Wavelet-based multiscale proper generalized decomposition

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Leon ANGEL, Anais BARASINSKI, Emmanuelle ABISSET-CHAVANNE, Elias CUETO, Francisco

CHINESTA Waveletbased multiscale proper generalized decomposition Comptes Rendus

-Mecanique - Vol. 346(7), p.485-500 - 2018

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Model reduction, data-based and advanced discretization in computational mechanics

Wavelet-based

multiscale

proper

generalized

decomposition

Angel Leon

a

,

Anais Barasinski

a

,

Emmanuelle Abisset-Chavanne

b

,

Elias Cueto

c

,

Francisco Chinesta

d

,

∗

a_GeM_Institute,_École_centrale_de_Nantes,_1,_rue_de_la_Noë,_BP_92101,₄₄₃₂₁_Nantes_cedex_3,_France

b_High_Performance_Computing_Institute_&_ESI_GROUP_Chair,_École_centrale_de_Nantes,_1,_rue_de_la_Noë,_BP_92101,₄₄₃₂₁_Nantes_cedex_3,_France c_I3A,_University_of_Zaragoza,_Maria_de_Luna_s/n,₅₀₀₁₈_Zaragoza,_Spain

d_PIMM_Laboratory_&_ESI_GROUP_Chair,_ENSAM_ParisTech,_151,_boulevard_de_{l’Hôpital,}₇₅₀₁₃_Paris,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received4September2017 Accepted7February2018 Availableonline4May2018 Keywords:

Wavelets

ProperGeneralizedDecomposition Multi-resolution

Multi-scalePGD

Separated representations at the heart of Proper Generalized Decomposition are con-structed incrementally by minimizing the problem residual. However, the modes involved in the resulting decomposition do not exhibit a clear multi-scale character. In order to recover a multi-scale description of the solution within a separated representation frame-work, we study the use of wavelets for approximating the functions involved in the sepa-rated representation of the solution. We will prove that such an approach allows separating the different scales as well as taking proﬁt from its multi-resolution behavior for deﬁning adaptive strategies.

1. Introduction

ModelOrderReduction–MOR–techniquesallow nowadayssolving,underreal-timeconstraints,complexmodels. In-tenseresearchactivitiesallowedreachingatpresentacertainmaturityinthedomainofmodelorderreduction.Amongthe numerousreferences, theinterested readercan refer tosome review papers andbooks[1–4], coveringthree majorMOR technologies:POD(ProperOrthogonalDecomposition),RB(ReducedBasis),andPGD(ProperGeneralizedDecomposition).

ProperOrthogonalDecomposition(POD)isageneraltechniqueforextractingthemostsignificantcharacteristicsofa sys-tem’sbehaviorandrepresentingtheminasetof“PODbasisvectors”.Thesebasisvectorsthenprovideanefficient(typically low-dimensional) representationofthe keysystembehavior,whichproves usefulinavarietyofways. Themostcommon useisto projectthesystem-governing equationsontothereduced-order subspacedefinedby thePODbasis vectors.This yields an explicitPODreducedmodelthat canbe solved inplaceoftheoriginal system. ThePOD basiscan alsoprovide alow-dimensionaldescriptioninwhichtoperformparametricinterpolation,infillmissingor“gappy”data,performmodel adaptation,ordefine hyper-reductionprocedures [5].There isan extensiveliteratureandPODhasseen broadapplication acrossfields.SomereviewofPODanditsapplicationscanbefoundin[6,7].

Another family of model reduction techniques lies in the use of Reduced Basis constructed by combining a greedy algorithm and“a posteriori”errorindicators.As forthePOD,the ReducedBasismethodrequires some amountofoﬄine work, but then the reduced basis model can be used online for solving different models with control of the solution

*

Correspondingauthor.

E-mailaddresses:[email protected](A. Leon),[email protected](A. Barasinski),[email protected]

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accuracy,becauseoftheavailabilityoferrorbounds.Whentheerrorisunacceptablyhigh,thereducedbasiscanbeenriched byinvokingagreedyadaptionstrategy[8,9].

Separatedrepresentations,attheheartoftheso-calledProperGeneralizedDecompositionmethods,areconsideredwhen solving at-hand partial differential equations by employing procedures based on the separation of variables. Then they were considered in quantumchemistry forapproximatingmultidimensionalquantum wave-function. Inthe 1980s,Pierre Ladevèze proposed the use ofspace-time separated representations of transient solutions involved in stronglynonlinear models,defininganon-incrementalintegrationprocedure[10,11].Later,separatedrepresentationswereemployedfor solv-ing multidimensionalmodelssufferingtheso-calledcurseofdimensionality[12,13],aswellasinthecontextofstochastic modeling [14]. Then, theywere extended tothe separationof spacecoordinates,making possiblethe solutionto models definedindegenerateddomains,e.g.,plateandshells[15],aswellasforaddressingparametricmodelswheremodel param-eterswereconsideredasmodelextra-coordinates,makingpossibletheofflinecalculationoftheparametricsolution,which can be viewedasa metamodelora computational vademecum tobe used onlineforreal-time simulation,optimization, inverseanalysis,andsimulation-basedcontrol[2,16].

1.1. Separatedrepresentations

WithinthePGDframework,fourkindsofseparatedrepresentationshavebeenwidelyconsidered.

(i) Space-timeseparatedrepresentations thatallowedtheconstructionofeﬃcientincrementalandnon-incremental integra-tors.

Withinthe standard ﬁnite element method, a space-time solutionu

(

x

,

t

)

,x

∈

⊂ R

3 andt

∈

I ⊂ R

, ofa transient problemisapproximatedfrom

u

(

x

,

t

)

≈

M

i=1

u

(

xi

,

t

)

Ni

(

x

)

(1)

where

M

isthenumberofnodesemployedforinterpolatingtheunknown ﬁeld,locatedatpositions xi,andNi

(

x

)

the

so-calledshapefunctions.Becauseoftheinterpolativepropertyoftheshapefunctions,theapproximationcoeﬃcients correspondtothenodalvalueoftheapproximatedﬁeld, u

(

xi

,

t

)

.Thus,ingeneral,whensolvinganonlinearproblem,

atleastalinearsystemofsize

M

mustbesolvedateachtimestep.Whenconsidering

P

timesteps(

P

canreachseveral millions),thecomplexitygrowsveryfast.

When considering POD-based model order reduction, the solution is projected into the reduced basis composed of functions

{φ1

(

x

),

· · · ,

φ

R

(

x

)

}

extractedfromsomecollected snapshotsoftheproblemsolution,withingeneral

R

M

, andconsequentlythesolutionapproximationreads

u

(

x

,

t

)

≈

R

i=1

ξ

i

(

t

)φ

i

(

x

)

(2)

whichrequirestheresolutionoflinearsystemsofsize

R

insteadoftheonesofsize

M

characteristicofﬁniteelement solutions.Theuseofareducedbasisimpliesinmanycasesimpressivecomputing-timesavings.

Approximations(1) or(2) implyafinitesumoftime-dependentcoefficientsandspacefunctions.Thelastareassumed known; they consist of the usual finite element shape functions or the modes extracted by applying,for example, ProperOrthogonalDecomposition–POD.Astepforwardcouldconsistinassumingspacefunctionstobealsounknown andincomputingbothtimeandspacefunctionsonthefly.Inthiscase,theapproximationreads

u

(

x

,

t

)

≈

N

i=1

Ti

(

t

)

Xi

(

x

)

(3)

Because both functionsinvolved in approximation (3) are unknown, it deﬁnes a nonlinear problem whose solution requires an appropriate linearizationstrategy. The interested readercan refer to [17] and the referencestherein for additionaldetailsontheseparatedrepresentationconstructor.

Expression(3) evidencesthatthesolutionprocedurerequirestheresolutionofabout

N

problems,with

N

M

and

N

∼ R

(infact abitmorebecauseofthenonlinearityinduced byseparatedrepresentations) involvingthespacecoordinates (in generalthree-dimensional –3D –and whoseassociateddiscrete systems are ofsize

M

) forcomputingthe space functions Xi

(

x

)

and about

N

one-dimensional –1D – problemsfor calculating the time functions Ti

(

t

)

. Due to the

fact that thecomputing costrelatedto the solutionof 1D problemsis negligiblewithrespect tothe solution of3D problems,theresultingcomputationalcomplexityreducesdrastically,scalingwith

N

insteadof

P

.

(ii) Spaceseparation allowedaddressingmulti-physicsproblemsdeﬁnedindegeneratedgeometriesinwhichatleastoneof itsdimensionsremainsmuchsmallerthattheotherones(e.g.,beams,plates,shells,laminates...)orprocessesinvolving additivelayers(e.g.,automatedtapeplacement,3Dprinting,oradditivemanufacturing).

(4)

Ifdomain

can be decomposedas

=

x

×

y

×

z,the solution u

(

x

,

y

,

z

)

could be approximated by usingthe separatedrepresentation u

(

x

,

y

,

z

)

≈

N

i=1 Xi

(

x

)

Yi

(

y

)

Zi

(

z

)

(4)

whichallowscalculatingthe3Dsolutionfromasequenceof1Dproblems.

Forsome geometries,astheonesassociatedwithplatesorshells,in-plane–out-of-planeseparatedrepresentation be-comesspeciallysuitable,

u

(

x

,

y

,

z

)

≈

N

i=1

Xi

(

x

,

y

)

Zi

(

z

)

(5)

wherethe3D complexity is reducedtothe one characteristicof2D problems,the onesrelatedtothe calculation of in-planefunctions Xi

(

x

,

y

)

.

(iii) Space–time–parameterseparatedrepresentations allowed constructing the so-called computational vademecums (also knownasabacus, virtual charts, nomograms...)eﬃciently consideredformultiple purposes: simulation,optimization, inverseanalysis,uncertaintypropagationandsimulation-basedcontrol,allofthemunderthereal-timeconstraint. Whentheunknown ﬁeld involvesspace, time,anda seriesofparameters

μ

1

,

. . . ,

μQ

,its associatedseparated

repre-sentationreads u

(

x

,

t

,

μ

1

, . . . ,

μ

Q

)

≈

N

i=1 Xi

(

x

)

Ti

(

t

)

Q

j=1

_ij

(

μ

j

)

(6)

(iv) Separatedrepresentationsofintrinsicallymultidimensionalmodels involvingdifferentialoperatorsapplyingontime,space, andaseriesofconformationcoordinatesc1

,

. . . ,

cC.Inthiscase,thesolutionisapproximatedaccordingto

u

(

x

,

t

,

c1

, . . . ,

cC

)

≈

N

i=1 Xi

(

x

)

Ti

(

t

)

C

j=1 C_ij

(

cj

)

(7) 1.2. Papermotivation

This work focuseson the separatedrepresentations previously discussed andcontributes intwo differentways. First, ingeneral thecalculation offunctionsinvolved inthe finitesumrepresentations isperformedon a givenapproximation basis. Thus, the possible multi-scale character ofthe individual modes (functions involvedin the finite sum) and/or the reconstructedsolution(thefinitesumitself)isnotaddressed.

EveniftodayitisacceptedwithinthePGDcommunitythatboththenumberoftermsintheﬁnitesum,andthenumber ofnodesusedfordiscretizingeachfunctioninvolvedintheformerareofmajorrelevanceforensuringaccuracy,andevenif differenterrorestimatorshavebeenproposedinthiscontext[18–20],themulti-scalenatureofboth,thetermsinthesum andthesumitself,remainalmostunexplored.AnadaptiveprogressivePGDstrategywasproposedin[21].

Thisworkconsiderstheuseofanaturallymulti-scaleapproach,a wavelet-basedapproximationtechnique,for approxi-matingthedifferentfunctionsinvolvedintheseparatedrepresentation.Then,recombiningthedifferentscalesinvolvedin thisapproximationallowsone tocapturethemulti-scalecharacter ofthesolution.Evenif,inourknowledge,thiskindof approximationswasemployedinaGalerkinsetting,asreferredtolater,itsconsiderationwithinthePGDframeworkremains unexplored.

The next section revisitsthe PGD constructor andthe wavelet-basedmulti-resolutionanalysis that willbe applied in Section 3 within the PGD constructor described above. Then Section 4 will illustrate some potential applicationsof the proposednumericaltechnique.

2. Methods

2.1. PGDconstructorataglance

ConsiderthesolutiontothePoissonequation

u

(

x

,

y

)

=

f

(

x

,

y

)

(8)

inatwo-dimensionalrectangulardomain

=

x

×

y

= (

0

,

L

)

× (

0

,

H

)

.WespecifyhomogeneousDirichletboundary

con-ditions forthe unknown ﬁeld u

(

x

,

y

)

, i.e.u

(

x

,

y

)

vanishes atthe domain boundary

. Furthermore,we assume that the sourceterm f isconstantoverthedomain

.

(5)

Forallsuitabletestfunctionsu∗,theweightedresidualformof(8) reads

x×y u∗

∂

2u

∂

x2

+

∂

2u

∂

y2

−

f

dx d y

=

0 (9)

OurgoalistoobtainaPGDapproximatesolutionto(8) intheseparatedform

u

(

x

,

y

)

=

N

i=1

Xi

(

x

)

·

Yi

(

y

)

(10)

We shalldosoby computingeach termoftheexpansiononeatatime,thus enrichingthePGDapproximation untila suitableconvergencecriterionissatisﬁed.Thus,ateachenrichmentstepn (n

≥

1),wehavealreadycomputedthen

−

1 ﬁrst termsofthePGDapproximation

un−1

(

x

,

y

)

=

n−1

i=1

Xi

(

x

)

·

Yi

(

y

)

(11)

Wenowwishtocomputethenextterm Xn

(

x

)

·

Yn

(

y

)

toobtaintheenrichedPGDsolution

un

(

x

,

y

)

=

un−1

(

x

,

y

)

+

Xn

(

x

)

·

Yn

(

y

)

=

n−1

i=1

Xi

(

x

)

·

Yi

(

y

)

+

Xn

(

x

)

·

Yn

(

y

)

(12)

BothfunctionsXn

(

x

)

andYn

(

y

)

areunknownatthecurrentenrichmentstepn,andtheyappearintheformofaproduct. Theresultingproblemisthusnon-linear,andasuitableiterativeschemeisrequired.Weshallusetheindexp todenotea particulariteration.Atenrichmentstepn,thePGDapproximationun,p obtainedatiteration p thusreads

un,p

(

x

,

y

)

=

un−1

(

x

,

y

)

+

Xnp

(

x

)

·

Ynp

(

y

)

(13)

andthealgorithmproceedsby(i)calculatingXnp

(

x

)

fromYnp−1

(

y

)

,then(ii)updatingYnp

(

y

)

fromthejust-computed Xnp

(

x

)

;

andﬁnally(iii)checkingtheconvergencebyevaluating

Xnp

(

x

)

·

Ynp

(

y

)

−

Xnp−1

(

x

)

·

Ynp−1

(

y

)

.

Intheﬁrststep,byintroducingtheapproximate

un,p

(

x

,

y

)

=

un−1

(

x

,

y

)

+

Xnp

(

x

)

·

Ynp−1

(

y

)

(14)

aswellastheweightfunction

u∗

(

x

,

y

)

=

X_n∗

(

x

)

·

Ynp−1

(

y

)

(15)

intotheproblemintegralform(9),afterintegratingover

y,oneﬁnds

x X∗_n

· α

xd 2_Xp n dx2

+ β

x_Xp n

dx

= −

x X_n∗

·

n−1

i=1

γ

_ixd 2_X i dx2

+ δ

x iXi

dx

+

x X_n∗

· ξ

xdx (16)

wherethecoeﬃcientsresultfromtheintegrationover

y offunctionsdependingonthey-coordinate.Formoredetails,the

interestedreadercanreferto[17].Theaboveone-dimensionalproblemisthendiscretizedbyusingastandardmesh-based discretizationtechnique,asforexampletheﬁniteelementmethod.

Havingthus computed Xnp

(

x

)

,we arenowreadytoproceed withthe secondstepofiteration p,whichupdatesYnp

(

y

)

fromthejust-computed Xnp

(

x

)

.Theprocedureexactlymirrors whatwehavedoneabove.Indeed,wesimplyexchange the

rolesplayedbyallrelevantfunctionsofx and y,startingfromthesolutionapproximate

un,p

(

x

,

y

)

=

n−1

i=1

Xi

(

x

)

·

Yi

(

y

)

+

Xnp

(

x

)

·

Ynp

(

y

)

(17)

whichﬁnallyleadstotheone-dimensionalprobleminvolvingtheunknownﬁeldYnp

(

y

)

:

y Y_n∗

· α

yd 2_Yp n d y2

+ β

y_Yp n

d y

= −

y Y_n∗

·

n−1

i=1

γ

_iyd 2_Y i d y2

+ δ

y iYi

d y

+

y Y_n∗

· ξ

yd y (18)

Itisimportanttorealizethattheoriginaltwo-dimensionalPoissonequationdeﬁnedover

=

x

×

y hasbeen

trans-formed within the PGD framework into a series of decoupled one-dimensionalproblems formulated in

x and

y. The

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Fig. 1. PGD error EM(uNM)as a function of the number of enrichment steps N for different numbers M of discretization points.

Aspreviously indicated,thefunctions Xn

(

x

)

andYn

(

y

)

are approximatedusingstandard finiteelementapproximations ororthogonalpolynomials(spectralapproximations).Thus,infirstapproximationonecouldexpectthattheaccuracyofthe computedsolutionshoulddependonthenumberoftermsinvolvedinthefinitesum

N

andthenumberofnodes

M

(orthe polynomialdegree)consideredforapproximatingthedifferentfunctionsinvolvedintheﬁnitesum, Xn

(

x

)

andYn

(

y

)

inthe previouscasestudy.

In [17], the authors considered the Poisson equation (8) on a two-dimensional rectangular domain

=

x

×

y

=

(

0

,

2

)

× (

0

,

1

)

with f

=

1,whichhasasexactsolution

uex

(

x

,

y

)

=

m,n odd 64

π

4

_4n2

₊

_m2

sin

_m

_π

_x 2

sin

(

n

π

y

)

(19)

Functions Xi

(

x

)

andYi

(

y

)

weresoughtonauniformone-dimensionalgridwith

M

points.Thesolutionconvergence(with

respecttotheexactone)wasevaluatedbyusingtheerror

EM

(

uNM

)

=

1

0 2

0

uex

(

x

,

y

)

−

uNM

(

x

,

y

)

2 dx d y (20)

whereintegralswereperformednumericallyanduN

Mreferstothesolutionobtainedusingaﬁnitesumcomposedof

N

terms andtheinvolvedone-dimensionalfunctionswereapproximatedbyusing

M

nodes.Fig.1depictsthePGDerrorEM

(

uNM

)

asa functionofthenumberofenrichmentsteps

N

fordifferentnumbers

M

ofdiscretizationpoints.Thisﬁgureprovesthatboth

M

and

N

arerelevant toensure convergence,andthat both mustbe considered together,because,fora givenmesh, and independentlyofthenumberoftermsconsideredinthefinitesum,theerrorreachesaplateau.Intheothersense,evena veryfinemeshisunabletocapturethesolutionifthenumberoftermsinthefinitesumisinsufficient.

2.2. Wavelet-basedmulti-resolutionanalysisandadaptiveapproximations

Multi-resolutionanalysisisbasedontheconstructionofaseriesofembeddedsubspaces Vj

⊂

Vj+1,

∀

j

∈ Z

,i.e.

{∅} · · · ⊂

Vj−1

⊂

Vj

⊂

Vj+1

· · · ⊂

L2

(

R)

(21)

whereeachsubspaceisspannedbytheintegertranslationofasinglefunction,thescalingfunction

φ (

x

)

[22].

If

φ (

x

)

∈

V0,thefunctions

φ

0,k

= φ(

x

−

k

)

constitutea basisof V0,andthefunctions

φ

j,k

(

x

)

=

2j/2

φ (

2jx

−

k

)

deﬁnea

basisofsubspaceVj.Thus,theprojectionofthe function f

(

x

)

inVj reads

P

jf

(

x

)

=

k=+∞

k=−∞

cj,k

φ

j,k

(

x

)

(22)

Thefactthat V0

⊂

V1 allowswriting

φ (

x

)

=

k=+∞

k=−∞

ak

φ (

2x

−

k

)

(23)

(7)

Vj+1

=

Vj

⊕

Wj (24)

withVj

⊥

Wj.Animportantconsequenceisthat

j∈Z Wj

=

L2

(

R)

(25) or V0

⊕

j∈N Wj

=

L2

(

R)

(26)

SpacesWjarealsospannedbytheintegertranslationofasinglefunction,thewaveletfunction

ψ(

x

)

,deﬁningthebases

ateachscale:

ψ

j,k

=

2j/2

ψ(

2jx

−

k

)

.BecauseW0

⊂

V1,wecanwrite

ψ (

x

)

=

k=+∞

k=−∞

bk

φ (

2x

−

k

)

(27)

Inwhat followsweareconsidering Daubechieswavelets[23] forapproximatingthedifferentfunctionsinvolvedinthe separatedrepresentationoftheproblemsolution.Thischoiceismotivatedbytheircompactsupportandregularity (approx-imationconsistency).Inoppositiontousualapproximationtechniques,itsuseallowsspaceandfrequencylocalization, the latterbecauseofitsmultiscalecharacterandtheformerduetoitscompactsupport(inoppositiontoapproximationsbased onFourierpolynomials).TheapproximationisfullydefinedbythecoefficientsakinEq. (23).Thecoefficientsbkdependon

akfrombk

= (−

1

)

ka1−k.

Thecoeﬃcientsak resultfromthefollowingconditions:

(i) thenormalizationcondition, ∞

−∞

φ (

x

)

dx

=

1 (28)

which,takingintoaccountEq. (23),leadsto

k

=∞ k=−∞

ak

=

2 (29)

(ii) theorthogonalityconditionbetween

φ (

x

)

anditsintegertranslatesinto ∞

−∞

φ (

x

)φ (

x

−

l

)

dx

= δ

0l (30)

with

δ

theKroeneckerdeltafunction.TakingintoaccountEq. (23),thepreviousequationresultsin

k

=∞ k=−∞

akak+2l

=

2

δ

0l

,

∀

l

∈ Z

(31)

(iii) if we consider a choice with

N

coeﬃcients, the previous conditions only provide

N /

2

+

1 equations. Thus, other

N /

2

−

1 conditionsshouldbeaddedinordertocomputethe

N

coeﬃcients.Onecommonrouteconsistsinenforcing thatthescalingfunctionexactlyrepresentspolynomialsoforder

M

=

N /

2,i.e.

f

(

x

)

=

a0

+

a1x

+ · · · +

aM−1xM−1 (32)

fromwhich,byusing

f

(

x

)

=

k

=∞

k=−∞

ck

φ (

x

−

k

)

(33)

andthefactthatthewaveletfunction

ψ(

x

)

isorthogonaltothescalingtranslates

φ (

x

−

k

)

,itresults

f

(

x

), ψ (

x

)

=

k

=∞ k=−∞

(8)

with

g

(

x

),

h

(

x

)

=

∞

−∞ g

(

x

)

h

(

x

)

dx (35)

Byreplacing f

(

x

)

byitsexpression(32) intoEq. (34) leadsto

a0

ψ(

x

),

1

+

a1

ψ(

x

),

x

+

aM−1

ψ (

x

),

xM−1

=

0 (36)

whichappliesforanyvalueofcoeﬃcientsa0

,

· · · ,

aM−1.Thus,bychoosingal

=

1 andam

=

0

,

∀

m

=

l,itresults

ψ (

x

),

xl

=

0

,

l

=

0

,

1

,

· · · ,

M

−

1 (37)

implyingthatthe

M

ﬁrstmomentsofthewaveletfunctionvanish.Thisconditioncanbeprovedtobeequivalentto

k

=∞

k=−∞

(

−

1

)

kakkl

=

0

,

l

=

0

,

1

,

· · · ,

M

−

1 (38)

Now,alltheconditionsaboveallowdeterminingthe

N

coeﬃcientsthatdeterminecompletelytheapproximation. Inordertodiscretizetheweakformofagivenproblem,one musthaveaccesstothescalingandwaveletfunctionand theirderivativesatanypoint(inparticularattheintegrationpointsconsideredforevaluatingtheintegralsinvolvedinthe problemweakform).Forthatpurpose,westartbyconsidering

φ (

x

)

=

a0

φ (

2x

)

+

a1

φ (

2x

−

1

)

+ · · · +

aN−1

φ (

2x

−

N

+

1

)

(39)

or,takingintoaccountthatthesupportofthescalingfunctionis

[

0

,

N

]

,itresults

⎧

⎪

⎨

⎪

⎩

φ (

0

)

=

a0

φ (

0

)

φ (

1

)

=

a0

φ (

2

)

+

a1

φ (

1

)

+

a2

φ (

0

)

φ (

2

)

=

a0

φ (

4

)

+

a1

φ (

3

)

+

a2

φ (

2

)

+

a3

φ (

1

)

+

a4

φ (

0

)

..

.

(40)

which deﬁnesan eigenvalueproblemgiving accesstothe value ofthe scaling functionat theintegers. Now,in orderto obtainitsvalueattheso-calleddyadicpoints,wemakeuseof

φ (

x

/

2

)

=

k

=∞

k=−∞

aK

φ (

x

−

k

)

(41)

andfromthosewecancompute

φ (

x

/

4

)

andsoon.

Now, for the derivatives, we proceed in a similar manner, by taking the derivative of Eq. (39), which allows us, by followingthesamerationaleandsolvingtheassociatedeigenproblem,tocalculatethederivativeattheintegerpoints,and fromthistheoneatthedyadicpoints,andsoon.

Approximationsbasedonwaveletshavebeensuccessfullyconsideredfordiscretizingpartialdifferentialequations,most ofthetimebyusingGalerkinformulations[22,24–27].

If,forawhile,weconsidertheone-dimensionalprobleminvolvingtheunknownﬁeldu

(

x

)

d dx

K

(

x

)

du dx

=

0 (42)

withx

∈ [

0

,

1

]

,u

(

x

=

0

)

=

0,u

(

x

=

1

)

=

1 andthemodelparameterK

(

x

)

deﬁnedfrom

K

(

x

)

=

Kmax if x

∈ [

0

.

15

,

0

.

85

]

Kmin elsewhere (43)

itisexpectedthatthecontinuousﬁeldu

(

x

)

exhibitsadiscontinuityinitsderivativeatpointsx

=

0

.

15 andx

=

0

.

85,because ofthecontinuityoftheﬂuxes,i.e.

K

(

0

.

15

−

)

du dx

x=0.15−

=

K

(

0

.

15

+

)

du dx

x=0.15+ (44)

(9)

Fig. 2. Adaptive approximation strategy.

Whenusingwavelet-basedapproximations,theunknownﬁeldcanbeexpressedatthelowestlevelbyusingthescaling functions

u0

(

x

)

=

k

c0,k

φ (

x

−

k

)

(45)

Theproblemsolution(describedlater)consistsofthesetofcoeﬃcientsc0,k.Thesolutionatthenextleveliswrittenby

combiningboththescalingandwaveletfunctionatthelowestlevel,i.e.

u1

(

x

)

=

k c0,k

φ (

x

−

k

)

+

k d0,k

ψ (

x

−

k

)

(46)

whosesolutionconsistsnowofthecoeﬃcientsc0,kandd0,k.

Becauseofthemulti-resolutionpropertyoftheemployedapproximation,one expectsthatthecoeﬃcientsd0,k become

higheratlocationsk wherethelowest-levelapproximation u0

₍

_x

₎

_based_on_the_use_of_the_lowest_level_scaling_functions_is

notabletorepresentthesolutionwithsuﬃcientaccuracy.Thus,wecoulddeﬁneathresholdvalue

κ

,allowingustoidentify twosets

S

₁− and

S

₁+,composedrespectivelybyintegersk suchthatd0,k

≤

κ

andd0,k

>

κ

.

Now, the next approximation level u2

₍

_x

₎

_is _deﬁned _by _adding _to _the _previous _one _u1

₍

_x

₎

_only _the _wavelet_functions

relatedtopoints in

S

₁+andtheoneshavinganaturalneighborin

S

₁+.Theresultingextendedsetisdenoted by

S

˜

₁+,and nowtheapproximationu2

₍

_x

₎

_reads

u2

(

x

)

=

u1

(

x

)

+

k∈ ˜S1+

d1,k

ψ

1,k

(

x

)

(47)

Aftercalculatingcoeﬃcientsd1,k,theyareclassiﬁedinsets

S

2− and

S

2+,andtheadaptiveapproximationcontinues.

Fig.2depictstheevolutionoftheapproximationbasisontheleft(withtheredlinebeingthecomputedsolution),where the pointsatthelowest level(blueones) arerelatedtotheuniformlowest-levelapproximation (basedontheuseofthe scaling functions). Then, thesecond level,red points, are the onesin

S

˜

₁+,and soon. On theright (Fig. 2),the different contributions to thesolutionare depicted: u1

(

x

)

(bluecurve) aswell asthe adaptivecontributions (forexample,the red curverepresentsu2

(

x

)

−

u1

(

x

)

,whosesupportisdeﬁnedby

S

˜

₁+).

Inthenextsection,thewaveletapproximationswillbeconsideredfordiscretizingPDEswithin theseparated represen-tationformatcharacteristicofPGDmethods.

3. Multi-scaleProperGeneralizedDecomposition

Forthesakeofsimplicity,weconsidertheheatequation

∇ (

K

(

x

,

y

)

∇

u

(

x

,

y

))

=

f

(

x

,

y

)

(48)

where K

(

x

,

y

)

representsthematerialconductivity,assumedisotropic.

Problem(48) isdeﬁnedin

=

x

×

y

= (

0

,

L

)

× (

0

,

H

)

andwherewithoutlossofgeneralitythehomogeneousDirichlet

boundaryconditionisenforcedonthedomainboundary

≡ ∂

. TheweakformrelatedtoEq. (48) reads

x×y

∇

u∗

· (

K

∇

u

(

x

,

y

)

+

f

(

x

,

y

))

dx d y

=

0 (49)

whosesolution,asdescribedinSection

1

,issearchedundertheseparatedform

un

(

x

,

y

)

=

n

i=1

(10)

whichimpliesthealternativeresolutionoftwoone-dimensionalproblems,theﬁrstoneforcalculatingXi

(

x

)

,andthesecond oneforcalculatingYi

(

y

)

,similarlytoEqs. (16) and(18).

When consideringwavelet-basedapproximations, thefunctions Xi

(

x

)

andYi

(

y

)

attheenrichment iterationn andthe nonlineariteration p,involvedin theone-dimensionalproblemsofthe PGDseparatedrepresentationconstructor,are ap-proximatedaccordingto:

⎧

⎪

⎨

⎪

⎩

Xnp

(

x

)

=

2j0/2

k Xp,n_j 0,k

φ

2j0_x

−

_k

+

J

j=j0 2j/2

k Xp,n_j,k

ψ

2jx

−

k

Ynp

(

y

)

=

2j0/2

k Yp,n_j 0,k

φ

2j0_y

−

_k

+

J

j=j0 2j/2

k Yp,n_j,k

ψ

2jy

−

k

X_n∗

(

x

)

=

2j0/2

k X∗,_jn 0,k

φ

2j0_x

−

_k

+

J

j=j0 2j/2

k X∗,_j,kn

ψ

2jx

−

k

Y_n∗

(

y

)

=

2j0/2

k Y∗,_jn 0,k

φ

2j0_y

−

_k

+

J

j=j0 2j/2

k Y∗,_j,kn

ψ

2jy

−

k

Xi

(

x

)

=

2j0/2

k Xi_j 0,k

φ

2j0_x

−

_k

+

J

j=j0 2j/2

k Xi_j,k

ψ

2jx

−

k

Yi

(

y

)

=

2j0/2

k Yi_j 0,k

φ

2j0_y

−

_k

+

J

j=j0 2j/2

k Yi_j,k

ψ

2jy

−

k

(51)

where j0 referstothelowestlevelandwherethecoeﬃcientsaffectingthedifferentscalingandwaveletfunctionsarethe

unknowns.

Itisimportanttonotethat,eveniftheapproximates(51),aswritten,involveallthetranslationsk ateachlevel j,the multi-resolutionanalysisallows considering,aspreviously discussed,only thewaveletfunctionslocated intheregions in whichthey contributeto thesolutionimprovement.Moreover,evenifthelargestscaleisassumedtobe thesameforall thefunctionsinvolvedintheseparatedrepresentation, J forallthem,ingeneraleachfunctionateachlevelcouldinvolvea differentnumberofscales.

As previously described, all these functions, as well as their derivatives, can be computed at dyadic points, until approaching suﬃciently the considered point. Thus, by injecting the previous approximates into the weak form of the one-dimensional problemsresultingfromthe PGDdiscretizationof Eq. (49) and usingan integration quadrature,the ap-proximationcoeﬃcientscanbeobtained.

It is important to mention that Eq. (49) also involves the conductivity K

(

x

,

y

)

, which should be separated. For this purpose,weproceedasdescribedinourformerworks(e.g.,[17])byconstructingitsseparatedrepresentation

K

(

x

,

y

)

≈

K

i=1

K

x i

(

x

)

K

y i

(

y

)

(52)

TheseparationcouldbeperformedbyusingastandardSVD–SingularValueDecomposition.Anotherpossibilityconsists inusingthePGD(seeChapter3in[17]),whichconsistsinsolvingtheproblem

˜

K

(

x

,

y

)

−

K

(

x

,

y

)

=

0 (53) with

˜

K

(

x

,

y

)

=

K

i=1

K

x i

(

x

)

K

y i

(

y

)

(54)

whoseintegralformreads

x×y

˜

K∗

(

x

,

y

)

˜

K

(

x

,

t

)

−

K

i=1

K

x i

(

x

)

K

y i

(

y

)

dx d y

=

0 (55)

Again,thisequationissolved byusingthePGDconstructorandnow, becausenoregularityisneededinthe represen-tationoftheconductivity,theHaar’s waveletis chosen,i.e.usedforapproximatingfunctionsinvolved intheconductivity separatedrepresentation

K

x_i

(

x

)

and

K

_iy

(

y

)

.

(11)

The main issue when considering the Daubechies andHaar representations ofthe temperatureﬁeld u

(

x

,

y

)

and the conductivity K

(

x

,

y

)

isthat the integralsin Eq. (49) involve the product of two or three scaling and waveletfunctions, of different nature (Daubechies and Haar) at different scales. This diﬃculty has been circumvented by considering the techniqueproposedin[28] andsummarizedinAppendixA.

Aftersolvingtheproblem, thatis,aftercomputingthecoeﬃcientsaffectingthedifferentscalingandwaveletfunctions appearingintheapproximationoffunctionsinvolvedinthesolutionseparatedrepresentation,onerealizesthateachmode

n

≤ N

, Xn

(

x

)

·

Yn

(

y

)

,involvesdifferentlevels.The GalerkinPGDconstructor doesnot allowassociatingmodeswithlevels; eachmodecontainsmanylevelsofdescription,andeachlevelofdescriptionispresentinmostofthePGDmodes.

However,inordertoextractthemulti-scalefeaturesofthesolution,afterhavingsolvedtheproblem,onecouldproceed with re-orderingthe modesby enforcing theﬁrst mode tocontain the lowest-levelcontribution (zero level), thesecond modetheﬁrstlevelcontribution,andsoon.

Tobetterdescribethere-orderingprocedure,weconsiderthesolutionapproximation

uN

(

x

,

y

)

=

N

i=1 Xi

(

x

)

Yi

(

y

)

(56) with

⎧

⎪

⎨

⎪

⎩

Xi

(

x

)

=

2j0/2

k Xi_j 0,k

φ

2j0_x

−

_k

+

J

j=j0 2j/2

k Xi_j,k

ψ

2jx

−

k

Yi

(

y

)

=

2j0/2

k Yi_j 0,k

φ

2j0_y

−

_k

+

J

j=j0 2j/2

k Yi_j,k

ψ

2jy

−

k

(57)

Solution(56) canberewrittenintheform

uN

(

x

,

y

)

=

J

i=0 Zi

(

x

,

y

)

(58) with Z0

(

x

,

y

)

=

N

i=1

2j0/2

k Xi_j 0,k

φ

2j0_x

−

_k

·

₂j0/2

k Yi_j 0,k

φ

2j0_x

−

_k

₍₅₉₎ Z1

(

x

,

y

)

=

N

i=1

2j0/2

k Xi_j 0,k

φ

2j0_x

−

_k

·

2j0/2

k Yi_j 0,k

ψ

2j0_y

−

_k

+

2j0/2

k Xi_j 0,k

ψ

2j0_x

−

_k

·

2j0/2

k Yi_j 0,k

φ

2j0_y

−

_k

(60)

andsoon.Here,eachtermZi

(

x

,

y

)

containsalevelofdescription,fromthecoarsestonetotheﬁnestone.

Itisimportanttonotethateachterm Zi

(

x

,

y

)

hasaseparatedrepresentation,aﬁnitesumoffunctionalproducts,with

oneoftheinvolvedfunctionsdependingonthex-coordinate,andtheotheroneonthe y-coordinate. 4. Numericalexamples

4.1. ThePoissonproblem:convergenceanalysis

Inordertochecktheproposedstrategy,weconsidertheproblemdiscussedinSection1,Eq. (8),with f

=

1 and homo-geneousboundaryconditionswhoseexactsolution,aspreviouslydiscussed,isavailable.

Fig. 3depictsthe reconstructedsolution andtheassociatedmodesin each direction.It was noticed thatthe approxi-mation of functionsinvolvedinhigher modes(e.g., XN

(

x

)

andYN

(

y

)

) requiredhigher levels.The difference betweenthe reconstructedandtheexactsolutionisdepictedinFig.4.

The convergenceanalysisconsideredincreasingthenumberofmodelsinvolvedintheseparatedrepresentationfor dif-ferent levels in the mode approximation andthe inverse,increasing the numberof levels fora ﬁxed number ofmodes. TheerroriscalculatedusingtheL2-norm.Asexpected,astrategyincreasingeitherthenumberofmodesorthenumberof levels rapidlyreachesaplateau,asFig.5atteststo.Onlybyincreasing simultaneouslybothofthem,theconvergencerate canbemaintained.

Fig. 6 compares theﬁrst three PGD models Xi

(

x

)

·

Yi

(

y

)

andthe re-ordered multi-scalemodes, referred toas WPGD

modes, Zi

(

x

,

y

)

,accordingtothestrategydescribedintheprevioussection.

Finally,Fig.7illustratesthecontributionofdifferentlevelstoeachPGDmode.ItcanbenoticedthattheﬁrstPGDmodes involvemainlylowerlevels,whereashigherlevelscontributemainlytothelastPGDmodes.

(12)

Fig. 3. Reconstructed solution and involved modes in each direction.

Fig. 4. Differencebetweenthereconstructedsolutionandtheexactonefordifferentnumberofmodelsconsideredinthesolutionseparatedrepresentation.

(13)

Fig. 6. PGD modes Xi(x)·Yi(y)versus WPGD modes Zi(x,y).

Fig. 7. Contribution of the different levels to each PGD mode.

4.2. Multi-scaleanalysis

Inthiscasestudy,weconsiderathermalproblem(heattransferequation)inthedomaindepictedinFig.8,consistingof twophaseswithdifferentthermalconductivities(bothofthemassumedisotropic).Asdiscussedbefore,Daubechieswavelets with

N =

6 areconsideredforapproximatingthefunctionscomposingthedifferentsolutionmodes,i.e.thefunctionsXi

(

x

)

andYi

(

y

)

,whiletheseparatedrepresentationoftheconductivityusesaHaarwaveletrepresentation.

First,theconductivityisapproximatedatafineenoughlevelbyusingaHaarwaveletrepresentation.Then,thesolution is approximatedatdifferentlevels usingDaubechieswavelets.Whenconsideringfine scales,asfineasthemicrostructure representation,thesolutionbecomes locallyrepresentativeoftheexactsolution,whereaswhenconsideringcoarser repre-sentations withrespectto themicrostructure length scale,thelatter isconsidered inan averaged sense inthelow-scale solutionapproximations.Fig.9representsthesolutionwhenconsideringdifferentrepresentationlevelsofthesolution.

Finally, Figs. 10 and 11 depict respectively the PGD modes Xi

(

x

)

·

Yi

(

y

)

and their associated multi-scale re-ordered modes Zi

(

x

,

y

)

described inthe previoussection, wherethemulti-scalecharacter ispointedout,withhigherfrequencies appreciatedathighermodes.

4.3. Parametricsolutions

Untilnow,separatedrepresentationswereassociatedwiththespacecoordinatesx and y.However,asdiscussedin Sec-tion

1

,oneappealingfeatureofseparatedrepresentationsconcernsthesolutiontoparametricmodels.In[2],thesolutionto parametricmodelswithinthePGDframeworkwaswidelydescribed.Thus,ifoneisinterestedincalculatingthetemperature inthedomain

foranyvalueofthematerialconductivityK

∈

I ⊂ R

+,thesolutionseparatedrepresentationreads

(14)

Fig. 8. Domain exhibiting two phases with different thermal conductivities.

Fig. 9. Reconstructed solution for different approximation levels.

Fig. 10. PGD modes: Xi(x)·Yi(y).

u

(

x

,

K

)

≈

N

i=1

Xi

(

x

)

Ci

(

K

)

(61)

which,injectedintheproblemintegralform,allowscalculatingthefunctionsXi

(

x

)

andCi

(

K

)

.However,suchaprocedureis toointrusivetobeconsideredforcalculatingaparametricsolutionwhenusinganexternalsimulationcodeabletoprovide thetemperatureﬁeldforagivenvalueoftheconductivity,thatis,u

(

x

;

K

)

.

(15)

Fig. 11. Multi-scale modes Zi(x,y)constructed from the wavelet-based PGD separated representation.

The last approachisbased onan appropriate samplingoftheparametric domainby choosingparticularvalues ofthe conductivitygroupedintheset

K = {

K1

,

K2

,

· · · ,

KK

}

andsolvingtheproblemforallthesevaluesinordertoobtainu

(

x

;

Ki

)

,

∀

i

∈

K

.Then, all thesesolutions can be adequately interpolatedfor deﬁning thesolution foranyconductivity value K

∈

[

K1

,

KK

]

.Thisisthekeyideabehindsurrogatemodels,responsesurfaces,etc.

Ifweimagineforawhilethatwesolvedtheproblemathandfortheminimumandmaximumvaluesofthe conductiv-ities,leadingtou

(

x

,

Kmin

)

andu

(

x

,

Kmax

)

,theconductivityforanyothervalueoftheconductivityK

∈ [

Kmin

,

Kmax]canbe

linearlyapproximatedaccordingto

u

(

x

,

K

)

=

u

(

x

,

Kmin

)

N1

(

K

)

+

u

(

x

,

Kmax

)

N2

(

K

)

(62)

wheretheapproximationfunctionsintheparametricspaceN1

(

K

)

andN2

(

K

)

read

N1

(

K

)

=

K

−

Kmin Kmax

−

Kmin (63) and N2

(

K

)

=

Kmax

−

K Kmax

−

Kmin (64)

Finerapproximationsrequireﬁnersamplingsandthesubsequentinterpolation.Themaindiﬃcultyishowchoosingthese pointstokeeptheaccuracyundercontrol.In[29],theauthorsproposedanon-intrusivesparsesubspacelearningapproach usinghierarchicalapproximationbases.

Another possibilityconsists intaking advantageof the multi-resolutionproperty ofwavelet-based approximationsfor controlling thesamplingrichness,that is,thenumberofelements intheset

K

.Thus,one cancontroltheconvergenceby analyzingthevalueoftheassociatedcoeﬃcientsrelatedtothewaveletfunctions.Thisappealingpropertyisabsentinusual polynomialapproximations.

Usingawaveletrepresentationoftheparametricevolutionemployingspline-wavelets,

φ (

x

)

=

1

2

φ (

2x

)

+ φ (

2x

−

1

)

+

1

2

φ (

2x

−

2

)

(65)

hastheadvantageofrecoveringcoeﬃcientsthatcorrespondtothevalueoftheapproximatedfunctionatitslocation. Then,assoonasthesolutioniscomputedattwoconsecutivelevelsbyapproximatingtheparametricfunctionsbyusing thescalingfunctions

φ

j,kand

φ

j+1,k,respectively,theirdifferencerepresentsthewaveletcontributionthatcanbeexpressed

inthewaveletbasis

ψ

j,k,andwhosehighercoeﬃcientsindicatethelocationsatwhichsamplingmustbereﬁned.

Thisprocedureallowedustoreducesigniﬁcantlythenumberofsamplingpoints,ofsomeordersofmagnitudeinsome ofournumericalexperiments.

5. Conclusions

In this work, we proved that separated representations involved in PGD solution procedures can be combined with wavelet-basedfunctionalapproximationsforextractingthemulti-scalebehaviorpresentinthesolutions.

(16)

Moreover,weprovedthatthemulti-resolutionpropertyinherenttowaveletrepresentationsallowsdefiningsimple adap-tiveprocedureswithmultipleapplications:modelrefinementorefficientsamplingforcalculatingparametricsolutions.

The connection between multi-scale solutions in complex microstructures andhomogenization constitutes a work in progress.

Appendix A. Calculatingintegralsinvolvingthreefunctionsatthesameandatdifferentlevels

Weconsidertheintegral

d1,d2,d3 l,m

≡

∞

−∞

φ

d1

(

x

) φ

d2

(

x

−

l

) φ

d3

(

x

−

m

)

dx (66)

whered1

,

d2 andd3 referstothederivativeorder.Integral(66) canbeexpressedas

d1,d2,d3 l,m

=

2d12d22d3 ∞

−∞

k ak

φ

d1

(

2x

−

k

)

r ar

φ

d2

(

2x

−

2l

−

r

)

p ap

φ

d3

(

2x

−

2m

−

p

)

dx (67)

Byconsideringthechangeofvariable y

=

2x

−

k,theprecedingexpressionreads

d1,d2,d3 l,m

=

2d1₂d2₂d3 2

k ak

r ar

p ap ∞

−∞

φ

d1

₍

_y

_{) φ}

d2

₍

_y

+

_k

−

_2l

−

_r

_{) φ}

d3

₍

_y

+

_k

−

_2m

−

_p

₎

_{d y} ₍₆₈₎ whichcanberewrittenas

d1,d2,d3 l,m

=

2d1₂d2₂d3 2

k ak

r ar

p ap

_2ld1₊,d_r2₋,d_k,3_2m₊_p₋_k (69)

whichdefinesarank-deficienteigenproblem.Therankdeficiencyiscircumventedbyaddingtherequirednumberofextra equations[28].

Nowweconsiderthattheintegralinvolvesfunctionsatdifferentscales,

d1,d2,d3 im,l

≡

∞

−∞

φ

d1

(

y

) φ

d2

(

y

−

m

) φ

d3

(

2 y

−

l

)

d y (70)

whichcanberewrittenas

d1,d2,d3 m,l

=

∞

−∞

k ak2d1

φ

d1

(

2 y

−

k

)

r ar2d2

φ

d2

(

2 y

−

2m

−

r

) φ

d3

(

2 y

−

l

)

(71) or

d1,d2,d3 m,l

=

2 d1₂d2

k ak

r ar ∞

−∞

φ

d1

₍

_z

_{) φ}

d2

₍

_z

+

_k

−

_2m

−

_r

_{) φ}

d3

₍

_z

+

_k

−

_l

₎

=

2d1₂d2

k ak

r ar

2m+r−k,l−k (72)

whichresultsagaininaneigenproblem.

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