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_{Arts et Métiers ParisTech, DynFluid laboratory, 151 Boulevard de l’Hopital, 75013 Paris, France} b_{Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 2, Delft, The Netherlands}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history:

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-mail addresses:[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 inSection5wepresenttheobtainedresultsandthediscussion.Finally,wegiveourconclusionsinSection6.

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

isavectorofd randominputparameters

ξ (

ω)

= (ξ

1

(

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 x orotherdeterministic 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 referstoa certainsimplex element.Furthermore,l

=

0

,

· · · ,

Nj isa localindexusedtocountthenumberofsamplesfromv involved 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 onthenearest-neighbor principle[38].Inthiscasethefirstd

+

1 pointsarethevertices

{ξ

_kj,0

,

· · · ,

ξ

_kj,d

}

ofthesimplex



j,

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

whichwouldsufficefor pj

=

1.Forhigherdegreeinterpolation,neighboring points

ξ

k areaddedbasedontheir proximity tothecenterofsimplex



j,i.e.basedontheirrankingaccordingto

ξk

− ξ

center,j2

,

(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

− ξk

j,0

ξ

kj,2

− ξk

j,0

· · · ξ

kj,d+1

− ξk

j,0



∈ R

d×d

_.

₍₁₁₎

Finally,theorderofconvergenceOjisgivenby[37]

Oj

=

pj

+

1

d

.

(12)

The simplexwiththehigheste

¯

j isselectedforrefinement.Toensureagoodspreadofthesample points,only randomly-selectedpointsinsideasimplexsub-element



subj areallowed.Theverticesofthissub-elementaredefinedas

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 )

=

min vj

∧

max ξ∈j wj

(ξ )

=

max vj

,

(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-lectedsimplicesiseitherrepeatedforamaximumofI iterations,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 rootmeansquare(RMS)errornorm

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

¯

i wehave

|¯

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

ξ

1,l

ξ

2,l

ξ

₁2_,l

ξ

1,l

ξ

2,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

+

1 vertices

ξ

₀

,

· · · ,

ξ

_N

∈ R

d_{arepolynomiallypoisedifftheyarenotasubsetofanyalgebraichypersurfaceof}

degree

≤

p.

Analgebraichypersurfacein

R

d isad

−

1 dimensionalsurfaceembeddedinad-dimensionalspaceconstrainedtosatisfy anequation f

(ξ

1

,

· · · ,

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_{,thereexistsp distincthyperplanesG}

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

+

1 verticesin

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

[ξ

0

,

ξ

1

]

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

/

dx

=

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)for d=2,3 and 1000 samples.

Md

= ξ

0

+

d



i=1 i



j=1 r 1 d−j+1 d−j+1

(ξ

i

− ξi

−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 expensiveforhighd and 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 a consequenceoftheexponentialnatureofnw (seeSection5.1.1), wealsoseeanexponentialincrease inthecomputation

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 informationcontainedinv regardingthediscontinuitylocationtocreateasmallsetofSC stencilsthatalsoresembleENO stencils,i.e.whichdonotcrossadiscontinuity.WewilldenotethesestencilsasSCENOstencils.Althoughmoresophisticated approachesareavailable [40],forreasonsofsimplicityweidentifythe



j throughwhichthediscontinuityrunsbysimply imposingathresholdvt onthemaximumjumpobservedinv ateachsimplex.Then,thesetofdiscontinuoussimplicescan bedefinedas

D

= {j

| |

max vkj,l

−

min vkj,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

, i.e.weaddalldiscontinuoussimplicestothesetofcoveredsimplices.Next,wespecifytheinitialsimplex



∗_jasthesimplex

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 the rangeofdimensions wherea Delaunay triangulationcan bemade. Forproblems ofhigherdimensionalitya different

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

Table 1

Examplesofill-conditionedsystems.Weshowthedimension

d,

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,wherex isapossiblephysicalcoordinatewhich 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

,

· · · , ξi

l

)

+ · · · +

f1···d

(ξ

i1

,

· · · , ξi

d

).

(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 numberofvariablesactingtogetherontheoutput.Ifhigh-ordercorrelationsareweak,thephysicalsystem f

(ξ )

canbe

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

,

· · · , ξd

)

=

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

M f

(ξ )

:=



Kd f

(ξ )d

μ

,

M(i1···il)_f

_{(ξ )}

:=



Kd−l f

(ξ )

⎡

⎣



j∈{/i1···il} d

μ

j

(ξ

j

)

⎤

⎦ .

(30)

Then,viaafamilyofprojectionoperators Pi1···il

:

Kd

→

Kl,thecomponentfunctionsarerecursivelydefinedasfollows[26]: f0

:=

P0f

(ξ )

=

M f

(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

,

· · · , ξi

l

)

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

=

d

ξ

1d

ξ

2

· · ·

d

ξ

d) 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

,

· · · ,

ηd

)

.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

,

· · · , ξi

l

)

−



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

ηi

,except inputs

ξ

i1

,

· · · ,

ξ

il.As an example,consider theunivariate function f

(i)

_(ξ

i

)

=

f

(

η

1

,

· · · ,

ηi

−1

,

ξ

i

,

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)

(ξ

1

,

ξ

4

,

ξ

6

)

.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 cjlE





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

=



Ji2



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

αi

>



1.Theonlyhigher-ordercomponentfunction thatwillbecomputedinthiscaseis f12

(ξ

1

,

ξ

2

)

,regardlessofthevalueofd.

Thedownsideof(38)isthatitishardtochoose



1 beforehand.Oneshouldfirstcreatethefirst-orderHDMRexpansion anddecideonan appropriatevalueaposteriori.Analternativeistouseaweightmeasuringtherelativecontributionof Ji withrespecttothesumofallfirst-ordermeans,i.e.

α

i

=



Ji2



d

k=1



Jk2

.

(39)

Nowonecanapriori choosea



1

∈

[0

,

1],andselectthesmallestsetofimportantdimensionsforwhichthesumoftheir

αi

isgreaterthan



1.

Dimensionadaptivityisextendedtohigherdimensionsaswellbydefiningaweightfor

|

u

|

>

1 as[20]

α

u

=



Ju



2





v∈Vcomp,|v|<|u−1| Jv



2

.

(40)

Here,theset

V

compsimplyholdsalltheindicesv thatwere computed.Furthermore,allsubsetsv ofcomponentfunctions whichare importantare addedto aset

V

imp. Thatway,ahigher-orderimportant u isadmissible ifallv

⊂

u requiredto compute(35)arealsoin

V

imp.Thisistheso-calledadmissibilitycondition,whichisgivenby

Fig. 9. Moutas function of p and ptobtained 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

α

usumtooneandwecanchoosea



1

∈ [

0

,

1

]

apriori.

Finally,arelativeerrormeasurebetweentwoHDMRexpansionsofconsecutiveorders p

−

1 andp isdefinedas

α

p

=





_|u|≤p Ju

−



_|u|≤p−1Ju



2





_|u|≤p−1Ju



2

.

(43)

Thealgorithmstopswhen

αp

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 p ofthesurroundingsatthenozzle 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

/

p willresultinsmoothflowthroughthenozzle,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

/

p isequaltothethresholdvalue,theflowischaracterized byanormal shocklocatedatthenozzleexit:upstream oftheshock, theflowissmooth,andverifiesadaptationconditionsintheexit section;immediatelydownstreamofit,theflowissubsonicandmatchestheoutsidepressure.Finally,whenpt

/

p isbelow thethresholdvalue,anormalshockwaveisformedsomewhereinsidethenozzle.Thisresultsinsubsonicflowattheexit, andanexitpressurethatisequaltop[2].

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

f

(x)

=

a

+

b tanh

(cx

−

d) . (44)

To test the SSC method, we specify two different ranges for the uncertain parameters such that two radically different responsesurfaceshavetobecreated.First,weprescribeawiderrangeforp suchthattheQoIishighlydiscontinuous,see

Fig. 9.Inthesecondcasewerestrictp toamorenarrowintervalsuchthattheQoIissmooth.Morespecifically,weprescribe

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

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

veryquicklyasd increasesinthecaseofthebaseline method,especiallyinthecaseofthesmoothQoI.Toexplainwhich elementisresponsibleforthehighcomputationtime,wealsoshowthepercentageofT thatisspentontheLECcondition, constructionofthestencils Sj,andQoIcalculation.

Sincethenozzlecodeisjustacheap test problem,Tables 3 and4show thatcomputingtheQoIsamples v onlytakes up a significant portion of T for low 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 forarbitraryd isnottrivial.Theworst-casenumberofsimplices inaDelaunaytriangulationisboundedbytheso-calledUpper-Bound theorem,whichstatesthatne isatmostof

O(

nds/2

)

. Inthebest-casescenario(pointsdistributeduniformlyatrandominsidetheunitsphere)ne scalesas

O(

ns

)

foranyd,with