Simplex-stochastic collocation method with improved scalability

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Wouter Nico EDELING, Richard P. DWIGHT, Paola CINNELLA - Simplex-stochastic collocation

method with improved scalability - Journal of Computational Physics - Vol. 310, p.301-328 - 2016

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Simplex-stochastic

collocation

method

with

improved

scalability

W.N. Edeling

a

,

b

,

∗

,

R.P. Dwight

b

,

P. Cinnella

a

a_{ArtsetMétiersParisTech,DynFluidlaboratory,151Boulevarddel’Hopital,75013Paris,France} b_{DelftUniversityofTechnology,FacultyofAerospaceEngineering,Kluyverweg2,Delft,TheNetherlands}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received25November2014

Receivedinrevisedform14August2015 Accepted15December2015

Availableonline18December2015

Keywords:

Simplex-stochasticcollocationmethod Uncertaintyquantification

Surrogatemodel

High-dimensionalmodelreduction techniques

Uniformsimplexsampling

The Simplex-Stochastic Collocation (SSC) method is a robust tool used to propagate uncertaininputdistributionsthroughacomputercode.However,itbecomesprohibitively expensive forproblemswithdimensionshigherthan5.Themain purposeofthispaper is to identifybottlenecks, and toimprove upon thisbad scalability. In order to do so, we propose an alternative interpolation stencil technique based upon the Set-Covering problem, and we integrate the SSC method in the High-Dimensional Model-Reduction framework.Inaddition,weaddressthe issueofill-conditionedsamplematrices, andwe presentananalyticalmaptofacilitateuniformly-distributedsimplexsampling.

1. Introduction

In orderto make reliablepredictions of aphysical systemusing acomputer code it isnecessary to understandwhat

effectthe uncertaintiesin theinputshaveonthe outputQuantity ofInterest (QoI).Attempts todo sowhilekeepingthe computationalcostlowcanbefoundin[30,22,14],whichrelyonSmolyak-typesparse-gridstochastic-collocationmethods.

Whereastraditional collocation-typemethods [4,18] usefull-tensor product formulasto extenda set ofone-dimensional

nodes to higher dimensions, sparse-grid methods build a sparse interpolant using a constrained linear combination of

one-dimensionalnodes.Thiscanprovideagridwithapotentialreductioninsupportnodesofseveralordersofmagnitude.

Althoughcomputationallymoreefficient thanfull-tensorapproaches, sparse-gridmethods addpoints equallyinall

di-mensions,irrespective ofwhetherthe responsesurface islocallysmooth ordiscontinuous.Thereforefurthergainscan be achievedthroughadaptivestochastic-collocationschemeswhichhavebeendevelopedinrecentyears.ForinstanceMaetal. [19]proposed anAdaptiveSparse-Grid(ASG)collocationmethodwheretheprobabilisticspaceisspannedbylinear finite-elementbasisfunctions.Duringeachiterationtheprobabilityspaceisrefinedlocallythroughanerrormeasurebasedupon thehierarchicalsurplus,definedasthedifferencebetweentheinterpolationofthepreviouslevelandthenewlyaddedcode sample. Although thespace is refined locally,unphysical oscillations can still occur dueto the lack ofsample points on someoftheedgesofthelocalsupportofthebasisfunctions.TheSimplex-StochasticCollocation(SSC)methodofWitteveen etal.[38] circumventsthisproblembydiscretizing thedomainintosimplicesby meansofa Delaunaytriangulation,and

enforcing the so-calledLocal-Extremum Conservinglimiter to suppressunphysical oscillations. Furthermore,it computes

EssentiallyNon-Oscillatory(ENO)stencils[39] ofthesample points whichallowsforhigh-order polynomialinterpolation.

*

Correspondingauthorat:DelftUniversityofTechnology,FacultyofAerospaceEngineering,Kluyverweg2,Delft,TheNetherlands.

E-mailaddresses:[email protected](W.N. Edeling),[email protected](R.P. Dwight),[email protected](P. Cinnella).

Further features include randomized sampling, the ability to deal with non-hypercube probability spaces and it can be extendedtoperforminterpolationwithsub-cellresolution[40].

Besides schemes which efficiently sample the probabilistic space, there are other means of dealing with the curse

of dimensionality, i.e.the exponential increase in the amountof computational cost withincreasing dimension. Physical

systems often have a low effective dimension, meaning that only a few coefficients are influential, and only low-order

correlations between these input coefficients have a significant impact on the output. To capitalize on this behavior,

High-DimensionalModel-Reduction (HDMR)techniquescanbe applied[27].InHDMRad-dimensionalfunction isexactly

represented as a hierarchical sum of 2d _{component functions of increasing dimensionality. In the case of low effective}

dimension,thed-dimensionalfunctioncanbeapproximatedwellbyatruncatedexpansion.Themainideaistosolve

sev-eral low-dimensional sub-problems instead of one high-dimensional one, which greatly reduces the computational cost.

A well-known memberofthisclassofdecompositions istheanalysisofvariance (ANOVA)decomposition.In [12],Foo et

al.successfullycoupledtheir Multi-ElementProbabilisticCollocationmethod[13] withaHDMR–ANOVAdecompositionto

problemswithupto600dimensions.Althoughtheyachievedasignificantreductioninthecomputationalcostcomparedto

approacheswithfull-tensorproducts,thenumberofpointsrequiredtosufficientlysampletheseextremelyhigh-dimensional spacesisstilloftheorderofmillionsorevenmore.Furthermore,in[20]Maetal.coupledtheirpreviouslymentionedASG

method with the so-called cut-HDMR technique of [27]. This approach is computationally more efficient than ANOVA–

HDMR, asit doesnot requirethe evaluationof multi-dimensionalintegrals.Besides truncating the cut-HDMR expansion

at a certain order,the authorsof [20] alsomade their approach dimensionadaptive through weights whichidentify the

dimensionsthat contributethemosttothemeanofthestochastic problem.Theyappliedthisapproachto several easily-evaluated testproblemsofveryhighdimensionality,i.e.uptoastochasticdimensionof500.Again,theirresultsrepresent a significantreductionintherequirednumberofcodesamplesfora certainerrorlevelcomparedtofulltensorgrids,but inabsolutetermsthenumberofsamplesisstillveryhigh.

In ourview,the interestof asurrogate modellingtechnique isto apply it tosome complexsimulation codewhich is

too expensivetointegrate bysimpleMonte Carlotechniques.Inthissettingit willbe intractableto sufficientlysample a probabilisticspaceofdimension

O

(100),

regardlessofthesurrogatemodellingtechniqueused.Thereforewewillinvestigate

meanstoefficientlycreatesurrogatemodelswithamoderatenumberofuncertaininputparameters,undertheassumption

that theQoI istheoutput ofa computationallyexpensivecode.Ofcourse,the term“moderatedimensionality”issystem

dependent. So to clarify, we consider the dimensionality moderate when the number of inputs parameters falls within

the range of5 to 10.Many problemsof engineeringinterest are simulated usingcodes with similar dimensionality,e.g.

turbulencemodels[11,15],groundwatermodels[28]orthermodynamicequations-of-state[8,21].

Dueto itsadaptivity,highpolynomialorderandRunge-phenomenonfree interpolation,theSSCmethodrequiresa

rel-atively low numberofcode samplesto attaina givenconvergence levelfor problemswithmoderatedimensionality. For

example,theSSCmethodhasbeenappliedin[3]tooptimization underuncertaintyofaFormula1tirebrakeintake.After aone-dimensionalperturbationanalysis,3variableswereselectedforanalysis.Furthermore,in[9,10]theSSCmethodis ef-ficientlycoupledwithDownhill-Simplexoptimization inasettingforrobustdesignoptimization.Severalexampleproblems

are considered,butagainthemaximumnumberofuncertainvariablesis3.WefoundthattheSSCmethoditself,without

considering thecost ofsamplingthecode,canbecome prohibitivelyexpensivewhen considering5uncertain parameters.

Furthermore,duetoexcessivememoryrequirementswewereunabletocreatesurrogatemodelsofdimension6orhigher.

In[38]theauthorsalsonotethatthecostoftheDelaunaytriangulationbecomesprohibitivelylargefromadimensionality

of 5onward. Theysuggestedtoreplacethe Delaunaytriangulation witha schemewheresimplicesare formedby

select-ing thenearestpointsfromrandomlyplacedMonteCarlo(MC)samples.UsingthisapproachtheSSCmethodwasapplied

to acontinuous QoIwith15uncertainparameters.However, inthiscaseindividual simplicescan overlapandthere isno guaranteethattheentireprobabilisticdomainiscovered.

Thispaperisaimedatidentifyingbottlenecks,andreducingthecomputationalburdenoftheSSCmethod,whileretaining the Delaunay triangulation. Weinvestigate two separate techniques.First we propose theuse ofnew alternativestencils

based upontheset-coveringproblem[16].The mainideaisto usethefastincreaseinnumberofsimplexelements with

polynomial order to createa smallset ofstencils whichcovers the entireprobabilistic domain. Afterwards,only thisset

is usedforinterpolation.ThisallowsforamoreefficientimplementationoftheSSCmethod.Ourresultsshow thatthese

stencils are capableof reducing the computational cost up to 8 dimensions. We furthermore presenta newmethod for

avoiding problems with the ill-conditioning of the sample matrix, andwe provide a new formulafor placing uniformly

distributedsamplesinasimplexofarbitrarydimension.Secondly,inspiredbytheworkofMaetal.[20],weintegratethe

SSCmethodintothecut-HDMRframework.ThisapproachcombinestheadvantagesofSSCandcut-HDMR,andavoidsthe

disadvantagesrelatedtotheASGmethodsuchaslinearinterpolationandthepossibleoccurrenceoftheRungephenomenon. Unliketheauthorsof[20],weapplyourmethodtoacomplexcomputercodeforwhichobtainingsamplesisexpensive.For bothproposed techniquesweperformadetailedanalysisoftheerrorandwegiveadiscussiononcomputationalcostasa functionofthenumberofinputparameters.

This paper is organized as follows: in Section 2 we presentthe baseline SSC method asdeveloped by Witteveen et

al. Next, in Section 3 we describe the Set-Covering stencils, our method for avoiding singular sample matrices andthe

analytic mappingforuniformly-distributedsimplexsampling.The followingsection describesthecut-HDMRapproachand

2. Simplex-stochasticcollocationmethod

InthissectionwegiveageneraloutlineoftheSimplex-StochasticCollocation(SSC)methodasdevelopedbyWitteveen etal.Foramoredetaileddescriptionwereferto[41,38,39,37].

2.1. GeneraloutlinebaselineSSCmethod

TheSSCmethodwasintroducedasanon-intrusivemethodintendedforrobustandefficientpropagationofuncertainty

through computer models. It differs fromtraditional collocation methods, e.g. [4,18],in two main ways. First, for

multi-dimensional problems it employs the Delaunay triangulation to discretize the probability space into simplex elements,

ratherthan relying on themore commontensorproduct ofone-dimensional abscissas [24]. Usinga multi-element

tech-niquehastheadvantage thatmesh adaptationcanbe performed,such that onlyregionsofinterest arerefined. Secondly,

theSSCmethodiscapableofhandlingnon-hypercubeprobabilityspaces[38].

TheresponsesurfaceoftheQoIu

(

ξ

)

isdenotedby w

(

ξ

)

anditisconstructedusingasetofnssamplesfromthe

com-putationalmodel,v

= {

v1

,

· · ·

,

vns

}

.Here,

ξ

isavectorofdrandominputparameters

ξ

(

ω

)

=

(ξ1(

ω

1),

· · ·

,

ξ

d

(

ω

d

))

∈

⊂

R

d.

Furthermore,we define

to be the parameter spaceand

ω

=

(

ω

1,

· · ·

,

ω

d

)

∈

⊂

R

d is a vector containing realizations

of theprobability space

(,

F

,

P

)

with

F

the

σ

-algebra ofevents and P a probability measure. The variables in

ω

are

distributeduniformlyas

U

(0,

1),andtheinputparameterscanhaveanydistribution fξ,althoughforthesakeofsimplicity werestrict ourselvesinthispaperto fξ

=

U

(

ξ

ai

,

ξ

bi

),

withthebounds

ξ

ai and

ξ

bi.We performallouranalysisontheunit

hypercubeKd

:= [

0,1

]

d,andweusealinearmapinordertogofromKdtotheparameterdomain

ξ

.Ourgoalisto

propa-gate fξ throughthecomputationalmodelinordertoassesstheeffectof fξ onthem-thstatisticalmomentofu

(

ξ

(

ω

)),

i.e.

wewishtocompute

μ

(um)

:=

u

(

ξ

)

m fξ

(

ξ

)

d

ξ

.

(1)

Notethat u canalsobea functionofa physicalcoordinate xorotherdeterministic explanatoryvariables,butforbrevity

we omit x from the notation. Since the SSC method discretizes the parameter space

into ne disjoint simplices

=

1

∪ · · · ∪

ne,themth statisticalmomentisapproximatedas

μ

(um)

=

u

(

ξ

)

mfξ

(

ξ

)

d

ξ

≈

μ

(wm)

:=

ne

j=1

j wj

(

ξ

)

m fξ

(

ξ

)

d

ξ

.

(2)

Here, wjisalocalpolynomialfunctionoforderpj associatedwiththe j-thsimplex

jsuchthat

w

(

ξ

)

=

wj

(

ξ

),

for

ξ

∈

j

,

(3)

andtheinterpolationconditionrequires

wj

(

ξ

kj,l

)

=

vkj,l

.

(4)

Thesubscript kj,l

∈ {

1,

· · ·

,

ns

}

isa globalindexwhichrefers tothe k-thaddedcomputational sample,while j refers toa

certainsimplex element.Furthermore,l

=

0,

· · ·

,

Nj isa localindexusedtocountthenumberofsamplesfromvinvolved

intheconstructionofwj.The Nj

+

1 numberofpointsneededford-dimensionalinterpolationoforderpjisgivenby Nj

+

1

=

(

d

+

pj

)

!

d

!

pj

!

,

(5)

andthelocalinterpolationfunction wj itselfisgivenbytheexpansion

wj

(

ξ

)

=

Nj

l=0

cj,l

j,l

(

ξ

).

(6)

Thechoiceofbasispolynomials

j,l,andthedeterminationoftheinterpolationcoefficientscj,l isdealtwithinSection2.2.1.

Notethat fora givend, themaximumallowable order pj basedon thenumberof samplesns can be inferred from(5).

The particular choice of pj will depend on the smoothness of the response, with the objective of avoiding the Runge

phenomenon.

WhichNj

+

1 pointsareusedin(6)isdeterminedbytheinterpolationstencilSj.Thestencilcanbeconstructedbased

Fig. 1.Delaunaytriangulationfortwostenciltypeswithadiscontinuityrunningalongthedottedline.(Forinterpretationofthereferencestocolorinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

whichwouldsufficefor pj

=

1.Forhigherdegreeinterpolation,neighboring points

ξ

k areaddedbasedontheir proximity

tothecenterofsimplex

j,i.e.basedontheirrankingaccordingto

ξ

_k

−

ξ

_center,j

2

,

(7)

wherethose

ξ

k ofthecurrentsimplex

jareexcluded.Thesimplexcenter

ξ

center,j isdefinedas

ξ

_center,j

:=

1 d

+

1 d

l=0

ξ

_k j,l

.

(8)

The nearest neighbor stencil (7) leads to a pj distribution that can increase quite slowly when moving away from

a discontinuity. An example of this behavior can be found in Fig. 1(a), which showsthe Delaunay triangulation with a

discontinuityrunningthroughthedomain.Analternativetonearest-neighborstencilsarestencilscreatedaccordingtothe EssentiallyNon-Oscillatory(ENO)principle[39].TheideabehindENOstencilsistohavehigherdegreeinterpolationstencils up toa thinlayerofsimplicescontaining thediscontinuity.Foragivensimplex

j,its ENOstenciliscreatedby locating

all thenearest-neighborstencils thatcontain

j,andsubsequentlyselectingtheone withthehighest pj.This leadsto a

Delaunay triangulation whichcapturesthediscontinuity betterthan itsnearest-neighbor counterpart.Anexample canbe

foundinFig. 1(b).Unlessotherwisestated,forallsubsequentbaselineSSCsurrogatemodelswewilluseENO-typestencils. The initialsamples,atleastinthecaseofhypercubeprobabilityspaces,arelocatedatthe2d corners ofthehypercube

Kd.Furthermore,onesample isplacedinthemiddleofthehypercube.Next,theinitialgridisadaptivelyrefinedbasedon

an appropriate errormeasure. Thiserror measure can eitherbe based onthe hierarchical surplusbetweenthe response

surfaceofthepreviousiterationandnewasamplevk,oronthegeometricalpropertiesofthesimplices.Thelatteroptionis

morereliableinmultiplestochasticdimensionsasitisnotbasedonthehierarchicalsurplusinasinglediscretepoint [37].

Thegeometricalrefinementmeasureisgivenby

¯

ej

:= ¯

j

¯

2Oj

j

.

(9)

Itcontainstheprobabilityandthevolumeofsimplex

j,i.e.

¯

j

=

j fξ

(

ξ

)

d

ξ

and

¯

j

=

j d

ξ

,

(10)

where

¯

=

ne_j=1

¯

j.Theprobability

¯

jcanbecomputedbyMonte-Carlosamplingand

¯

j viatherelation

¯

j

=

1 d

!

|

det

(

D

)

|

,

D

=

ξ

_k j,1

−

ξ

kj,0

ξ

kj,2

−

ξ

kj,0

· · ·

ξ

kj,d+1

−

ξ

kj,0

∈

R

d×d

.

₍₁₁₎

Finally,theorderofconvergenceOjisgivenby[37] Oj

=

pj

+

1

d

.

(12)

The simplexwiththehigheste

¯

j isselectedforrefinement.Toensureagoodspreadofthesample points,only

Fig. 2.Thesubsimplex(dottedline)ofatwo-dimensionalsimplex.Uponrefinementonesampleisplacedatarandomlyselectedlocationinsidethesub simplexinordertoavoidclusteringofpoints.

ξ

_sub j,l

:=

1 d d

l∗=0 l∗=l

ξ

_k j,l∗

,

(13)

seeFig. 2fora two-dimensionalexample.Inorderto placerandomsamplesuniformlyinan arbitrarysimplexwe derive ananalyticalmap Md

:

Knξ

→

j,seeSection2.2.2.TheSSCalgorithmcanbeparallelizedbyselectingtheN

<

ne simplices withtheN largeste

¯

j forrefinement.

Notethat by using(13)onlysimplex interiorswill be refined(see againFig. 2), andtheboundariesofthehypercube willneverbesampledoutsidetheinitial2d _{points.Asaconsequence,discontinuitiesthatcrossahypercubebordercannot}

be capturedaccurately atthat border.Toavoidthis, we donot use(13) ifasimplex element locatedatthe boundaryis

selectedforrefinement.Instead,werandomly placesamplesatthelongestsimplex edgewhichisattheboundary,

±

10%

fromtheedgecenter.

Notethat(9)isprobabilisticallyweighted through

j andthatitassigns high

¯

ej to thosesimpliceswithlow pj since

ingeneral

¯

j

1.Effectivelythismeansthat(9)isasolution-dependentrefinementmeasurewhichrefinessimplicesnear

discontinuities sincetheSSCmethodautomatically reducesthepolynomial orderifa stencil Sj crossesa discontinuity.It

achieves thisby enforcingthe so-called Local Extremum Conserving (LEC)limiter to all simplices

j in all Sj.The LEC

conditionisgivenby min ξ∈j wj

(

ξ

)

=

minvj

∧

max ξ∈j wj

(

ξ

)

=

maxvj

,

(14)

wherevj

= {

vkj,0

,

· · ·

,

vkj,d

}

arethesamplesattheverticesof

j.Ifwj violates(14)inoneofits

j

∈

Sj,thepolynomial

order pj ofthatstencilisreduced, usuallyby1.Sincepolynomial overshootsoccur whentryingto interpolatea

disconti-nuity, pj isreducedthemostindiscontinuousregions. Interpolatinga functionona simplexwith pj

=

1 andvj located

atitsverticesalwayssatisfies(14) [37].Thisensures that w

(

ξ

)

isabletorepresentdiscontinuities withoutsufferingfrom theRungephenomenon.Inpractice,(14)isenforcedforall

j inall Sj viarandomsamplingofthe wj.Ifforagivenwj (14)isviolated foranyoftherandomly placedsamples

ξ

_j,thepolynomial orderofthecorresponding stencilisreduced. Again,howwe samplethed-dimensionalsimplicesisdescribed inSection 2.2.2.Thecomputational costofenforcing(14) isinvestigatedinSection5.

The procedureof enforcingthe LECcondition, computing arefinement measure andsubsequentlyrefining certain

se-lectedsimplicesiseitherrepeatedforamaximumofIiterations,nsmax samplesorhaltedwhenasufficientlevelofaccuracy

is obtained. Thislevel ofaccuracy can be estimatedthrough an error measure basedupon the hierarchicalsurplus [37]. As mentioned, this is the difference betweenthe response surface wj and the newly added code sample vkj,ref at the

refinementlocation

ξ

kj,ref,i.e.

(

ξ

_k

j,ref

)

:=

wj

(

ξ

kj,ref

)

−

vkj,ref

.

(15)

Thisis a point estimate ofthe error,located atwhat will be avertex inthe newrefined Delaunay grid. Toassign error estimatestothesimplicesratherthantovertices,theerror

˜

jisintroduced.Foreach

j,

˜

jissimplytheabsolutevalueof (15)ofitsmostrecentlyaddedvertex

ξ

k∗.SinceaddingverticeswillchangetheDelaunaydiscretizationwerelatetheerror

oftheprevioussimplextothenewonevia

ˆ

j

≈ ˜

j

¯

j

¯

k∗,ref

_Oj (16) [38].Theratio

¯

i

/

¯

k∗,ref representsthe changeinvolumefromits old size

¯

k∗,re f,i.e.thevolume ofthesimplexwhich

wasrefined by

ξ

k∗,toitsnewsize

¯

j.Finally,eachindividual

ˆ

j iscombinedina globalerrorestimateviathefollowing

ˆ

rms

=

ne j=1

j

ˆ

2_j

.

(17)

ThecompletebaselineSSCmethodisgiveninpseudocodeinAppendix A.

2.2. ImprovementsonthebaselineSSCmethod

BeforediscussingournewstencilselectiontechniqueinSection3.1,weintroducetwoimprovementstothebaselineSSC methodnotdiscussedintheoriginalreferences[41,38,39,37].

2.2.1. Poisedsamplesequence

Theauthorsof[35]write(6)inmatrixform,constraining

j,l totheclassofmonomials,andsubsequentlysolve

explic-itly forthecoefficientscj,l.Theynote thatalthoughthey hadno difficultiesinsolvingthissystem,the matrixcouldhave

a highcondition number.Thisposesnorealproblemford

≤

3, butforhigherdimensionsitcan becomeproblematic. To

copewiththisweimposeanadditionalconditionontheconstructionofthestencils Sjsuchthattheinterpolationproblem

is poised,meaningthat thesample matrix isnon-singular[23].Inthe followingdiscussionwe dropthe subscript j until furthernoticetomakethenotationmoreconcise.

To constructtheinterpolating monomials,let usdefine thecollection consistingof N

+

1d-dimensionalmulti-indices

¯

i

:=

(

i1,

· · ·

,

ik

,

· · ·

,

id

),

whereforall

¯

iwehave

|¯

i

|

:=

i1

+· · ·+

id

≤

pjandeachikisanintegerbetween0andd.Furthermore,

for agiven vertex

ξ

_l

=

(ξ

1,l

,

· · ·

,

ξ

d,l

)

belongingto stencil S, letus defineits

¯

i-thpowerto be

ξ

¯l i

:=

ξ

i1

1,l

× · · · ×

ξ

id d,l. The

samplematrix

,

amulti-dimensionalVandermondematrix,canthenbewrittenas

=

⎡

⎢

⎢

⎢

⎣

ξ

¯₀0

ξ

¯₀1

· · ·

ξ

¯₀N

ξ

¯₁0

ξ

¯₁1

· · ·

ξ

¯₁N

..

.

..

.

..

.

ξ

¯_N0

ξ

¯_N1

· · ·

ξ

¯_NN

⎤

⎥

⎥

⎥

⎦

∈

R

(N+1)×(N+1)

.

(18)

As anexample,thel-throwof(18)inlexicographicalorderforpj

=

2 willlooklike[ 1

ξ1

,l

ξ2

,l

ξ

₁2_,l

ξ1

,l

ξ2

,l

ξ

₂2_,l].The

coefficientscl in(6)cannowbeobtainedbysolvingthesystem

⎡

⎢

⎢

⎢

⎣

ξ

¯₀0

ξ

¯₀1

· · ·

ξ

¯₀N

ξ

¯₁0

ξ

¯₁1

· · ·

ξ

¯₁N

..

.

..

.

..

.

ξ

¯_N0

ξ

¯_N1

· · ·

ξ

¯_NN

⎤

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎣

c0 c1

..

.

cN

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

v0 v1

..

.

vN

⎤

⎥

⎥

⎦

,

(19)

where

{

v0,

· · ·

,

vN

}

arethecodesamplesbelongingtostencil S.Oncethecl areknown,wecaninterpolatetoanypoint

ξ

inthedomainspannedby S.

Wedefine

≡

det

(),

andnotethatthewholeapproachhingesonthewell-poisednesscondition

=

0.Thiscondition

is relatively easy violated during the SSC procedure in higher dimensions. For instance, if for d

=

4 we determine the

maximumallowable p using(5)ontheinitialDelaunay gridweobtain pmax

=

2.However,manystencilsinthiscasewill

have

=

0.Alsosituationswherea stencilhastoomanyverticeslocatedinthesameplane (e.g.duetoedgerefinement

at theboundary of Kd), canlead toa zerodeterminantof (18). Thus,ford

>

1 thepoisednesscondition

=

0 imposes

constraintsonthegeometricaldistributionofthe

ξ

l.From[23,7]weknow

Theorem1.TheN

+

1vertices

ξ

₀

,

· · ·

,

ξ

_N

∈

R

d_{arepolynomiallypoisedifftheyarenotasubsetofanyalgebraichypersurfaceof}

degree

≤

p.

Analgebraichypersurfacein

R

d isad

−

1 dimensionalsurfaceembeddedinad-dimensionalspaceconstrainedtosatisfy anequation f

(ξ1,

· · ·

,

ξ

d

)

=

0.Thedegreeisgivenby f.

The authorsof[7]devisedaGeometric Characterization(GC) conditionwhichallows ustodetectifasetofverticesis poised,i.e.:

Definition1.GCcondition:Foreach

ξ

_l inasetofN

+

1 verticesin

R

d_{,thereexistspdistincthyperplanesG}

1,l

,

· · ·

,

Gp,lsuch

thati)

ξ

l doesnotlieonanyoftheseplanes,andii)allother

ξ

k,k

= {

0,

· · ·

,

N

}\{

l

}

lieonatleastoneofthesehyperplanes.

Mathematicallyspeakingi)andii)amountto

ξ

_i

∈

p

k=0

Fig. 3.When selecting nodeξ1, there exists one (p=1) plane which contains all other points exceptξ1. This is true for all nodes in the simplex. Theorem2.Let

{

ξ

l

}

beasetofN

+

1verticesin

R

d.If

{

ξ

l

}

satisfiestheGCcondition,then

{

ξ

l

}

admitsauniqueinterpolationofdegree

≤

p[7].

Duetoitsgeometricalconfiguration,asinglesimplex

j in

R

d alwayssatisfiestheGCconditionforp

=

1,seeFig. 3for

athree-dimensionalexample.Foragivenvertex

ξ

l

∈

j,wealwayshaveonehyperplanecontainingthefaceofthesimplex

madeupbyallverticesexcept

ξ

_l.Thus,Theorem 2impliesthatsimplex

j willleadtoa

with

=

0 andpj

=

1.

Weusethisresulttoobtainasetofwell-poisedENOstencilsSj

∀

j

=

1,

· · ·

,

ne,inawaythatissimilartotheconstruction

oftheENO-stencils asdescribed in[39].Onlyifduringthe enforcementoftheLECcondition (14)we encounterastencil

Sj forwhich

=

0,we collecta setofk candidatenearest-neighborstencils

{

Sj,i

}

k_i=1 whichall contain simplex

j.We

thenselectthe Sj whichhasthehighestpj and

=

0.Intheworstcasescenariowe getpj

=

1,where Sj containsonly

theverticesof

jitselfandforwhich

=

0 isguaranteedbyTheorem 2.IfwehavemultipleSjwithpj

>

1 whichsatisfy

theseconditions,weselecttheonewiththesmallestaverageEuclideandistancetothecell-center

ξ

center,j.

2.2.2. Simplexsampling

Simplicesarerefinedbyrandomlyplacingapointinsidethesub-simplex(13).Also,torandomlysamplethe wj during

theLECenforcementweneedtoplacerandompointsinsided-dimensionalsimplices.Ifwewouldliketouniformlysample alinesectionwiththeendpoints

[

ξ0,

ξ1

]

wewouldusethemapping

M1

=

ξ

0

+

r1

(

ξ

1

−

ξ

0

),

(21)

wherer1

∼

U

[

0,1

]

.Generatingpointsinsideatrianglecanbedonewith

M2

=

ξ

0

+

r 1/2

2

(

ξ

1

−

ξ

0

)

+

r 1/2

2 r1

(

ξ

2

−

ξ

1

)

(22)

whichmapspoints

{

r1,r2

}

inside theunitsquare K2 topoints insideatriangledescribedby thevertices

{

ξ

0

,

ξ

1

,

ξ

2

}

[33]. Theworkingprincipleof(22)isshowninFig. 4(a).Theparameterr₂1/2selectsalinesegmentparalleltotheedge

[

ξ

0

,

ξ

1

]

, whiler1 selectsapointalongthechosen linesegment.Theexponent1/2 ensuresthatuniformlydistributedpoints inthe square yielduniformlydistributedpoints inthetriangle.Thiscan beshownby consideringthe lengthofthe chosenline segment,whichincreaseslinearlywhenr₂1/2 movesfrom

ξ

0 to

ξ

1.Since werequireauniformdistributionofpoints, and consideringr1

∼

U

[

0,1

]

,thepdfofr12/2 shouldbelinearaswell.Ifwehavetherandomvariable X

=

r1/τ withr

∼

U

[

0,1

]

and

τ

∈

N

>0,wefindthecumulativedistributionfunction(cdf)ofX as

FX

(

x

)

=

P

(

X

≤

x

)

=

P

r1/τ

≤

x

=

P

r

≤

xτ

=

xτ

.

(23) Andthuswehavethepdf fX

(

x

)

=

dFX

/d

x

=

τ

xτ−1

∼

Beta(

τ

,

1).Therefore,inordertohavealinearpdfforr1/τ,wemust

set

τ

=

2.

Itissuggestedin[33] that(22)canbeextendedtohigherdimensions,althoughnospecific formulasare given.Hence, we usethe same principleto selectuniformly distributedpoints inside a tetrahedron, seeFig. 4(b).Here, theparameter

r1₃/3 selectsatriangleparalleltothebaseofthetetrahedron.Fromthereweuser1₂/2 andr1 asbeforetoselectapointon thistriangle.Theexponent1/3 againensuresthatthepointdistributionwillbeuniform.Notethattheareaoftheselected trianglesincreasesquadraticallyasr₃1/3movesfrom

ξ

0 to

ξ

1.Hence,itmustbedistributedasBeta(3,1).Wecannowderive anexpressionforM3usingthegeometricalsimilaritiesbetweenthebasetriangleandtheselectedparalleltriangle,which givesus M3

=

ξ

0

+

r 1/3 3

(

ξ

1

−

ξ

0

)

+

r 1/3 3 r 1/2 2

(

ξ

2

−

ξ

1

)

+

r 1/3 3 r 1/2 2 r1

(

ξ

3

−

ξ

2

).

(24)

When comparing (21), (22) and (24) we see a pattern emerge which suggests that the map from a d-dimensional

Fig. 4.Selecting a point inside a triangle and tetrahedron.

Fig. 5.An example of the map(25)ford=2,3 and 1000 samples.

Md

=

ξ

0

+

d

i=1 i

j=1 r 1 d−j+1 d−j+1

(

ξ

i

−

ξ

i−1

),

(25)

where againtherq are distributedas

U

[

0,1

]

.Ourproof that (25)producesuniformly distributedsamplesinthesimplex

canbefoundinAppendix D.

Tonumericallytest(25)in2and3dimensionswecansimplyplotsamplespoints,anexampleofwhichcanbefound

inFig. 5.Wehaveperformedsimilartestsupto8dimensions. 3. SSCSet-Coveringmethod

Inthissectionwedescribealternativeinterpolationstencils,whichresultsinacomputationalspeedupinhigher dimen-sions.

3.1. Setcoveringstencils

Section 5 willshow thatthe enforcementofthe LECcondition canbecome computationally expensiveforhighdand

pj. Thisis especially trueforsmooth responsesurfaces ofthe QoI. Formanystencils of ourdiscontinuous problem, the

LEC condition is violated and pj is reduced which in turn significantly lowers the total required number of surrogate

model evaluations(nw) neededtocheck (14).Thisdoesnot happenvery oftenwhen theresponse surface issmooth. As

309

Fig. 6.Two stencils which overlap each other. The dark simplices are shared by both stencils.

timeneededtoconstructthesurrogatemodel.NotethatthisincreaseisduetotheSSCprocedure,andthusisadditionalto

thetimeneededtosamplethecomputercode.

However,theproblemliesnot onlywiththeexponentialincreaseofnw,butalsointheextremelylargeoverlapofthe

stencils Sj.NotethatthebaselineSSCmethodenforcestheLECconditionforallsimplices

jinallstencilsSj.Hence,ineach

simplex

j, wjisevaluatedthesamenumberoftimesas

j appearsinallstencils Sj.Foratwo-dimensionalexamplesee Fig. 6.There aretwo stencils,denotethem Sr and Sq,associatedto twodifferentsimplices

r and

q.Thedarkcolored

simplicesaretheoneswhichappearinbothstencils.Thus,whentheLECconditionischeckedforbothstencils,wr butalso

wq isevaluated inthedarksimplices. Moreover,sincethere arene stencils,theoverlapwillbe large,andmanydifferent

wj willbe evaluated inthe same simplex element. Thisis no bottleneckfor problemsof low-dimensionality, butifthe

dimensionincreasesthisoverlapwillmaketheLECconditionverycostlytoenforce,seeSection5.1.1.

Weproposeanalternativetechniqueforproblemswithhigherd,usingSet-Covering(SC)stencilsbasedonthewell-know set-coveringproblem[16],statedasfollowsinSSCterminology:

SetCoveringproblem.LetXj

= {

j,0,

· · ·

,

j,K

}

bethesetofallsimplicesthatareinsidethedomainspannedbytheverticesof

stencilSj.Then,giventheset

X

= {

X1,

· · ·

,

Xne

}

,andthesetofallsimplices

U

= {

1,

· · ·

,

ne

}

,findthesmallestsubset

C

⊆

X

that

covers

U

,i.e.forwhich

U

⊆

Xj∈C

Xj

holds.

Itisshownin[16]thattheset-coveringproblemisNP-complete,andthusnofastsolutionisknown.Wecould approxi-mate

C

bythegreedyalgorithm,whichateachstepsimplyselectsthe Xjwiththelargestnumberofuncoveredsimplices.

WethenwouldhavetochecktheLECconditionforallstencilsin

S

sc,definedasthesetofSjcorrespondingtothe Xj

∈

C

.

For(high-dimensional) problemswitha maximumpolynomial order pmax

>

1, thenumber ofstencils in

S

sc willbe

sig-nificantlylower than ne.However, thisapproach wouldstill requireto constructall Xj

∈

X

.Also, many ofthe Xj could

potentially cross a discontinuity,leading to a violationof the LECcondition and thesubsequent reduction insize of Xj.

When this happensthe SC property of

C

can nolonger be guaranteed. Thus, an iterativeapproach would be necessary

whichrunsuntil

S

scsatisfiesboththeSCandLECproperty.

Forreasonsofcomputationalefficiency,wewanttoavoidthisiterativeapproachasmuchaspossible,andthusnotrely completelyontheLECconditiontoturnasetofnearest-neighbor stencilsintoasetofENOstencils.Hencewewillusethe informationcontainedinvregardingthediscontinuitylocationtocreateasmallsetofSC stencilsthatalsoresembleENO stencils,i.e.whichdonotcrossadiscontinuity.WewilldenotethesestencilsasSCENOstencils.Althoughmoresophisticated approachesareavailable [40],forreasonsofsimplicityweidentifythe

j throughwhichthediscontinuityrunsbysimply

imposingathresholdvt onthemaximumjumpobservedinvateachsimplex.Then,thesetofdiscontinuoussimplicescan

bedefinedas

D

= {

j

| |

maxvkj,l

−

minvkj,l

| ≥

vt

,

l

=

0

,

· · ·

,

d

,

j

=

1

,

· · ·

,

ne

}

(26)

Forthenozzleflowcasewe setthethresholdvalueto vt

=

1.0.Atwo andthree-dimensionalvisualization ofthe

j

∈

D

can be found in Fig. 7.We furthermore redefinethe set

C

as theset containing all thesimplices

j that are currently

coveredbyastencil Sj,ratherthanthetruesmallestsubset

C

⊆

X

oftheSCproblem.

ThegeneraloutlineforconstructingtheSCENOstencilsisnowasfollows.Forthe

j

∈

D

weset pj

=

1 and

C

=

C

∪

D

,

310

Fig. 7.Discontinuous simplices identified by(26).

fromtheset

U

\

C

withthelargestvolume.Fortheselectedsimplexwegrowits stencilby addingneighboring

j which

are notcoveredyet,i.e.whicharenot in

C

.Thiswillyieldaset

C

whereeverysimplex appearsonlyonce,i.e.asetwith zerooverlap.Notethattorelax thiscondition onecaneasily allowfortheadditionofneighboring simplices whicharein

C

\

D

.Ineithercasewecontinuegrowingthestenciluntiltherearenomoreavailableneighbors oruntil Sjislargeenough

toallowinterpolationoforder pmax.Wethenmovetothenext

∗_j andrepeatuntil

C

coverstheentireprobabilisticspace

U

.ForagraphicalrepresentationofthestencilconstructionwerefertoFig. 8.Itisimportanttonotethatourmaingoalis tofindaset

C

withacardinality

|

C

|

significantlylessthatne,whichisaneasiertaskthanapproximatingthetrueminimal

C

oftheSCproblemascloselyaspossible.InAppendix BthealgorithmforconstructingtheSCENOstencilsisdisplayedin

pseudocode.

Thisapproachassuresthatwehavearelativelysmallset

S

sc forwhich:i)

|

S

sc

|

ne,ii)thatnotallne nearest-neighbor

Xi

∈

X

needbe calculated,iii)that no Xj crossesadiscontinuity,andiv)the

j

∈

D

are interpolatedlinearly.Theresult

isthatthenumberoftimestheLECconditionneedstobecheckedisreducedsignificantly.Onlyforthose Sjassociatedto

the Xj

∈

C

\

D

itisstillnecessarytocheckforinterpolationovershoots,sincethe

j

∈

D

areguaranteedtobeLECdueto

theirlinearinterpolation.ThepropertyofSCENOstencilsmentionedunderiii)alsomeansthatthenumberoftimestheLEC condition isviolated isreduced,althoughnot alwaystozeroduetoreasonsofillconditioningofthesamplematrix(18). Thisisespeciallytrueforhighd.AnapproachasdescribedinSection2.2.1wouldrendersomeoftheadvantagesmentioned under i)–iv) void. Reducing pj for ill-conditionedstencils will increase the cardinality of

S

sc, andall Xj

∈

X

should be

calculatedinordertolookforalternativestencils.Insteadwe directlysolveanill-conditionedsystem(19)inthenon-null

subspace ofthe solution asdescribed in [17]. This method utilizes Gauss–Jordan eliminationwith complete pivoting to

identifythenullsubspaceofasingularmatrix

,

i.e.

nullcnull

=

0.Thispartitionsthelinearsystemasdepictedbelow,

range

· · ·

· · ·

null crange cnull

=

_v

..

.

,

(27)

where

rangecrange

=

visthenon-nullsubspaceinwhichwecanobtainaccuratesolutions.Inthecaseofanill-conditioned

system, thenullsubspaceiscloselyapproximatedbyaspacewherethepivots

ψ

ii areverysmallbutnot exactlyequalto

zero.Thestartofthis‘near-null’subspaceisidentifiedbythefirstpivot

ψ

ii forwhichthecondition

|

ψ

ii

/

η

c

|

<

holds,where

η

c isthelargestpivotof

and

isaverysmallparameter,whichwesetequalto10−14.Inboththeill-conditionedand

singular casethedetrimentaleffectof

null onthesolutioniseliminatedbyaso-calledzeroingoperation,whichbasically

replaces

null byan identitymatrixofequaldimensionandsetscnull

=

0.Thus,effectivelyspeakingthosecoefficientscj,l

which havebeenoverwhelmedby round-offerrorare automaticallycut out oftheexpansion (6). Inourexperimentswe

found thatthedimension of

null,i.e.thenullityof

,

issmallcomparedtothedimensionofthefull

,

seeTable 1for

sometypicalexamplesatd

=

6.

If the systemof equations is well-posed, the algorithm amounts to regular Gauss–Jordan elimination withcomplete

pivoting.Inanycase,thequalityoftheresponsesurfaceischeckedviatheLECcondition. 4. High-DimensionalModel-Reduction

As willbe showninSection5,theuseofSCENOstencilsmakestheSSCmethodmorecomputationallyefficientwithin

Fig. 8.A two-dimensional example of the SC stencil construction.

Table 1

Examplesofill-conditionedsystems.Weshowthedimensiond,thepolynomialorderofthe stencil,thenullityandconditionnumberofthesamplematrix,andfinallythecondition numberofthenon-nullrange.

d pj Dimension Nullity Cond. Cond.range

6 2 28×28 1 1.36e+17 9.39e+3

6 3 82×82 1 1.31e+17 2.40e+4

6 3 84×84 2 2.85e+17 2.66e+3

approach is required. In physical systems it is often found that only a few parameters are influential, and only

low-order correlationsbetweenthe input parameters havea significant impact on theoutput. Tocapitalize onthis behavior,

High-DimensionalModel-Reductiontechniquescanbeapplied,seethereferencesofRabitzandAli ¸s[27,26].OurQoIis rep-resentedbyad-dimensionalfunction f

(

ξ

,

x

)

definedonthehypercubeKd,wherexisapossiblephysicalcoordinatewhich

wewillagainomitfromthenotationforthesakeofbrevity.Then,theHDMRexpansionisanexactandfinitehierarchical

expansionofcomponentfunctionsofincreasingdimension,givenby

f

(

ξ

)

=

f0

+

i fi

(ξ

i

)

+

i1<i2 fi1i2

(ξ

i1

, ξ

i2

)

+ · · · +

i1<···<il fi1···il

(ξ

i1

,

· · ·

, ξ

il

)

+ · · · +

f1···d

(ξ

i1

,

· · ·

, ξ

id

).

(28)

Here,thei1,

· · ·

,

id areintegerssatisfying1

≤

i1

<

i2

<

· · ·

<

id

≤

d.Thezero-thordercomponentfunction f0 isaconstant andrepresentsthemeaneffect.Thefirst-orderfunction fi

(ξ

i

)

isaunivariatefunction,generallynonlinear,whichrepresents

theeffectofindependentlyvaryinginputparameter

ξ

i.Higherorderfunctionsrepresentthecooperativeeffectsofincreasing

ef-ficientlyrepresentedbyatruncatedL-thorderexpansion,whereL

<

d.Thisacalledaproblemwithloweffectivedimension, which occursfrequently inproblemsof physicalnature[12].Thus, the generalideais tosolve multiplelow-dimensional

subproblems inplace of a single high-dimensional one. The resultant computational effort to determine the component

functionswillscalepolynomially,ratherthanthetraditionalexponentialincreasewithd[26].

Ameasure

μ

forthemeasurespace

(

Kd

,

B

(

Kn

),

μ

),

where

B

istheBorel

σ

-algebraonKd,isdefinedas

d

μ

(

ξ

)

:=

d

μ

(ξ

1

,

· · ·

, ξ

d

)

=

d

i=1 d

μ

i

(ξ

i

),

K1 d

μ

i

(ξ

i

)

=

1

,

d

μ

(

ξ

)

=

g

(

ξ

)

d

ξ

=

d

i=1 gi

(ξ

i

)

d

ξ

i

.

(29)

Here, g

(ξ

i

)

isthemarginal densityoftheinput

ξ

i.It istheparticular formchosen forthe gi

(ξ

i

)

that willdetermine the

form of the componentfunctions. In order to compute these functions, let usalso define unconditional andconditional

meanwithrespecttoagroupofinputvariablesas

Mf

(

ξ

)

:=

Kd f

(

ξ

)

dμ

,

M(i1···il)f

(

ξ

)

:=

Kd−l f

(

ξ

)

⎡

⎣

j∈{/i1···il} dμj

(ξ

j

)

⎤

⎦

.

(30)

Then,viaafamilyofprojectionoperators Pi1···il

:

Kd

→

Kl,thecomponentfunctionsarerecursivelydefinedasfollows[26]: f0

:=

P0f

(

ξ

)

=

Mf

(

ξ

)

fi

(ξ

i

)

:=

Pif

(

ξ

)

=

M(i)f

(

ξ

)

−

P0f

(

ξ

)

fi j

(ξ

i

, ξ

j

)

:=

Pi jf

(

ξ

)

=

M(i j)f

(

ξ

)

−

Pif

(

ξ

)

−

Pjf

(

ξ

)

−

P0f

(

ξ

)

..

.

fi1···il

(

ξ

)

:=

Pi1···ilf

(

ξ

)

=

M (i1···il)_f

(

ξ

)

−

j1<···<jl−1⊂{i1···il} Pj1···jl−1f

(

ξ

)

− · · · −

P0f

(

ξ

)

(31)

Thecomponentfunctions fi1,···il and fj1···jk areindependentandorthogonal,thusaslongasoneindexbetween

{

i1,

· · ·

il

}

and

{

j1

· · ·

jk

}

differswehave

Kd

fi1,···il

(ξ

i1

,

· · ·

, ξ

il

)

fj1···jk

(ξ

j1

,

· · ·

, ξ

jk

)

d

μ

=

0 (32)

The correlation interpretation of fi1···il is associated with the chosen form of the measure

μ

. If gi

=

1, i

=

1,

· · ·

,

d,

the Lebesgue measure(d

μ

=

dξ1dξ2

· · ·

dξd) is retrievedand(28) together with(31) becomes thewell-know Analysis Of

Variance (ANOVA) decomposition.Computing the componentfunctions inthe ANOVA decompositioninvolves evaluating

multi-dimensionalintegrals,whichcanbedonebyforinstanceMCtechniques[31].Analternativewhichismore computa-tionallytractableisthecut-HDMRdecompositionproposedin[27,26].Inthiscasethemeasureisdefinedas

d

μ

=

d

i=1

δ(ξ

i

−

η

i

)

d

ξ

i

,

(33)

i.e.gi

(ξ

i

)

=

δ(ξ

i

−

η

i

),

aDiracmeasurelocatedatthe‘cutcenter’

η

=

(

η

1,

η

2,

· · ·

,

η

d

).

Thischoiceremovestheneedfor

eval-uatingmulti-dimensionalintegrals,anditexpresses f

(

ξ

)

asasuperpositionofitsvaluesalonglines,planesandhyperplanes

passingthroughthecutcenter

η

.Thecomponentfunctions(31)nowbecome

f0

:=

P0f

(

ξ

)

=

f

(η)

fi

(ξ

i

)

:=

Pif

(

ξ

)

=

f(i)

(ξ

i

)

−

P0f

(

ξ

)

fi j

(ξ

i

, ξ

j

)

:=

Pi jf

(

ξ

)

=

f(i j)

(ξ

i

, ξ

j

)

−

Pif

(

ξ

)

−

Pjf

(

ξ

)

−

P0f

(

ξ

)

..

.

fi1···il

(

ξ

)

:=

Pi1···ilf

(

ξ

)

=

f (i1···il)

(ξ

i1

,

· · ·

, ξ

il

)

−

j1<···<jl−1⊂{i1···il} Pj1···jl−1f

(

ξ

)

− · · · −

P0f

(

ξ

).

(34) Here, f(i1···il)

(ξ

_i

1

,

· · ·

,

ξ

il

)

isthe conditional mean(30)taken withrespect tomeasure (33),andthus it equals f with its

inputs

ξ

i set to

η

i,except inputs

ξ

i1

,

· · ·

,

ξ

il.As an example,consider theunivariate function f(i)

(ξ

i

)

=

f

(

η

1,

· · ·

,

η

i−1,

ξ

i

,

η

i+1,

· · ·

,

η

nξ

).

Theauthors of[20]used thecut-HDMR frameworkcoupledwiththeir Adaptive Sparse-Grid(ASG) collocationmethod [19],wheretheychose

η

asthemeanoftherandominputvector.Besidestruncating(28)atacertainorder,theyalsomade

theirapproachdimensionadaptivebasedonweightswhichidentifytheimportantdimensions.AlthoughtheirASGmethod

usesonlyalinearfinite-elementbasis,interpolationovershootscanstilloccur.Thus,motivatedbytheirworkin[20]wewill

alsoemployadimensionadaptivecut-HDMRapproach,exceptwewillcoupleitwiththeSSCmethodutilizingtheSCENO

stencilstoavoidthementioneddownsidesofASG.

Ifwedefine

K

:= {

1,2,

· · ·

,

d

}

,theHDMRexpansion(28)canbewritteninshort-handnotationas[20]

f

(

ξ

)

=

u⊆K fu(

ξ

u

)

=

u⊆K

v⊆u

(

−

1

)

|u|−|v|f(v)

(

ξ

v

),

(35)

where in the first equality we sum over the powerset of

K

, i.e. over all possible subsets u

⊆

K

. We furthermore set

f_∅

=

f0.Thesecondequalityisobtainedbyexpandingeachcomponentfunction fu

(

ξ

u

)

asindicatedin(34).Notationwise,

iffor instancev

= {

1,4,6

}

, then f(v)

(

ξ

v

)

=

f(146)

(ξ1,

ξ4,

ξ6).

Each individual

|

v

|

-dimensional subproblem f(v)

(

ξ

v

)

can be

approximatedbyaSSCsurrogate(6).Inthatcase(35)becomes

f

(

ξ

)

≈

w

(

ξ

)

=

u⊆K

v⊆u

(

−

1

)

|u|−|v| ne

j=1 Nj

l=0 cjl

jl

(

ξ

v

).

(36)

Inordertoassesstheconvergenceofeachindividual f(v)

(

ξ

v

),

theauthorsof[20]usethehierarchicalsurplus.Thisisalso

possibleinthecaseoftheSSCmethod,see(15).Alternatively,theRMSerrorestimate(17)canusedforthispurpose.Since (17)isaglobalerrorestimate anditalsoincludesinformationfromthedistributionofthe inputparameters, weusethe RMSerrortoassesstheconvergence.

Furthermore,themeanofeachcomponentfunction,definedas Ju,canalsobecomputedfromthesurrogatemodel

Ju

=

v⊆u

(

−

1

)

|u|−|v| ne

j=1 Nj

l=0 cjl

E

jl

(

ξ

v

)

.

(37)

We compute (37) via random sampling, which can be performed quickly since it requires only sampling the surrogate

model.

Inordertoidentifytheimportantdimensions,allfirstordercomponentfunctions fi

(ξ

i

)

arecomputed.Again,theseare

one-dimensionalfunctionswhichmeasuretheimpactofasingleindependentinputparameterontheoutput.Next,a weight

isdefined

α

i

=

Ji

2

f0

2

,

(38)

which measures the contribution of each individual

ξ

i on the mean of all first order component functions[20].We

al-ways take the L2 norm

·

2 over the spatial domain. Equation (38)can be considered asa sensitivity index, andonly

those dimensionsforwhich (38)islarger than a user-prescribederror threshold

1 are considered important.All higher order fv

(

ξ

v

)

wherev contains indicesof dimensionswhichdid not make thecut will not be computed. Consider e.g. a

d-dimensionalproblemon Kd,whereonly v

= {

1

}

andv

= {

2

}

satisfy

α

i

>

1.Theonlyhigher-ordercomponentfunction thatwillbecomputedinthiscaseis f12(ξ1,

ξ2),

regardlessofthevalueofd.

Thedownsideof(38)isthatitishardtochoose

1 beforehand.Oneshouldfirstcreatethefirst-orderHDMRexpansion anddecideonan appropriatevalueaposteriori.Analternativeistouseaweightmeasuringtherelativecontributionof Ji

withrespecttothesumofallfirst-ordermeans,i.e.

α

i

=

Ji

2

d

k=1

Jk

2

.

(39)

Nowonecanapriorichoosea

1

∈

[0,1],andselectthesmallestsetofimportantdimensionsforwhichthesumoftheir

α

i

isgreaterthan

1.

Dimensionadaptivityisextendedtohigherdimensionsaswellbydefiningaweightfor

|

u

|

>

1 as[20]

α

u

=

Ju

2

v∈Vcomp,|v|<|u−1| Jv

2

.

(40)

Here,theset

V

compsimplyholdsalltheindicesvthatwere computed.Furthermore,allsubsetsvofcomponentfunctions

whichare importantare addedto aset

V

imp. Thatway,ahigher-orderimportant uisadmissible ifallv

⊂

urequiredto

compute(35)arealsoin

V

imp.Thisistheso-calledadmissibilitycondition,whichisgivenby

Fig. 9.Moutas function ofpandptobtained by MC sampling, with the geometrical constants fixed to their nominal value.

Similartothefirst-ordercase,wecandefinearelativecounterpartof(40)as

α

u

=

Ju

2

v∈Vcomp,|v|=|u|

Jv

2

,

(42)

suchthatthe

αu

sumtooneandwecanchoosea

1

∈ [

0,1

]

apriori.

Finally,arelativeerrormeasurebetweentwoHDMRexpansionsofconsecutiveorders p

−

1 andpisdefinedas

α

p

=

_|u|≤p Ju

−

_|u|≤p−1Ju

2

_|u|≤p−1Ju

2

.

(43)

Thealgorithmstopswhen

α

p becomessmallerthananotherused-definedthreshold

2.AnoverviewoftheHDMRalgorithm

isdepictedinAppendix C. 5. Resultsanddiscussion

5.1. ComparisonENO–SCENOstencils

WepresenttheresultsobtainedwiththebaselineSSCmethodwithENOstencils,versustheSSCmethodwiththeSCENO

stencils.Asatestcaseweuseaquasi-1Dnozzlecaseupto5dimensionsandanalgebraictestfunctionuptod

=

8.

5.1.1. Nozzleflow

Asatestcaseweusethesolverfrom[25],whichcomputestheflowthroughaquasi-1Ddivergingnozzle.Weprescribe theflowtobesonicatthenozzleinlet,i.e.Min

=

1.Fromfluidmechanicsweknowthattheflowisdrivenbythepressure

ratio,i.e.bytheratiobetweenthetotalpressure pt attheinletandthestaticpressure pofthesurroundingsatthenozzle

exit. Depending on the value of pt

/

p, the flow can show very different behavior. If pt

/

p exactly equals the adaptation

value, the flowreachesthe staticpressureof thesurroundings atthenozzleexit andthejet exhaustssmoothly into the atmosphere.Astrongerpt

/

pwillresultinsmoothflowthroughthenozzle,whichissupersonicatthenozzleexit.Inorder

to matchtheoutsidepressure p,theflowundergoesa supersonicexpansion attachedtothe nozzleexit(under-expanded

nozzle). A smaller pt

/

p, butstill above a thresholdthat depends onthe ratioof the exitto thethroat area, still results

insmooth flowthrough thenozzle,butthisisnowover-expanded andiscompressedto theoutsidepressurethroughan

obliqueshockattachedtothenozzleexit.When pt

/

pisequaltothethresholdvalue,theflowischaracterized byanormal

shocklocatedatthenozzleexit:upstream oftheshock, theflowissmooth,andverifiesadaptationconditionsintheexit section;immediatelydownstreamofit,theflowissubsonicandmatchestheoutsidepressure.Finally,whenpt

/

pisbelow

thethresholdvalue,anormalshockwaveisformedsomewhereinsidethenozzle.Thisresultsinsubsonicflowattheexit, andanexitpressurethatisequaltop[2].

Giventhepressureratio,theflowiscompletelycharacterized bytheshapeofthenozzle[2].Asin[25],weconsiderthe followinghyperbolictangentforthenozzleshape

f

(

x

)

=

a

+

btanh

(

cx

−

d

) .

(44)

To test the SSC method, we specify two different ranges for the uncertain parameters such that two radically different

responsesurfaceshavetobecreated.First,weprescribeawiderrangeforpsuchthattheQoIishighlydiscontinuous,see Fig. 9.InthesecondcasewerestrictptoamorenarrowintervalsuchthattheQoIissmooth.Morespecifically,weprescribe

the uniform input distributions forthe 6 uncertain parameters described in Table 2.Furthermore, we choose Mout (the

Table 2

Uncertaininputparametersofthediscontinuous(D)andsmooth(S)case.

d Parameter Mean (D) Range (D) Mean (S) Range (S) 1 p[bar] 0.55 [0.5, 0.6] 0.625 [0.60, 0.65] 2 pt[bar] 1.0 [0.9, 1.1] 1.0 [0.9, 1.1] 3 a[–] 1.75 [1.575, 1.925] 1.75 [1.575, 1.925] 4 b[–] 0.7 [0.63, 0.77] 0.7 [0.63, 0.77] 5 c[–] 0.8 [0.72, 0.88] 0.8 [0.72, 0.88] 6 d[–] 4.0 [3.6, 4.4] 4.0 [3.6, 4.4] Table 3

ThecomputationalcostofthediscontinuousQoI.

Type d[–] ns[–] T[min] LEC [%T] Sj[%T] v[%T] Baseline 2 50 0.56 3.56 3.16 87.3 3 100 2.09 24.39 11.46 39.32 4 150 10.95 73.42 15.37 6.22 5 200 119.29 85.21 11.26 0.58 SCC-SC 2 50 0.54 1.45 1.24 82.46 3 100 1.33 1.2 2.33 54.75 4 150 1.37 5.56 12.34 42.99 5 200 4.75 4.88 17.2 11.47 Table 4

ThecomputationalcostofthesmoothQoI.

Type d[–] ns[–] T[min] LEC [%T] Sj[%T] v[%T] Baseline 2 50 0.73 2.28 2.87 89.9 3 100 2.52 20.37 16.42 42.07 4 150 22.86 62.18 30.87 3.95 5 200 731.5 58.31 40.99 0.13 SCC-SC 2 50 0.7 1.28 0.31 85.64 3 100 1.65 4.0 0.43 61.14 4 150 1.63 16.76 1.26 49.01 5 200 4.68 13.62 1.41 15.45

isentropicrelationsonce Mout isknown[2].Whenconstructingthesurrogatemodels,wewillusealineartransformation

for each input to map points from

[

0,1

]

in the stochastic domain to points in the physical domain with the range as

specifiedinTable 2.Thissimplifiestheconstructionofthesurrogatemodelsasitallowsustoalwaysworkinthestandard

d-dimensionalhypercubeKd.

Fornow, wewillconsiderjustthefirst5uncertainparametersofTable 2.InTables 3 and4weshowthecomputation

time T in minutes versus the dimension d,in case ofthe discontinuous and smooth QoI for both the baseline andthe

methodbased on SCENOstencils. Thisisof course dependentupon the available computational resources, inour casea

24coreworkstation.We usethesecoresto parallelize theLEC condition,code samplingandENO stencils.Ouralgorithm

forthe construction oftheSCENO stencilsis not implementedin parallel,anduses just1 core.We can seethat T rises veryquicklyasdincreasesinthecaseofthebaseline method,especiallyinthecaseofthesmoothQoI.Toexplainwhich elementisresponsibleforthehighcomputationtime,wealsoshowthepercentageofT thatisspentontheLECcondition, constructionofthestencils Sj,andQoIcalculation.

Sincethenozzlecodeisjustacheap test problem,Tables 3 and4show thatcomputingtheQoIsamples vonlytakes

up a significant portion of T forlow d. Forthe baseline SSC method the construction ofENO-type stencils makes up a

significant partofthecomputationalcost, butthe enforcementoftheLECcondition isthe mostexpensivecomponentin

higherdimensions.Thus,forthebaselinemethod,mostofthecomputationaleffortisputintoenforcingtheLECcondition. ForthatreasonthecomputationalcostoftheLECconditionisinvestigatedinmoredetail.

As explained in Section 2.1, the LEC condition (14) is enforced by a MC approach, for all simplices in Sj at all j

=

1,

· · ·

,

ne. Thus,forthe baseline SSCmethodthe numberoftimes thesurrogate modelisevaluated in each iterationi is

boundedby

nwi

=

ne

×

ne,Sj

,

i

=

1

,

· · ·

,

I (45)

wherene,Sj isthe numberofsimplicesina singlestencil Sj with p

=

pmax,andI is thetotalnumberorofiterationsof

theSSCalgorithm.Hereweassumedthatper Sj,onesampleisplacedineachsimplexusing(25).Thenumberofpointsin

theDelaunaygridisgivensimplyby(5),butestimatingne forarbitrarydisnottrivial.Theworst-casenumberofsimplices

inaDelaunaytriangulationisboundedbytheso-calledUpper-Bound theorem,whichstatesthatne isatmostof

O

(

nds/2

).

Fig. 10.neasfunctionofnsforthesmoothQoI.ThediscontinuousQoIgivesasimilarfigure.Theslopedne/dnsiscomputedviaaleast-squareregression line.

Fig. 11.Examples of the exponential growth of SSC components.

a constantfactorthatisexponentialwiththedimension[1].Tofindoutwhereinbetweenthesetwoboundsourspecific

problemresides,weplotne versusnsinFig. 10ford

∈ {

3,4,5

}

.Theseresultsindicatethatweareclosetothe

O

(

ns

)

bound,

since thene

(

ns

)

aredescribedquitewell bythelinearregressionalsoshowninFig. 10.However, theexponentialincrease

ofdne

/d

nsmeansthatforamoderatenumberofsampleswecanstillhavealargenumberofsimplicesifdishighenough.

Notethatotherthanlimitingthenumberofsamplesns,wehavenomeansofcontrollingthemagnitudeofne.

The termne,Sj in(45)grows exponentiallywith pj foragivend. Thiscanbe seen inFig. 11(a), wherewe plotne,Sj

versus the localpolynomial order pj ford

=

5.Unlike ne, we obviously havesome control over the magnitudeof ne,Sj

through theinclusionofa maximumallowablecutoff valuefor pj.Notehoweverthatlimiting pj willaffecttheorderof

convergence(12).Theupperbound(45),addedoveriterationsi isplottedasafunctionofns inFig. 11(b)ford

=

2,

· · ·

,

5.

Itshowsarapidincreasewithbothdandpj.

Bycomparingthecomputationaltime T oftheSSCmethodwiththatoftheSSC-SCmethod(Tables 3–4),itisclearthat

theSSC-SCmethodisseveralordersofmagnitudemoreefficientforthedimensionsconsideredinthisexample.Toclearly

showwhytheSSC-SCmethodiscomputationallymoreefficientthanthebaselinemethod,considerFig. 12.Herewedisplay

thefractionofthevolume

¯

thatiscoveredbySCENOstencilsofdifferentpolynomialorder pj,forthediscontinuousand

smooth casewithd

=

5.Alsothenumberofstencils

|

Sj

|

per orderisshown. Notethat forthediscontinuous QoI,just7

high-orderSCENOstencils(stencilswithpj

>

1)alreadycover76.1%ofthedomain.Inthecaseofthebaselinemethodthe

numberofstencils(andtherebyLECiterations)equalsne,whichis11 034inthisexample.ForthesmoothQoI(Fig. 12(b))

we required13fourth-order stencilstocoverthe entiredomain.Withthebaseline methodwe wouldhavea setof9451

Fig. 12.The volume coverage of SCENO stencils per polynomial order.

Table 5

Therelativeerrors(46)ofthediscontinuousQoIforthebaselineSSCandSSC-SCmethod.

Type d ns μ σ w

Baseline 2 50 3.536e−02 3.669e−02 2.001e−01 3 100 5.271e−02 7.475e−02 2.408e−01 4 150 2.532e−02 1.425e−01 2.828e−01 5 200 2.006e−02 2.320e−01 3.253e−01 SCC-SC 2 50 1.590e−02 4.397e−02 1.597e−01 3 100 3.975e−03 7.452e−02 2.108e−01 4 150 1.199e−03 1.329e−01 2.547e−01 5 200 3.368e−03 1.803e−01 2.876e−01

Table 6

Therelativeerrors(46)ofthecontinuousQoIforthebaselineSSCandSSC-SCmethod.

Type d ns μ σ w

Baseline 2 50 1.131e−06 5.137e−06 1.088e−05 3 100 7.276e−07 1.572e−05 1.416e−05 4 150 1.015e−06 3.784e−06 2.566e−05 5 200 2.446e−05 3.376e−04 2.228e−03 SCC-SC 2 50 8.042e−07 1.952e−05 1.916e−05 3 100 8.734e−07 3.019e−07 7.075e−06 4 150 1.006e−06 2.234e−06 2.104e−05 5 200 3.034e−06 8.186e−06 8.218e−05

AsstatedinSection2.1,ourprimaryinterestiscomputingthestatisticalmomentsoftheQoI,inparticularthemeanand standarddeviation.ToassesstheaccuracyoftheSSCmethodweusedareferencesolutionforeachconsidereddimensiond. TocomputetheerrorswedefinethefollowingrelativeL2errormeasuresforthemean,standarddeviationandinterpolation surface

μ

=

μ

w

−

μ

ref

2

μ

ref

2

,

σ

=

σ

w

−

σ

ref

2

μ

ref

2

,

w

=

w

(

ξ

_ref

)

−

vref

2

vref

2

.

(46)

Here, the subscript w denotes a quantity computed withthe surrogate model,and ref isthe exact value computedvia

randomsampling. Intheinterpolation surface error,vref is avector containing 104 MCcode samplesandw

(

ξ

ref

)

arethe

surrogate model outputs evaluated at the same MC locations

ξ

ref.The values of the errormeasures (46) for both QoIs

andboth surrogatemodelscanbefoundinTables 5–6.Notethat theerrorlevelsare roughlythesameforbothsurrogate models.

FromTables 5–6wenotethattheerrorsofthediscontinuouscaseareconsiderablehigherthanforthesmoothcase.This can beattributedto thesmearing ofdiscontinuities, i.e.thelinear interpolationofa discontinuityover a simplex,which especiallycontributestotheerrorofthesurrogatemodelinhigherdimensions. SeeforinstanceFig. 13,whichdepicts2D projectionsofa3D surrogatemodelalongwithreferencedataonan ordereduniformgrid.EspeciallyinFig. 13(a)wecan clearlyidentifyregionswherethesmearingofthediscontinuitycontributestotheerror.Forthisparticularcase,weplotted

Fig. 13.3Dsurrogatemodeldisplayedin2dimensionsbyfixing1dimensiontoaparticularvalue.Thegreendotsarereferencedataonanordereduniform grid.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

thedifferencebetweenthesurrogatemodelandthereferencedatainFig. 13(b),whichalsoidentifiessharpregionsofhigh error.Thissituationgetsprogressivelyworseasdincreases.

In[40,36]Witteveenetal.apply asub-cellresolutionapproachtotheSSCmethodforthecasewhenthediscontinuity intheprobabilisticspaceisafunctionofaphysicaldiscontinuitywithrandomlocation.Ourresultsindicatethatforhighd

sub-cellresolutioncouldprovetobebeneficial,evenifthe