energy points are agood choice tointegrate

functionsviatheQMCmethodongeneralmanifolds.Asprovedin[13],weonlyknowthatthegreedypointsareuniformly

distributedonthed-dimensionalunitsphere

S

d,d

≥

1.

In thiswork,we test thesegreedypoints forthe integrationon differentmanifoldsvia QMCmethod, makinga

com-parison withlow-discrepancy sequences (like Halton points or Fibonaccilattices) mapped tothe manifold by a measure

preserving map aimedtomaintain their uniformdistributionwithrespecttotheHausdorff measureofthemanifold (see

Section3),andalsowiththeMCmethodbytakingrandompointsonthemanifolditself.

Thepaperisorganized asfollows.Aftersomenecessarydeﬁnitions,notationsandresultson MCandQMCintegration,

recalled in the next section, in Section 3 we presentthe mapping technique from the unit square

[

0

,

1

]

2 to a general

manifold

M

⊂ R

2_._In_Section₄ _we_introduce_other _set_of_points,_that _is_the_minimal _s-Riesz_energy_points,_the_weighted

(

w

,

s

)

-Rieszpoints andthegreedyminimal

(

w

,

s

)

-energypoints.InSection5weprovideextensivenumericalexperiments

forcomparingthesesetofpointsforQMCintegrationondifferentfunctionsonclassical manifolds:cone,cylinder, sphere

and torus.WeconcludeinSection6bysummarizingtheresultsandproposingsomefutureworks.

2. Preliminaries

Let X beacompactHausdorffspaceand

μ

aregularunitBorelmeasureonX .

Deﬁnition1.Asequenceofpoints

S = (x

n

)

n≥1inacompactHausdorffspace X isuniformlydistributedwithrespecttothe

measure

μ

(or

μ

-u.d.)ifforanyreal-valuedboundedcontinuousfunction f

:

X

→ R

wehave

lim N→∞

N n=1f

(

xn

)

N

=

X f

(

x

)

d

μ

(

x

).

This deﬁnitiontells usthat, ifwe havea sequence uniformlydistributed withrespect toa given measure

μ

,we can

approximate

_X f d

μ

byusingtheQMCmethod.

Theorem1(cf.[12]).Asequence

(x

n

)

n∈Nis

μ

-u.d.inX ifandonlyif

lim

N→∞

#

(

J

;

N

)

N

=

μ

(

J

)

holdsforall

μ

-continuitysets J

⊆

X .

Here,by#

(

J

;

N

)

wemeanthecardinalityoftheset J

∩ {

xn

}

nN=1.

Anequivalentwaytodescribetheuniformdistributionofasequenceisintermsofthediscrepancy

#(J;N)

N

−

μ

(

J

)

.

For

[

0

,

1

)

dand

μ

= λ

d(theLebesguemeasure)itiscommonlyusedthefollowingdeﬁnitionofthediscrepancyofapoint

set.

Deﬁnition2. Let P

= {

x0

,

. . . ,

xN−1

}

denote a ﬁnite point set in

[

0

,

1

)

d and

B

a nonempty familyof Jordan measurable

subsetsof

[

0

,

1

)

d_._Then DN

(

B

;

P

) :=

sup B∈B

#

(

B

;

_NN

;

P

)

− λ

d

(

B

)

.

http://doc.rero.ch

(3)

Dependingonthefamily

B

wecandistinguishthediscrepanciesasfollows.

•

Stardiscrepancy:D∗_N

(

P

)

=

DN

(J

∗

;

P

)

,where

J

∗ isthefamilyofallsubintervalsof

[

0

,

1

)

d oftheform

di=1

[

0

,

ai

)

.

•

Extremediscrepancy:DN

(

P

)

=

DN

(J ;

P

)

,where

J

isthefamilyofallsubintervalsof

[

0

,

1

)

d oftheform

id=1

[

ai

,

bi

)

.

•

Isotropicdiscrepancy: JN

(

P

)

=

DN

(C;

P

)

,where

C

isthefamilyofallconvexsubsetsof

[

0

,

1

)

d.

Thestar-discrepancyisimportanton

[

0

,

1

)

d_in_estimating_the_error_of_the_QMC_method_by_the_{Koksma–Hlawka}_inequality

(seee.g.[12]).

Theorem2(Koksma–Hlawkainequality).If f hasmultivariateboundedvariationV

(

f

)

on

[

0

,

1

)

d(inthesenseofHardyandKrause) thenforeveryP

= {

x1

,

. . . ,

xN

}

⊂ [

0

,

1

)

d

1 N N

n=1 f

(

xn

) −

[0,1)d f

(

x

)

dx

≤

V

(

f

)

D∗N

(

P

),

(2)

whereD∗_N

(

P

)

isthestar-discrepancyofP .

Forthe QMC method,thanks to this inequality it is natural to take low-discrepancy sequences. Low-discrepancy

se-quences are those whose star discrepancy has decay order log

(

N

)

d

/

N, which is the best known order of decay. Some

examplesoflow-discrepancysequencesare:Halton,Hammersley,SobolandtheFibonaccilattice(fordetailsseee.g.[5]).

Onthe other hand,if we integrate a function usingthe QMC methodon convex subsetsof

[

0

,

1

)

d _we _have _a _similar

inequalityduetoZaremba(see[18])wheretheisotropicdiscrepancy, JN,isused.

Theorem3.LetB

⊆ [

0

,

1

)

dbeaconvexsubsetand f afunctionwithboundedvariationV

(

f

)

on

[

0

,

1

)

dinthesenseofHardyand Krause.Then,foranypointsetP

= {

x1

,

. . . ,

xN

}

⊆ [

0

,

1

)

d,wehavethat

1 N N

i=1 xi∈B f

(

xi

) −

B f

(

x

)

dx

≤ (

V

(

f

) + |

f

(

1

)|)

JN

(

P

),

(3) where1

= (

1

,

. . . ,

1

)

.

Onasmoothcompactd-dimensionalmanifoldthereisaKoksma–Hlawkalikeinequality(see[4]).

Theorem4.Let

M

beasmoothcompactd-dimensionalmanifoldwithanormalizedmeasuredx.Fixafamilyoflocalcharts

{

ϕ

k

}

kK=1,

ϕ

k

: [

0

,

1

)

d

→

M

,andasmoothpartitionofunity

{ψ

k

}

kK=1subordinatetothesecharts.Then,thereexistsanumberc

>

0,which dependsonthelocalchartsbutnotonthefunctionf orthemeasure

μ

,suchthat

M f

(

y

)

d

μ

(

y

)

≤

c

D

(

μ

)||

f

||

Wd,1_(M)

,

(4)

where

D(

μ

)

=

sup_U_∈_A

_Ud

μ

(y)

,

A

isthecollectionofallimagesofintervalsin

M

and

||

f

||

Wn,p_(M)

=

1≤k≤K

|α|≤n

⎛

⎜

⎝

[0,1)d

_∂

∂

_xαα

(ψ

k

(

ϕ

k

(

x

))

f

(

ϕ

k

(

x

)))

pdx

⎞

⎟

⎠

1/p

,

withWn,paSobolevspace.

Notice that, if d

μ

=

_N1

_x_∈_X_N

δ

x

−

dx in (4), we havethe analogue of the Koksma–Hlawka inequality formanifolds.

Therefore,inordertominimizetheerror,wehavetominimizethediscrepancy.

Wealsoremarkthatingeneralisnoteasy tocomputean estimateoftheerrorusingthisinequalitysincewe haveto

computethesupremumofthecollectionofallimagesofintervalsinthemanifold.

(4)

Thecaseofthespherehasbeensolvedbyadifferentapproach.Let

S

2 _be_the_2-sphere,_then_(see_e.g._[14]₎_it_is_proved

that,theworstcaseerror

sup f

1 N

x∈XN f

(

x

) −

1 4

π

S2 f

(

x

)

d

σ

(

x

)

isproportionaltothedistance-basedenergymetric

EN

(

XN

) =

⎛

⎝

4 3

−

1 N2

xi∈XN

xj∈XN

|

xi

−

xj

|

⎞

⎠

1/2

.

TheproofisbasedontheStolarsky’sinvarianceprinciple (seee.g.[16,17,15]).

Thus,ifwewanttominimizetheworstcaseerrorwehavetomaximizethesumofdistanceterm

_x

i∈XN

xj∈XN

|

xi

−

xj

|

whichiseasiertocheckcomputationallythanthecalculationofthesphericalcapdiscrepancy.

Thisis thereasonwhyinthe nextsection we explorethe useofsequences uniformlydistributedwithrespectto the

Hausdorffmeasureonagivenmanifold

M

.

3. Measurepreservingmapson2-manifolds

Let

S = (

XN

)

N≥1 be uniformlydistributedwithrespecttotheLebesgue measureona rectangle

U ⊂ R

2,

M

aregular

manifoldofdimension2 and

amapfrom

U

to

M

.

Take A

⊂

M

.Themeasure

μ

ofA in

M

isdeﬁnedas

μ

(

A

) := λ

2

(

−1

(

A

)) =

−1₍_A₎

d

λ

2

.

Byconstruction,thesequence

(S)

isthenuniformlydistributedwithrespecttothemeasure

μ

.

Now,letusconsidertheHausdorffmeasure

H

2 onthemanifold

M

which,bymeansoftheareaformula[8,p. 353]is

U

g

(

x

)

dx

,

(5)

with g a densityfunctionthatdependsonthe parametrization

of

M

.Welookforachangeofvariables fromanother

rectangle

U

⊂ R

2_to

_U

_such_that

:

U

−→

U

x

−→

(

x

) =

x

.

(6)

Then,wewishthat

g

((

x

))|

J

(

x

)| =

g

(

x

) =

1

.

(7)

This is equivalent to equalize the “natural” measure

μ

◦ (which comes from the parametrization) and the Hausdorff

measure

H

2onthemanifold

M

:

H

2

(

M

) =

U g

(

x

)

dx

=

U g

((

x

))|

J

(

x

)|

dx

=

U dx

=

μ

◦

(

M

) .

Summarizing, starting from a sequence

S

uniformly distributed with respect to the Lebesgue measure on a rectangle

U

_{⊂ R}

2_, _using _the _change _of _variables ₍₆₎_, _we _will _get _the _sequence

_((S

₎₎

_that _will _be _uniformly _distributed _with

respecttothemeasure

H

2 on

M

.

Proposition1.Let

U

beareferencerectanglein

R

2and

:

U → M

thecorrespondingmeasurepreservingmap.Themeasure preservingmapsforthecone,cylinderandsphereare:

cone:

U

= [

0

,

1

] × [

0

,

2

π

]

(

u

, θ) = (

√

u cos

(θ),

√

u sin

(θ),

√

u

)

cylinder:

U

= [−

1

,

1

] × [

0

,

2

π

]

(

u

, θ) = (

cos

(θ),

sin

(θ),

u

)

sphere:

U

= [−

1

,

1

] × [

0

,

2

π

]

(

u

, θ) = (

1

−

u2_cos

_(θ),

₁

₋

_u2_sin

_(θ),

_u

_).

(5)

Proof. Itisaneasyexercise.

2

Notice that anotherpossible wayof computingthe integral isby takingany coordinatechart for theparametrization

ϕ

:

U → M

ofthemanifoldandthenintegratewiththeQMCmethoddirectlyin

U ⊂ R

2withalow-discrepancysequence

multiplyingtheintegratingfunctionbythedeterminantoftheJacobianof

ϕ

.Unfortunately,withthisapproachwewillnot

havepointswhichwilllieonthemanifold.Infact,mappingthepoints directlywiththeparametrization

ϕ

willnotgivea

sequenceofpointsuniformlydistributedwithrespecttotheHausdorffmeasureofthemanifold.Thisapproachalsodepends

onthedeterminantoftheJacobiansinceitwillmaketheintegratingfunctionadifferentfunction.

4. MinimalRiesz-energypoints

Westartbyintroducingthes-Rieszenergyofasetofpoints.

Deﬁnition3(cf.[10]).Let XN

= {

x1

,

. . . ,

xN

}

⊂

A

⊆ R

d beasetof N distinctpoints.Foreachreals

>

0,thes-Rieszenergy

ofXN is Es

(

XN

) :=

x,y∈XN x=y 1

|

x

−

y

|

s

,

(8)

where

|

· |

denotestheEuclideandistancein

R

d.TheN-pointminimals-energyover A isthen

E

s

(

A

,

N

) :=

inf XN⊂A

Es

(

XN

)

(9)

Notethatin(9),byconvention,thesumoveranemptysetofindicesistakentobezeroandtheinﬁmumoveranempty

setis

∞

.Notice alsothat

E

s

(

A

,

N

)

=

E

s

( ¯

A

,

N

)

and

E

s

(

A

,

N

)

=

0 if A isunbounded.Hence, withoutloss ofgenerality, we

couldrestrictourselvestothecasewhen A iscompact.

AContinuousandPositiveontheDiagonal(CPD)weightfunctionisdeﬁnedasfollows.

Deﬁnition4.Let A

⊂ R

d bean inﬁnitecompactsetwhosed-dimensionalHausdorffmeasure

H

d

(

A

)

isﬁnite.Asymmetric

functionw

:

A

×

A

→ [

0

,

+∞)

iscalledaCPDweightfunctionon A

×

A if

(i) w iscontinuousasfunctionon A

×

A at

H

d-almosteverypointofthediagonalD

(

A

)

= {(

x,x)

:

x

∈

A

}

,

(ii) thereissomeneighborhoodG of D

(

A

)

suchthatinfGw

>

0,

(iii) w isboundedonanyclosedsubset B

⊂

A

×

A suchthat B

∩

D

(

A

)

= ∅

.

Thisdeﬁnitionallowsustodeﬁnetheweighted Rieszs-energyofapointset XN,ofasubset A

⊂ R

d andtheweighted

HausdorffmeasureofBorelsets B

⊂

A.

Deﬁnition5.Lets

>

0 and XN

= {

x1

,

. . . ,

xN

}

⊂

A.TheweightedRieszs-energyof XN is

Esw

(

XN

) :=

1≤i=j≤N

w

(

xi

,

xj

)

|

xi

−

xj

|

s

,

theN-pointweightedRieszs-energyof A is

E

w

s

(

A

,

N

) :=

inf

{

Ews

(

XN

) :

XN

⊂

A

,

#XN

=

N

} ,

andtheweightedHausdorffmeasure

H

s,w

d onBorelsetsB

⊂

A isthen

H

s,w d

(

B

) :=

B

(

w

(

x

,

x

))

−d/s_d

_H

d

(

x

).

Weneed another propertyfor theset A. Aset A issaid to be d-rectiﬁable (see e.g. [7]) ifand onlyifthere exists a

Lipschitzfunction

φ

mappingsomeboundedsubsetof

R

d onto A,i.e.thereexistsaconstant L andacompactset B

⊂ R

d

suchthat

|φ(

x

) − φ(

y

)| ≤

L

|

x

−

y

| , ∀

x

,

y

∈

B

and

φ(

B

)

=

A.

Theconnectionbetweenthes-RieszenergyandasequenceuniformlydistributedwithrespecttotheHausdorffmeasure

isgivenbythe WeightedPoppy-seedBagelTheorem (cf.[3,10]).

(6)

Theorem5.LetA

⊂ R

d_be_{a compact}_subset_of_a_{d-dimensional}

_C

1_-manifold_in

_R

d_,_d

_<

_d_,_and_{w is}_a_CDP_weight_function_on_A

_×

_A. Then lim N→∞

E

w d

(

A

,

N

)

N2_{log N}

=

Vol

(

B

d

₎

H

d,w d

(

A

)

.

(10)

Furthermore,if

H

d

(

A

)

>

0 andX∗NisasequenceofconﬁgurationsonA satisfying(10),with

E

dw

(

A

,

N

)

replacedbyEdw

(

X∗N

)

,then

1 N

x∈X∗_N

δ

x

(·)

−→

∗

H

d,w d

(·)

|A

H

d,w d

(

A

)

as N

→ ∞.

(11)

AssumenowthatA

⊂ R

disaclosedd-rectiﬁableset.Thenfors

>

d,

lim N→∞

E

w s

(

A

,

N

)

N1+s/d

=

Cs,d

(

H

s,w d

(

A

))

s/d

,

(12)

whereCs,d isaﬁnitepositivenumberindependentof A andd.Moreover,if

H

d

(

A

)

>

0,anysequence X∗N ofconﬁgurationson A satisfying(12),withEw

s

(

X∗N

)

insteadof

E

sw

(

A

,

N

)

,satisﬁes(11). 4.1. GreedyminimalRiesz-energypoints

The computation of an approximation of these minimal Riesz s-energy points can be done by the following greedy

algorithm whichprovidesagoodapproximationoftheminimalsetandwhichattainsthecorrectasymptoticmaintermfor

theenergyfors

<

d.

Algorithm1.Letk

:

X

×

X

→ R

∪ {∞}

beasymmetriclower-semicontinuouskernelonalocallycompactHausdorffspace X ,

andlet A

⊂

X beacompactset.Asequence

(

an

)

∞_n₌₁

⊂

A suchthat

(i) a1isselectedarbitrarilyon A; (ii) forn

≥

1,an+1 ischosensothat

n

i=1 k

(

an+1

,

ai

) =

inf x∈A n

i=1 k

(

x

,

ai

),

for every n

≥

1

.

Thesequence

{

an

}

n≥1 iscalleda

greedy

minimal

k

-energy

sequence

on

A (or

Léja-Gorski

points

).

TheRieszkernelin X

= R

d_,_which_depends_on_the_parameter_s

_{∈ [}

₀

_,

_+∞)

_,_is_the_radial_kernel

ks

(

x

,

y

) :=

Ks

(

x

−

y

),

x

,

y

∈ R

d

,

where

·

istheEuclideannorm,with

Ks

(

t

) :=

t−s if s

>

0

−

log

(

t

)

if s

=

0

.

Hence, fork

=

Ks we generatethe greedyminimalks-energypoints, whiletakingk

=

wKs we get the greedyminimal

(

w

,

s

)

-energypoints.

Asprovedin[13]fortheunitsphere

S

d_,_the_greedy_minimal

₍

_w

_,

_d

₎

_-energy_points _are_{asymptotically}_distributed_as_the

realonessuggestingthattheycanbeagoodchoiceforintegrationonmanifolds.

Theorem6.Assumethatw

: S

d

× S

d

→ [

0

,

+∞)

isacontinuousfunctionsuchthatw

(x,

x)

>

0 forallx

∈ S

d_._Let

_{

_Xw N,d

}

bean arbitrarygreedy

(

w

,

d

)

-energysequenceon

S

d_,_d

_≥

_1._Then

lim N→∞ Ew d

(

XNw,d

)

N2_{log N}

=

Vol

(

B

d

₎

H

d,w d

(S

d

)

,

andfurthermore 1 N

x∈Xw_N_,_d

δ

x

(·) →

H

d,w d

(·)

|Sd

H

d,w d

(S

d

)

.

Inparticular,anygreedykd-energysequence

(

XN,d

)

on

S

disasymptoticallyd-energyminimizingon

S

d.

(7)

Unfortunatelythisresultisnotvalidfors

>

d.Ontheotherhand,foranycompact A

⊂ R

d _with

_H

δ

(

A

)

>

0 (where

δ

is

arbitrary),thefollowingorderofgrowthforE_sw

(

X_Nw_,_s

)

holds(cf.[13]).

Theorem7.Let

H

∞_δ

(

A

) :=

inf

i

(

diam Gi

)

δ

:

A

⊂ ∪

iGi

,

0

< δ ≤

d

.

AssumeA

⊂ R

d_be_compact_with

_H

_δ

₍

_A

₎

_>

_0._Let_{w be}_a_bounded_lower_{semicontinuous}_CDP_weight_function_on_A

_×

_{A and}_consider anarbitrarygreedy

(

w

,

s

)

-energysequence

(

X_Nw_,_s

)

⊂

A,fors

≥ δ

.ThenforN

≥

2

Esw

(

XNw,s

) ≤

Ms,δ,A

||

w

||

H

∞_δ

(

A

)

−s/δN1+s/δ

,

s

> δ

M_δ,A

||

w

||

H

∞_δ

(

A

)

−1N2log N

,

s

= δ,

(13)

whereMs,δ,A

,

Mδ,A

>

0 areindependentofw andN,and

||

w

||

=

sup

{

w

(x,

y)

:

x,y

∈

A

}

.

The previous theorem leads us to the following Corollary which is helpful to understand the use of greedy

(

w

,

s

)

-sequencesforintegrationonmanifolds.

Corollary1.LetA

⊂ R

d_be_a_{d-rectiﬁable}_set._Suppose_s

_>

_{d and}_{w is}_a_bounded_lower_{semicontinuous}_CDP_weight_function_on_A

_×

_A. Consideranarbitrarygreedy

(

w

,

s

)

-energysequence

(

X_Nw_,_s

)

⊂

A.Then

(

X_Nw_,_s

)

isdenseinA.Ifs

=

d andA isassumedtobeacompact subsetofad-dimensional

C

1-manifold,thesameconclusionholdsforanygreedy

(

w

,

d

)

-energysequence.Takingw

=

1 theresultis applicabletogreedyks-energysequences.

Remark.Wedonotknowyetifgreedyminimal

(

w

,

d

)

-energy(orks)sequencesareagoodchoiceforintegratingfunctions

onamanifoldwithrespecttothemeasure

H

w

d asitisfortheminimalenergypoints. Orwhetherwe shouldpreferthem

toalowdiscrepancysequencesmappedonthemanifoldwithmeasurepreservingmaps.Thisiswhatwetrytounderstand

bythenumericalexperimentsinthenextsection.

5. Numericalexperiments

In this section we present some numerical tests showing that the greedy minimal ks-energy sequences are a good

choice for integratinga function when a measure preserving map is available, whereas they have a similar behavior of

low-discrepancysequencesifinsteadagenericmapisused.

Thefunctionsweconsideronthecone,cylinder,sphereandtorusare

f1

(

x

,

y

,

z

) :=

(

1

+

z

)(

1

−

z

)

cos

_x 2

+

y 3

+

z 5

,

f2

(

x

,

y

,

z

) :=

cos

(

30xyz

)

if z

<

1₂

(

x2

₊

_y2

₊

_z2

₎

3/2 _{if z}

_≥

1 2

,

f3

(

x

,

y

,

z

) :=

e−sin(2x 2₊_3y2₊_5z2₎

,

f4

(

x

,

y

,

z

) :=

e−x2₊_y2₊_z2 1

+

x2 cos

(

1

+

x 2

₎

_sin

₍

₁

₋

_y2

₎

_e|z|

_.

(14)

Tocomputetheintegrals

1

H

d

(

M

)

M

fi

(

x

)

d

H

d

(

x

),

i

=

1

, . . . ,

4

weusetheQMCmethodwith

(a) lowdiscrepancypointsmappedonthemanifolds,

(b) greedyminimalks-energypoints.

Toemphasize the signiﬁcanceof theQMC approach we did alsoa comparisonwith the MC methodtaking N points

randomlydistributedontherectangleandthenmappedonthe manifolds.Becauseoftherandom natureofthesepoints,

wecomputed10 timestheintegralswithMCandaveragedthem.

About(b),tocompute N greedyminimalks-energypointswestartedfromauniformmeshonarectangleconsistingof

N2

/

2 pointsandmappedthem,ifavailablebyusingthecorrespondingmeasurepreservingmap,tothemanifold.Thenwe

(8)

Table 1

Mapforthetorus(d).

[0,2π] × [0,2π] (u,v) → ⎧ ⎪ ⎨ ⎪ ⎩ x= (2+cos(u))cos(v) y= (2+cos(u))sin(v) z=sin(u) (15) Table 2

Exactvaluesoftheintegrals.

Cone Cylinder Torus Sphere

f1 6.378e−01 7.125e−01 3.435e−01 7.295e−01

f2 0.130e+01 5.784e−01 0.470e+01 2.809e−01

f3 0.116e+01 0.132e+01 0.131e+01 0.1340e+01

f4 1.458e−01 −1.160e−01 −1.269e−02 8.950e−02

extractedN greedyminimalks-energypointsfromthismappedmesh.Heres

=

2 becauseofthedimensionofthemanifold

(asurfaceimmersedin

R

3).

Forthetorusweusedthemap(Table 1) whichdoesnotpreservetheLebesguemeasure.

Tocomparetheresultswiththegreedyminimalk2-energypoints,wecomputedtheintegrals(Table 2) byQMCmethod

using Halton points andFibonacci lattices mapped on the manifolds. Because of the peculiarity of Fibonacci points, we

generated forall sequences anumber ofpoints like theFibonaccisequence, starting from144 (i.e. the twelfth Fibonacci

number)upto2584,theeighteenthFibonaccinumber (Figs. 1–4). Butinthetablesbelow(Tables 3–19) theresultsareonly

presentedfor144,610 and2584 points.Theexactvalueoftheintegralhasbeenconsidered,afteravariablechange,using

thebuilt-inMatlabfunctiondblquad withtoleranceof10−11_._The_relative_errors_are_then_computed_referring_to_this_value.

Hereonlysome experimentsonthecone,cylinder,sphereandthetorusare presented.Moreexperimentsareavailable in

theMaster’sthesisofthesecondauthor(see[6]).Allthetestshave beenperformedonalaptopwithIntelCorei3-3120M

@2.50GHz,4GB ofRAM,Windows10andMATLAB8.5.0.197613.

TheMatlab packageGMKs(GreedyMinimalkspoints)allowstoreproduceall theexperimentspresentedhereand

inter-estedpeoplecandownloaditathttp://www.math.unipd.it/~demarchi/software/GMKs.

Fig. 1. 610 points on the cone.

(9)

Table 3

Relativeerrorsfor f1ontheconewithFibonacci,Halton,Greedy

Min-imalk2-energypointsandMCmethod.

N Halton Fibonacci GM k2 MC

144 1.215e−02 5.352e−03 2.097e−01 1.325e−02 610 4.939e−03 1.270e−03 1.470e−01 6.137e−03 2584 1.241e−03 3.029e−04 9.817e−02 4.850e−03

Table 4

144 9.101e−03 6.250e−03 2.366e−01 3.498e−02 610 5.277e−03 1.173e−03 1.764e−01 2.294e−02 2584 6.766e−04 3.678e−04 1.212e−01 8.212e−03

Table 5

144 1.059e−02 1.416e−03 7.048e−02 4.324e−02 610 3.763e−04 3.389e−04 7.172e−02 2.050e−02 2584 1.289e−04 8.026e−05 3.790e−02 1.613e−02

Table 6

144 2.767e−02 1.384e−02 2.333e−01 8.209e−02 610 9.623e−03 3.230e−03 1.782e−01 1.708e−02 2584 3.068e−03 7.594e−04 1.313e−01 1.818e−02

Fig. 2. 610 points on the cylinder.

(10)

Table 7

Relativeerrorsfor f1onthecylinderwithFibonacci, Halton,Greedy

Minimalk2-energypointsandMCmethod.

144 1.092e−03 5.264e−04 2.240e−01 3.043e−02 610 3.131e−04 6.400e−05 1.603e−01 9.933e−03 2584 1.029e−04 6.929e−06 9.533e−02 5.645e−03

Table 8

144 6.513e−02 8.764e−03 3.830e−01 1.597e−01 610 2.900e−02 2.964e−03 2.267e−01 4.026e−02 2584 1.436e−03 6.975e−04 1.502e−01 2.273e−02

Table 9

144 1.792e−02 2.930e−04 1.150e−01 4.584e−02 610 2.359e−03 1.125e−05 8.751e−02 1.633e−02 2584 3.287e−04 6.267e−07 4.895e−02 7.310e−03

Table 10

144 6.611e−03 1.506e−04 3.879e−02 2.047e−01 610 1.457e−02 8.382e−06 2.524e−02 8.602e−02 2584 4.730e−04 4.671e−07 2.339e−02 5.468e−02

Fig. 3. 610 points on the sphere.

(11)

Table 11

Relativeerrors for f1 on thespherewith Fibonacci,Halton,Greedy

144 2.833e−03 6.448e−04 1.595e−03 1.974e−02 610 1.001e−03 7.411e−05 1.202e−03 7.526e−03 2584 1.017e−04 8.504e−06 1.324e−03 4.495e−03

Table 12

144 1.476e−01 1.089e−02 8.406e−02 1.281e−01 610 4.415e−02 8.045e−05 4.872e−03 1.080e−01 2584 6.847e−04 3.115e−06 5.177e−03 4.666e−02

Table 13

144 3.282e−03 1.025e−04 2.834e−03 4.531e−02 610 1.755e−03 5.727e−06 2.251e−03 1.244e−02 2584 7.294e−05 3.190e−07 1.325e−03 8.174e−03

Table 14

144 1.549e−03 1.425e−03 1.912e−03 1.113e−01 610 6.631e−03 7.970e−05 3.702e−03 4.988e−02 2584 2.725e−04 4.442e−06 4.756e−03 1.955e−02

Fig. 4. 610 points on the torus.

(12)

Table 15

Relativeerrorsforf1onthetoruswithFibonacci,Halton,Greedy

144 2.152e−01 1.777e−01 6.894e−02 2.030e−01 610 1.888e−01 1.780e−01 5.367e−02 1.768e−01 2584 1.788e−01 1.780e−01 4.014e−02 1.687e−01

Table 16

144 1.218e−01 1.690e−01 3.081e−02 1.778e−01 610 1.453e−01 1.410e−01 4.728e−02 1.272e−01 2584 1.414e−01 1.411e−01 2.297e−02 1.562e−01

Table 17

144 3.033e−02 3.426e−02 4.949e−03 4.531e−01 610 2.716e−03 8.821e−03 1.349e−02 2.811e−01 2584 8.763e−03 6.453e−03 1.673e−03 1.049e−01

Table 18

144 6.015e−01 5.339e−01 2.435e−01 5.692e−01 610 5.109e−01 5.237e−01 1.874e−01 4.779e−01 2584 5.252e−01 5.238e−01 1.319e−01 5.238e−01

Table 19

Timeinsecondstocomputethegreedyminimalk2-energypoints.

N Cone Cylinder Torus Sphere

144 0.217 0.218 0.248 0.208

610 20.067 21.046 19.340 19.284 2584 1519.112 1513.211 1571.449 1511.768

5.1. Greedyminimalks-energypoints:tuningtheparameters

Intheprevioussectionwesets

=

2 becauseofthedimension,butwecanconsideratuningofs andseehowtheerrors

change asa function of s. The script demo2 of the Matlab package GMKs, previously mentioned,allows us to makethe

experimentsthatwearegoingtopresent.

InFigs. 5–8, weseethebehavioroftherelativeerrorsoftheintegralsfors

∈ [

0

,

10

]

,withstep0

.

05,using200greedy

minimalks-energypoints.

6. Conclusion

In this paper we tested the QMC integration on manifolds by mapped low-discrepancy points and greedy minimal

ks-energypoints.

Analyzingtherelativeerrorsweobservedthat ifwehaveameasurepreservingmapitisbettertouselow-discrepancy

sequences, especiallytheFibonacciones, thangreedyminimal k2-energypoints. Ontheother hand,ifwedonot dispose

ofa measurepreserving map,asinthecaseofthetorus, orwe usemappedpoints byanotherparametrization, thebest

approximation are withthe greedy minimal k2-energypoints (as inthe case of the functions f1 and f2). The time for

extractingthegreedyminimal ks-energypoints grows asthe numberofpoints. Therefore,inthecaseofgreedyminimal

energypointsitisadvisabletouselesspointsespeciallyincaseofreducedcomputationalresources.Moreoverbyusingan

increasingnumberofgreedyminimalenergypointstheerrorsdecay,butslowerthanmappedlow-discrepancypoints.

(13)

Fig. 5. Relative errors using 200 points for the function f1.

(14)

(15)

Wehavealsonoticedthat keepingthe samenumberofpointsbuttuningtheparameter s,valuesbelowthemanifold

dimension(inourcases

=

2)shouldbeavoidablesincetherelativeerrorsareworsethanthosefors

>

2.Indeedfors

>

2

therelativeerrorsshowedtobe almostofthesameorderand, assuggestedby thetheory,the“optimal”s is aroundthe

manifolddimension.Theonlyexceptionisthesphere,whereweobtainedrelativeerrorsofthesameorderforallvaluesof

theparameter.

We wish also to underline that the QMC approach forintegration on manifolds is preferable to the MC method as

conﬁrmedby all tests. Producingpoints well distributed ona manifold could be usefulnot only forQMCintegration on

manifoldsbutalsoforother approximationmethodsinvolvingsequences ofpoints onmanifoldssuch asradialbasis

func-tions (RBF)approximation ormeshless approximation ofPDEs. Theseare future worksthat we wish to investigatemore

deeply.

Acknowledgements

Wearegratefulto oneunknown refereeforthesuggestions thathavesigniﬁcantlyimproved thepaper.Thisworkhas

beensupported by the University of Padovaex 60% andthe GNCS-INdAM “Visiting Professors 2015”funds. The authors

wishtothankProf.EdwardB.SaffofVanderbiltUniversity andProf.RobertoMontiofUniversityofPadovaforthevaluable

discussions.ThisresearchhasbeenaccomplishedwithintheRITA“ResearchITaliannetworkonApproximation”.

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