Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

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Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

Space-time

multilevel

Monte

Carlo

methods

and

their

application

to

cardiac

electrophysiology

S. Ben Bader

a

,

∗

,

P. Benedusi

a

,

A. Quaglino

a

,

b

,

P. Zulian

a

,

R. Krause

a

a_Center_for_{Computational}_Medicine_in_Cardiology,_Institute_of_{Computational}_Science,_Università_della_Svizzera_italiana,_Lugano,_Switzerland b_NNAISENSE_SA,_Lugano,_Switzerland

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline5February2021 Keywords:

UncertaintyQuantiﬁcation Multilevelmethods

Space-timeﬁniteelements(3D+1) Cardiacelectrophysiology Monodomainequation

We present anovel approach aimedat high-performanceuncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loève expansion and compute statisticsofhigh-dimensionalquantities-of-interest,suchasthecardiacactivationpotential. OurmethodologyreliesonacloseintegrationofmultilevelMonteCarlomethods,parallel iterativesolvers,and aspace-timediscretization.Thiscombinationallowsforspace-time adaptivity,time-changingdomains,andtotakeadvantageofpastsamplestoinitializethe space-timesolution.Theresultingsequenceofproblemsisdistributedusingamultilevel parallelizationstrategy,allocatingbatchesofsampleshavingdifferentsizestoadifferent numberofprocessors.Weassesstheperformanceoftheproposedframeworkbyshowing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology.Specifically,westudythe effectofspatially-correlatedperturbationsof theheartfibers’conductivitiesonthemeanandvarianceoftheresultingactivationmap. As shownby the experiments,thetheoretical rates ofconvergence ofmultilevel Monte Carloare achieved. Moreover,thetotalcomputational work foraprescribed accuracy is reducedbyanorderofmagnitudewithrespecttostandardMonteCarlomethods.

1. Introduction

Manyphenomenainscience,engineering,andmedicinecanbemodeledasinitialboundaryvalueproblemsforan un-known function.Theirnumericalsolutionisa computationallydemanding task.Inadditionto that,the inputdataorthe parameters ofthesemodelsmightnot be knownexactly,i.e., they aresubjecttoa possiblylarge uncertainty.Inorderto guarantee that the outcome of simulations can be relied upon, it istherefore imperative to quantify the effects ofsuch input uncertaintiesontheoutput quantities-of-interest.Thisprocessiscalledforwardpropagationandispartoftheﬁeld ofUncertaintyQuantiﬁcation(UQ).Thiscanbeachievedvia,e.g.sampling methodsbysolving,numeroustimes,a computa-tionallyexpensiveproblem.Thissettingrequiresthatoneshouldnotonlyrelyonhugecomputationalresources(e.g.parallel implementationonsupercomputers),butalsoonmathematicalstate-of-the-art techniquesforreducing thecomputational load.

Inthe caseofcardiacelectrophysiology,it hasbeenwell establishedthat theheart muscle-ﬁbersplay amajor role in the electrical conductivity.From CTand MRIimaging, the ﬁbersorientations aswell asthe conductivitiesthey carry are

*

Correspondingauthor.

E-mailaddress:seif.ben.bader@usi.ch(S. Ben Bader). https://doi.org/10.1016/j.jcp.2021.110164

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extremelyhardtodetermineprecisely.Moreover,thesecanvaryfromonepatienttoanother,leadingtohugeuncertainties if standard orientations or conductivities are used in patient-specific simulations. In this work, we focus only on fiber conductivities andmodelit asaspatially-correlated randomperturbation applied tothediffusion field.In thiscontext,a single patient-tailored simulationcan takeseveralhours on alarge cluster, forexamplewhen themonodomainmodelis employed [26].It followsthat,e.g.,UQ forsuchmodelsis currentlyunfeasible, orextremelyinefficient, withplainMonte Carlomethods.

A standardapproach toovercome thesedifficultiesis touse modelreduction strategies,such asreducedorsurrogate models [7,19].Theseapproximationsarebuild upontheobservationthat inmanyapplicationsitisclearly possibleto dis-tinguishbetweenoffline andonline phasesoftheworkflow,wheretheformeristypicallyveryexpensiveandencompasses everything thatcan beprecomputedinadvance,e.g.,training asurrogatemodelbeforeanypatient-specificdatabecomes available, whilethelatterisverycheap andconsistsonly ofmodelevaluations.Unfortunately, thisapproach hastwo sig-nificantdrawbacks.Ontheonehand,complexerrorestimatesmustbeprovidedtoensurethattheapproximationerroris within acceptable bounds.On theother hand,theoffline training shouldbe performedonlyonce,which isunrealisticin caseswhere,e.g.,thegeometryisnot knowna priori.Alternativetothat,physics-basedreduction strategieswhichbypass theprecomputationarealsopossible.Inthecaseofelectrophysiology,theactivationtimecanbecomputedviatheeikonal equation,whichprovidesaphysiologically-meaningfulsolution [31],stronglylinkedtothebidomainequation [6].

Amorerecentideaistoallowreducedmodelstobeinaccurate,inthesense ofnotprovidinganapproximationwithin certainerrorbounds.Accuracyisreplacedbycorrelation [29] orevenonlystatisticaldependence [22] withthehigh-fidelity model. This approach has a twofold advantage. Onthe one hand,it is the statistical dependence, ratherthan the error bounds of coarse models based on coarse meshes,that is crucial to ensure that propagating uncertainties via the low-fidelitymodelsprovideuseful informationonthestatisticsofthehigh-fidelityquantity-of-interest.Ontheotherhand, low-fidelitymodelsare notrestrictedtolow-resolutiongeometries,thereforesolvingthemcanbeseveralordersofmagnitude faster than a coarse model. However, some computational effort (

∼

100 runs of the high-ﬁdelity model) is needed for estimatingthestatisticaldependencebetweenthemodels.Moreover,thereisnoestablished approachfortreating vector-valued quantities-of-interest [28], whichmay requirea signiﬁcantly differentnumberof samplesofeach model foreach vectorcomponent.Anapproachbasedonminimizinganormoftheerrorwasrecentlyproposedin[34].

Inthespecificcontextofelectrophysiology,theuseofBayesianmultifidelitymethodshasbeenadvocatedin [33],where a Bayesian regression mappingthe low- and high-fidelity outputs iscreated. The Bayesiannature of thisapproach error automaticallyaugmentstheestimate ofthesolutionstatisticswithfullprobabilitydistributionsandcredibleintervals, ob-tainedindependentlyofthedegree ofstatisticaldependencebetweenmodelsandthenumberofsamplesemployed.This makesitaverysuitable approachforscenarioswhereresourcesarescarce,suchasclinicalapplications.However,this ap-proachislimitedtoscalarorlow-dimensionalquantities-of-interest.Moreover,itdoesnotyieldasymptoticconvergenceto point estimatesasthe numberofsamplestendstoinfinity, soit isunsuitableforscenarioswhereaccurate estimatesare desired.

All ofthe above methodologiesare characterized by a clearsegregationamong sampling, models,discretizations, and solutionmethods.Here,wetake theoppositerouteandintertwinethedifferentcomponents.Thecentral pointofour ap-proach is to consider the multilevelMonte Carlo method [13], which extends the idea ofcontrol variables [9]. The key idea istoperform mostofthe simulationson asequenceoflow-resolution models.Similarly tothecaseofmultifidelity, thedirectuseofthehighest-resolutionlevelguaranteesconvergence,whileasignificantportionofthecomputationalload is offsetto thelow-resolution hierarchy.However, in thiscase, nooffline preprocessingis neededtofind thecorrelation acrossmodelsandvector-valuedoutputsarenaturally handledbytheframework.Inordertoallowforarbitrarycoarsening in spaceandtime, we further rely on a space-timediscretization ofthe monodomain equation. Bydoing so, italso be-comespossibletousethealreadycomputedsamplesfortheinitializationofNewton’smethod,whichsolvesthespace-time equations,therebysignificantlyreducingtheamountofiterationsrequiredforconvergence.

Theoutlineofthepaperisasfollows:Section2introducestheelectrophysiologyequationsandtheuncertaintyrelated totheﬁberconductivities,Section

3

detailstheproposedmethodusingtheheatequationasamodelproblem,andSection4

presentstheresultsfromthenumericalexperiments.

2. Modeling theuncertaintyintheﬁberconductivities 2.1. Thedeterministicmonodomainequation

The heartassumesa pumpingfunctionthat istheresultofa verycomplexcontractionandrelaxationcycleoccurring in thecardiac cells. Thisprocess is controlled bya non-trivialpattern ofelectrical activationwhich isat thecoreof the heartfunction.Indeed,severalheartdiseasesarecloselyrelatedtodisturbancesoftheelectricalactivity.Asaconsequence, understandingandmodeling thisactivityisimportantforabetterclinicaldiagnosisandtreatmentofpatients.

Thepropagationoftheelectricalpotentialinsidethecardiacmuscleisinitiatedinthesinoatrialnode,thatisontopof theleft andrightatriums.Itgeneratesa stimulusthat givesriseto atraveling wave throughtheheart.Its cells havethe abilitytorespondactivelytothiselectricalstimulusthroughvoltage-gatedionchannels.

Startingfromtheﬁrstmodelsforionicchannels[20],alargefamilyofmodelshasbeendevelopedbynow,describingthe electrical activityatthesubcellularlevel.Combinedwithdiffusioninspace,theso-calledmonodomainequation hasbeen

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derived[21,25] fortheelectricalpropagationinthecardiactissue.Themonodomainequationiswrittenasanon-stationary reaction-diffusionequation

∂

u

(

x

,

t

)

∂

t

− ∇ · (

G

(

x

)

∇

u

(

x

,

t

))

+

Iion

(

u

(

x

,

t

))

=

Iapp

(

x

,

t

),

∀(

x

,

t

)

∈

D

× (

0

,

T

],

G

(

x

)

∇

u

(

x

,

t

)

·

n

(

x

)

=

0

,

∀(

x

,

t

)

∈ ∂

D

× (

0

,

T

],

(1) u

(

x

,

0

)

=

0

,

∀

x

∈

D

,

where u

=

u

(

x

,

t

)

is the electrical potential, D

⊂ R

d a domain representing the heart, T

∈ R

+ the endtime, Iapp

:

D

×

[

0

,

T

]

→ R

an appliedstimulusmodelingthe sinoatrialnodeactivation,and Iion

: R

→ R

an ionchannelmodel.We rely hereontheFitz-HughNagumo(FHN)model[10],forwhichwehave:

Iion

(

u

)

:=

α

(

u

−

urest

)(

u

−

uth

)(

u

−

upeak

).

(2)

The values urest, uth and upeak are characteristicpotential values representing theaction potential behavior that a heart cellgoesthroughduring itsactivation. Thesearerespectivelytherestingpotential urest (cellisunactivated),thethreshold potentialuth(cellistriggered)andthepeakvalueupeak(cellisactivated).Here,

α

isanon-negativescalingparameter.

Finally,G in(1) representstheconductivitytensormodelingthefibers,alongwhichelectricalpotentialpropagationtakes place,allovertheheartmuscle.Thetensor G playsamajorroleinourmodelasitmayrepresentasourceofuncertainty withregardstothefibersorientationsandconductivitiesitmodels.We hereassume that G isa scalarisotropicdiffusion fieldandfocusexclusivelyonthefiberconductivities.

2.2. Thestochasticmonodomainequation

The mathematicalandnumericalinstruments developedindecades ofresearch intheelectrophysiology ﬁeld allow in principle forvirtual therapyplanning. Themonodomainequation isby nowanestablished modelincardiac electrophysi-ology,describingwithhighaccuracytheelectricalactivityinthemyocardium.Nonetheless,patient-speciﬁcsimulationsare stillnotemployedasaroutinetoolinthetreatmentofpatients.

Aparticularreasonforthiscan befoundinthedatawhich isacquiredinclinical practice.Forinstance,theﬁber con-ductivitiesstillrepresentagreatchallengetobedeterminedfromanimagingpointofaview.Oneshouldthereforeaccount forpossibleuncertaintiesofthediffusionﬁeldG in(1).

Modelingtheuncertaintyrequirestoaccountforanadditionalstochasticvariableintheformulationofthemonodomain equation. Letus denote this stochastic variable with

ω

∈

where

is the set ofall possible outcomes, which in our applicationwouldrepresentthesetofalldiffusionﬁeldsmodeled bytheuncertainty.Wearenowinterestedinestimating thestatistics ofu

(

x

,

t

,

ω

)

,i.e.

E

[

u

(

x

,

t

)

]

=

u

(

x

,

t

,

ω

)

d

P (

ω

)

,asasolutiontothemonodomainstochastic PDE,forwhich themainequationreads:foralmostevery

ω

∈

:

∂

u

(

·,

ω

)

∂

t

− ∇ · (

G

(

·,

ω

)

∇

u

(

·,

ω

))

+

Iion

(

u

(

·,

ω

))

=

Iapp

(

·),

in D

× (

0

,

T

],

(3)

where we omittedthe spaceandtime variables

(

x

,

t

)

forthe sake of readability. Forevaluating

E

_[

u

(

x

,

t

)

]

given(3), we relyonMonte-CarloandMultilevelMonte-CarlotechniquesasdescribedinSection3.Thesemethodsimplyadeterministic reformulationof(3), andsolvingthemonodomainequation fordifferentindependentandidenticallydistributedsamples. Naturally,parallelizationplaysamajorroleintheperformanceofbothmethods,giventheimportantnumberofsamplesto consider.

3. Space-timemultilevelMonteCarlo 3.1. MultilevelMonteCarlo

Asintroducedintheprevioussection,weareinterestedinequationsthataresubjecttouncertainorunreliabledatain theaimofevaluatingtheinformationderivingfromtheprocessofsimulations.Thisconcept hadbeensummarizedunder theterminologyofuncertainty quantiﬁcation (UQ)andisconcerned aboutextractingrobustandqualitativeinformation de-spitethepresenceofthisvariability.Inthenextsubsection,webrieﬂydetailtheproblemweintendtosolve,andintroduce thereadertothenotationwerelyon.

3.1.1. ProblemsettingandUncertaintymodeling

From a mathematical point ofview, theunderlying taskresidesin computingthe solution ofa PDE withcoeﬃcients dependingonarandominputbelongingtoastochasticspace.Fortheproblem(3) tobewell-deﬁned,weneedtointroduce thefollowingsetting.

(4)

Fig. 1. Exampleofrandomﬁeldsforcubeandventriclegeometries.(Forinterpretationofthecolorsintheﬁgure(s),thereaderisreferredtothewebversion ofthisarticle.)

Let

(,

,

P )

be acompleteandseparableprobabilityspace, with

⊂

2 _a

_σ

_-ﬁeld_and

_P

_a _probability_measure._We considerabounded,LipschitzdomainD

⊂ R

d _with_d

₌

₁

_,

₂

_,

_3._We_also_require_the_Bochner_space

L2

(, ,

P;

B

)

:=

v

: →

B

|

v strongly measurable such that

v

_L2₍_;_B₎

<

∞

,

where B

=

L2

(

[

0

,

T

];

H1

(

D

))

isitselfaBochnerspace(

[

0

,

T

]

denotesthetimedomain)and

v

_L2₍_;_B₎

:=

v

(

·,

ω

)

2_Bd

P (

ω

)

1/2

.

(4)

ThenormontheBochnerspaceB

=

L2

₍

_[

₀

_,

_T

_];

_H1

₍

_D

₎₎

_is_in_turn_deﬁned_as

v

(

·,

ω

)

B

:=

[0,T]

D v

(

x

,

t

,

ω

)

2dxdt

1/2

.

(5)

Forthe sakeofbrevity, we willalways referto L2

(,

,

P

;

B

)

with L2

(

;

B

)

. Havingeverythingathand,we introducea simpliﬁedequationmodelof(3),givenby

ﬁndu

˜

∈

L2

₍

_;

_L2

₍

_[

₀

_,

_T

_];

_H1

₍

_D

₎₎₎

_such_that_for_almost_every

_ω

_∈

_,

∂

tu

˜

(

X

,

ω

)

−

div

(

G

(

X

,

ω

)

∇ ˜

u

(

X

,

ω

))

=

f

(

X

)

in D

× [

0

,

T

],

(6) where X

= (

x

,

t

)

andG

(

X

,

ω

)

isthediffusionﬁeldsubjecttotheuncertainty

ω

∈

,thatcanbeinterpretedasaparticular diffusionﬁeldfromallthepossibleonesweallowfortheproblemfor.Noticethatiftheforcing function f wasdepending onu

˜

(

X

)

andwrittenas f

(

X

)

=

Iapp

(

X

)

−

Iion

(

u

˜

(

X

,

ω

))

,werecoverthestochasticmonodomainequation(3).Wewill how-everlateronforerrorestimates(cf.subsection

3.1.2

)relyonaclassicalforcingfunction f

(

X

)

,independentfromu,

˜

treating thereforetheconvergenceratesforthequadraturemethodsonthestochasticheatequation.ThePDEfurtherencodes suit-ableboundaryandinitialconditions.Weareinterestedincomputingthestatisticsofthesolutionu tothestochasticPDE (6),i.e.

E

[˜

u

](

X

)

=

u

˜

(

X

,

ω

)

d

P (

ω

)

∈

L2

(

[

0

,

T

];

H1

(

D

))

.

Morespeciﬁcally,every

ω

∈

expressesadifferentdiffusionﬁeld G

(

X

,

ω

)

.Inourparticularcase,itismoreconvenient tothinkofthesedifferentdiffusionﬁeldsasasuperpositionofameandiffusionﬁeldG0

(

X

)

andarandomperturbationall overthedomain D (randomﬁeld).

Wefurthermoreassumethediffusionuncertaintiestobespatially correlated,sinceneighboringregionsare morelikely tohavethesameconductivityinahealthyheart.Wethereforetalkaboutspatiallycorrelatedrandomﬁelds[17].Examples ofrandomﬁeldsofcubeandidealizedventriclegeometriescanbefoundinFig.1.

The followingareobtainedby expressingthestochastic diffusioncoeﬃcient G

:

D

×

→ R

asa truncated Karhunen-Loève(KL)expansion [33] G

(

x

;

ω

)

=

G0

(

x

)

+

s M

k=1

λ

k

φ

k

(

x

)ψ

k

(

ω

) ,

(7)

where

λ

kand

φ

k

(

x

)

arerespectivelyeigenvaluesandeigenvectorsofthecovariancematrixinducedbyaspatialcorrelation kernelk. Regarding theKLtruncation,thestochastic dimension M is obtainedbylow rankapproximating thecovariance matrix withpivoted Choleskydecomposition [15,17].The

{ψ

k

(

ω

)

}

Mk=1 are stochastically uncorrelatedrandom variables of

(5)

thejointdensityfunction

ρ

(

y

)

=

M_i₌₁

ρ

k

(

yk

)

.Weusehereauniformdistribution,i.e.

ψ

i

∼

U(−

1

,

1

)

.Wearethereforeable toidentifythestochasticspace

withitsimage

[−

1

,

1

]

M _using_the_mapping

ψ

: → [−

1

,

1

]

M

,

(8)

ω

→ (ψ

1

(

ω

),

· · · , ψ

M

(

ω

)).

(9)

Likewisebyinsertingarealizationy

= (

y1

,

· · · ,

yM

)

= (ψ

1

(

ω

),

· · · ,

ψ

M

(

ω

))

,wecouldreformulatethetruncatedKL expan-sioninadeterministicformforG

:

D

× [−

1

,

1

]

M

→ R

as

G

(

x

,

y

)

=

G0

(

x

)

+

s M

k=1

λ

k

φ

k

(

x

)

yk

.

(10)

Finally,s

>

0 in(10) ischosensuchthatforallx

∈

D s

M

i=1

λ

i

φ

k

(

x

)

yk

L∞([−1,1]M₎

≤ δ

G0

(

x

),

for

δ

=

0

.

5,inordertoenforcethediffusioncoeﬃcient tobe uniformlyellipticandbounded, i.e.for all x

∈

D andforall y

∈ [−

1

,

1

]

M

0

<

Gmin

(

x

)

≤

G

(

x

,

y

)

≤

Gmax

(

x

) <

∞.

The arguments aboveallow also for rewriting the stochastic equation (6) in aparametric form: ﬁndu

∈

L2

(

[−

1

,

1

]

M

;

L2

₍

_[

₀

_,

_T

_];

_H1

₍

_D

₎₎₎

∂

tu

(

X

,

y

)

−

div

(

G

(

X

,

y

)

∇

u

(

X

,

y

))

=

f

(

X

)

in D

× [

0

,

T

],

(11) forally

∈ [−

1

,

1

]

M.

Theexpectationofthesolutionforthestochasticheatequation(11) canberewrittenasahighdimensionalintegral

E[

u

](

X

)

=

[−1,1]M u

(

X

,

y

)

ρ

(

y

)

dy

≈

˜

u

(

X

,

ω

)

d

P (

ω

)

= E[˜

u

](

X

).

(12)

3.1.2. Methodsanderrorestimates

Inordertonumericallyapproximatetheintegral(12),oneshouldrelyonaquadratureformulaofthetype

Q

u

(

·) =

Nl

i=1

wiu

(

·, ξ

i

)

ρ

(ξ

i

) .

(13)

Approximatingastochasticintegrallikewisecanfurtherbeunderstoodasasamplingmethod,wherethesamplesrepresent the quadrature points. A well known samplingmethod is the Monte-Carlo estimator. It ischaracterized by a fairly easy implementation and isstraightforward to use,but it has a low convergence rateand hence requires a large number of samples,i.e.simulations.Inthiswork,weconsideramultilevelapproachasanalternativetostandardMonte-Carlo.

TheMonte-Carlo(MC)estimatorrepresentsadirectmethodforcalculatingthesolutiontotheparametricproblem(12). Itsquadratureformulacanbeexpressedas

Q

MCu

(

·) =

N_l−1 Nl

i=1

ui

(

·),

(14)

i.e.themeanoversolutionsamplesui,i

=

1

,

..,

Nl correspondingto Nl independentandidenticallydistributedrealization. The definitionofthe MCestimator (14) wouldbe completelyauto-sufficient ifasolution ui _to_the _sample _{i could}_be computedexactly.However,inthecontextthemonodomainequation,anexactandanalyticalsolutioncannotbeprovided andwerelyonfiniteelementstonumericallyapproximateit.Letusconsideranestedsequenceofshaperegularand quasi-uniformtetrahedralizations

{

T

l

}

l≥0ofthespatialdomainD,eachofmeshsizehl

∼

2−l.Wedeﬁnethecontinuous,piecewise linearﬁniteelementspaces

S

l

= {

vl

∈

C0

(

D

)

:

vl|T

∈ P

1

(

T

),

∀

T

∈

T

l

}.

Wenowintroducethenotation

Q

_MC,lu

(

·) =

Nl−1 Nl

i=1

(6)

whereu_liistheﬁniteelementapproximationofthesolutionuiforthesamplei,lyinginthespace

S

l.Therefore,thesolution obtainedthroughtheMCmethodissubjectnotonlytothestochasticerrordependingonthenumberofsamplesusedas quadraturepoints,butalsoonthediscretizationerrorinherenttosolvingthePDEwithinthecontextofﬁniteelements.

ThestochasticerroroftheMCquadratureintegrationisboundedby [3]

E[

u

] −

Q

_MCu

_L2₍_[−₁_,₁_]m_;V)

≤

N_l−1/2

u

_L2₍_[−₁_,₁_]m_;V)

,

(16)

where

V =

L2

₍

_[

₀

_,

_T

_];

_H1

₍

_D

₎₎

_with_the_norm_deﬁned_as_in₍₅_)._On_the_other_hand,_using_linear_ﬁnite_elements_and_a_second ordermethodintime(e.g.Crank-Nicolson)wegetthefollowingerrorestimatefora H2–regularsolutioninspace [1,11,14,

35]

u

−

ul

_V

≤

C

(

hl2

+

tl2

)

f

V

,

(17)

forsomeconstantC

>

0 thatdependsontheinitialconditionandthenumberoftimestepsconsidered,butnotonhl and

tl whicharerespectivelythespatial andtimediscretizationparametersforagivenlevell.Putting(16) and(17) together resultsinhavingthegeneralerrorbound [3]

_E[

u

] −

Q

_MC,lu

_L2₍_[−₁_,₁_]m_;V)

≤

C

(

N−

1/2

l

+

h

2

l

+

t2l

)

f

L2₍_[−₁_,₁_]m_;V)

.

(18)

The setting for a MC estimation istherefore closely relatedto that of the following errorestimate, asit underlinesthe relationshipbetweenthemeshsizeandthenumberofsamplestoconsideronagivendiscretizationlevel.Indeed,inorder to not have an error that is neither dominated by discretization in stochastics or discretization in space and time, the numberofsamplestoconsideronalevell issuchthat

N_l−1/2

∼

O

(

h_l2

)

=⇒

Nl

∼

O

(

h−l 4

)

=

O

(

2 4l

_),

wherewe assumethathl

∼

2−l andhl

∼

tl.TheconvergencerateoftheMCmethodcanthereforebe derivedfrom(18) andisgivenby2−2l.Thereadermightatthislevelrealizethelimitationofsuchamethod,asthenumberofsamplestobe takengetstremendouslylargewhenconsideringdiscretizationlevelswithsmallhl and

tl.Inaddition,eachoneofthese samplesalsorequiresamoresigniﬁcantcomputationalloadwhichplacestheMCmethodintherangeofnumericallycostly methodsforPDEsrequiringahighresolutionsolution.

TheMultilevelMonteCarlo(MLMC)estimatorrepresentsanalternativefortheMCestimator.Wewouldliketosolvethe equation foran initiallydiscretized domain D

× [

0

,

T

]

,whichwe willrefer toasbeingtheﬁnediscretization level L.Let usestablishahierarchyofnestedcoarsediscretizationlevelsl

=

0

,

1

,

..,

L

−

1 obtainedbyuniformcoarseninginspaceand time.WedeﬁnetheMLMC-estimatorasfollows:

Q

MLMC,Lu

(

·) =

L

l=0

Q

L−l MC

(

ul

(

·) −

ul−1

(

·)),

(19) where

Q

l MC

L

l=0isasequenceofMCquadratureswithincreasingprecisionoforder2−

2l _[₁₆_]._We_also_deﬁne_u −1

≡

0. To clarify the above, the underlying idea about the MLMC-estimator is to perform the dominant number of MLMC samples on a coarse level l

=

0, and to obtain a solution forlevel l

=

L by adding corrections betweenevery different successivelevelsl

−

1 andl. TheMLMCanditsconvergenceratehavebeendiscussedindetailsin [5,16,36] forthecaseof thestochasticPoissonproblem.Forthestochasticequation,given(16),(17) andtheabovespeciﬁed2−2l increasingprecision forthequadraturerule,asimilarresultcanbeshown:

_E[

u

] −

Q

_MLMC,lu

_L2₍_[−₁_,₁_]m_;V)

≤

C 2−2LL

f

L2₍_[−₁_,₁_]m_;V)

,

(20)

forsomeconstant C

>

0.TheMLMChasthereforethesameconvergenceratethanMC uptoalogarithmicnumber,butis designedtobelesscomputationallyexpensive.

3.2. Space-timediscretizationforthelinearproblem

The heat equation (6) can be numerically solved by means ofﬁnite elements (FE)and ﬁnite differences(FD), which are well established forsuch problems.A classical approachin thatsense isto performa FEdiscretization inspace and FDintime, i.e.using asequentialtime-steppingmethod.The parallelscalabilityofthisapproachiseventually limitedby the timeintegrationprocess. Infact,when theparallelizationinspacesaturates,sequential timeintegrationbecomesthe naturalbottleneckforthescalabilityofthesolutionprocess.Thiscanbeovercomebyparallel-in-timemethods.Thosewere developedinthelastdecades [27] toovercomethisbottleneckandenhanceparallelisminbothspaceandtime.Inparticular, weemployamonolithicapproachinspaceandtime,whereweassemblealargespace-timesystemthatissolvedinparallel [24].Foracomprehensivereviewofparallel-in-timemethods,see[12].

(7)

We use Lagrangian FEof ﬁrst orderfor the discretizationin the spacevariable. Assuming that the solutionu

(

x

,

t

)

is suﬃciently regularon D, wederive theweak formulationfrom(6) (inwhichwe neglectthestochastic variable

ω

) using testfunctionsv

∈

H1₀

(

D

)

:

D

∂

u

(

x

,

t

)

∂

t v

(

x

)

dx

+

D

(

G

(

x

)

∇

u

(

x

,

t

))

∇

v

(

x

)

dx

=

D f

(

x

,

t

)

v

(

x

)

dx

.

Then,wedeﬁnethesemi-discreteproblem:

Mh

∂

u

(

t

)

∂

t

+

Khu

(

t

)

=

f

(

t

).

(21)

Here,Mh

∈ R

n×nandKh

∈ R

n×narethestandardmassandstiffnessmatricesobtainedusingn linearnodalbasisfunctions

{ψ

i

}

i_i=₌n₁

⊂ P

1,i.e. Mh

:=

⎡

⎣

D

ψ

i

(

x

)ψ

j

(

x

)

dx

⎤

⎦

n i,j=1

,

Kh

:=

⎡

⎣

D

(

G

(

x

)

∇ψ

i

(

x

))

∇ψ

j

(

x

)

dx

⎤

⎦

n i,j=1

,

(22) f

(

t

)

:=

⎡

⎣

D f

(

x

,

t

)ψ

i

(

x

)

⎤

⎦

n i=1

,

(23)

arisingfromtheapproximation:

u

(

x

,

t

)

n

i=1

ui

(

t

)ψ

i

(

x

),

with u

(

t

)

= [

u1

(

t

),

u2

(

t

),

· · · ,

un

(

t

)

]

T

.

Considerauniformpartitionofthetimeaxis

[

0

,

T

]

inm intervals,suchthat

t

=

T

/

m andtk

=

k

t,withk

=

0

,

· · · ,

m.We applythesecondorderCrank–Nicolsonmethodforthetimediscretizationof(21) andobtain,fork

=

0

,

· · · ,

m

−

1

Mh

+

t 2 Kh

uk+1

+

−

Mh

+

t 2 Kh

uk

=

fk and uk

:=

u

(

tk

),

(24) with fk

:=

t 2

f

(

tk+1

)

+

f

(

tk

)

.

(25)

Ifwedeﬁne At,h

:=

Mh

+

2tKh andBt,h

:= −

Mh

+

2tKh thesystemofequations(24) canbesummarizedinthecompact form: At,h Bt,h At,h Bt,h At,h

⎤

⎥

⎦

⎡

⎢

⎣

⎛

⎜

⎝

u1 u2

..

.

um

⎞

⎟

⎠

=

⎛

⎜

⎝

f1 f2

..

.

fm

⎞

⎟

⎠

⇐⇒

C u

=

f

,

(26)

whereC

∈ R

nm×nm_is_a_large_space-time_system_that_can_be_distributed_and_solved_in_parallel_and u

:= [

u1

,

u2

,

· · · ,

um

]

T and f

:= [

f1

,

f2

,

· · · ,

fm

]

T

.

3.3. Discretizationofthenonlinearproblem

The discretization of (1) is an extension of the assembly procedure described in Section 3.2. In particular, the linear system(26) ismodiﬁedtocontainthediscretizationofthenon-linearreactiontermIion:

C u

+

r

(

u

)

=

f

,

(27)

wherer

(

u

)

∈ R

nmisgivenby

r

(

u

)

:= (

t IN

⊗

Mh

)

Iion

(

u

)

with Iion

(

u

)

:= [

Iion

(

u1

),

Iion

(

u2

),

· · · ,

Iion

(

unm

)

]

T

,

wheren andm arerespectivelythespaceandtimedegreesoffreedom.

TheJacobianJ

(

u

)

∈ R

nm×nm ofthenon-linearoperatorinlefthandsideof(27) isgivenby J

(

u

)

=

C

+ (

t Im

⊗

Mh

)

·

diag

(

Iion

(

u1

),

Iion

(

u2

),

· · · ,

Iion

(

unm

)).

(8)

3.4. Transferofdiscreteﬁelds

ThetransferofinformationbetweenfiniteelementspacesinourmultilevelMonteCarlomethodisperformedbymeans ofboth L2–projectionsandstandardfiniteelementinterpolationmethods.Inthecasewhenmeshhierarchiesaregenerated byrefinement,theelementsoftherefinedmeshformpartitionsoftheelementsoftheoriginalcoarsemesh.Thisallowsus toemploystandardfiniteelementinterpolationfortransferringfromthecoarsespacetotherefinedspace.However,ifthe finemeshisnotgeneratedbyrefinementwecanuseL2_{–projections}_for_the_same_task._For_transferring_discrete_fields_from the finespace to acoarser spacewe also employ the L2–projection. L2–projectionsare proven tobe optimal andstable andin generalsuperiorto interpolation [18].In particular,we employ alocalapproximation ofthe L2_{–projection,} _which is constructedby exploiting thepropertiesof thedual basis [30,37]. Thislocal approximationallows usto construct the transferoperatorexplicitlyinsuchawaythatitcanbeappliedbymeans ofasimplesparsematrix-vectormultiplication. Inaparallelcomputingenvironment,wheremeshesarearbitrarilydistributed,theassemblyofthemassmatricesrelatedto

L2_{–projection} _is_not _trivial._The_construction _of_the_discrete _L2_{–projection}_requires_us_to _detect_and_compute _{intersections} betweentheelements ofthecoarsemeshandtheelementsoftheﬁnermeshwhichmightbestoredindifferentmemory addressspaces(e.g.,onasuper-computingcluster).Forthispurposeweusetheparalleltree-searchalgorithmsandassembly routinesdescribedin [23].

In thiswork we extend theabove to multilevel space-timediscretizations. By exploitingthe tensor-productstructure of (26) wecan simplifythe implementationandbeneﬁtfrombettercomputationalperformance. Asbefore,letusassume that the spatial mesh does not change at each time-step (i.e., no adaptive mesh reﬁnement). In this case, the tensor-product structureofthe space-timegridallows ustoconstructspace-time operatorsin aconvenientwaywhich requires theassemblyofthespatialtransferoperatortobeperformedonlyonce.

Forl

=

0

,

...,

L

−

1 let ul

∈ R

nlml _be_the_coeﬃcients_of_the_discrete _space-time_solution_on_level_l,_where_n_l _(resp._m_l) _is

thenumberofspatialnodes(resp.timesteps)onl.WithPl,h

∈ R

nl×n wedenotethespatialdiscreterepresentationsofthe

L2–projectionfromtheﬁneleveltoacoarserlevell, andwith Pl,t

∈ R

ml×m thetemporalrestrictionoperator,obtainedby standardbisection.Thespace-timetransfermatrixPl

∈ R

nlml×nm isconstructedwiththefollowingtensorproduct

Pl

=

Pl,t

⊗

Pl,h

,

andwewrite ul

₌

_P

lu.Thespace-timeinterpolation(prolongation)operatorfromanycoarselevell

−

1 toaﬁnerlevell is constructedinthesameway,usingthetensor-productbetweentransferoperators.

3.5. Parallelsolver

Thenonlinearproblem(27) issolvedwithNewton’smethod.Foreachofthearisinglinearproblemsintheform(26), we employ a space-time parallelPGMRES, withthe preconditionergiven byan ILU(0)factorizationwithin the PETSc [2] framework.Ifmultipleprocessorsareused,thecorrespondingblock-Jacobipreconditionerisemployed,andthePGMRESis applied,inparallel,tothelocalblocks.Theusage ofthelatterPGMRESismotivatedbythespectral analysisofthe space-timesystemin(26),carriedoutin [4].AsdefaultbyPETSc,theGMRESsolverisrestartedafter30iterations.Itisimportant torecallthattheﬁrstiterateplaysanimportantrolefortheconvergenceofNewton’smethod.Forthisreasonwe usethe unperturbedreferencesolutionasinitialguessforeachMonte-Carlosample.Thisguaranteesfastconvergenceinoursetting.

3.6. Parallelizationstrategy

The MLMCmethoddevelopedabove reducesundoubtedlythe computationalcomplexity thatiscarriedby anaive MC approach. However, to fullyexploit the multilevelalgorithm, itis necessary to relyon a parallel environment that takes advantageofalldifferentlayersunderwhichconcurrent workcanbeexecuted.TheMLMCrequirescomputingsamplesat differentdiscretizationlevels, requiringadifferentamount ofresources.Therefore,we could thinkofinitially distributing therequiredamountofworkforevery levelasaﬁrstparallelization layer.Onevery oneoftheselevels,multiplesamples needtobecomputed.Thosearecompletely independentandwecanconsiderdistributingworkacrosssamplesasasecond parallelizationlayer.Thethirdlayer,inthecontextofﬁniteelements,wouldnaturallybeaparallelizationacrossthe tempo-spatialgrid,usingspace-timeparallelmethod.

We therefore rely on a so-called three level parallelization strategy largely inspired by [8]. We furthermore adapt it to standard allocationprocedures oncurrentHigh Performance Computing (HPC)system, in whicha requestfor a small amount ofresources ismore likelyto be granted ina shorteramount oftime. We issueconsequently differentjob calls involvingdifferentamountofresources forevery paralleltaskconsistingofbatches ofsamplesto solve(cf.Fig.2). Asan example,let ussuppose wehave anMLMCsetting of L

+

1 levels.We start withinitiating p0

,

. . . ,

pL processes carrying thetasksforeveryoneoftheselevels.Thesamplestobecomputedforeverylevell areequallydistributedinbatchesover the pl processes createdforthat level. The differentnumberof samples Nl on each levelandthe ﬂexibility inchoosing the numberofprocesses pl are suchthat the batch size mayvaryacross thelevels, asshowninFig. 3.Every sample is furthersolvedinparallelwithdomaindecompositiontechniques,fulﬁllingthethirdparallelizationlayer.Detailsaboutthe parallelsolverwereliedonareprovidedattheendofsubsection

2.2

afterintroducingourapplication,i.e.themonodomain

(9)

Fig. 2. MLMC tasks scheduling and job requests step. tl,idenotes the i-th task of level l.

Fig. 3. The different processes executing concurrently different batches of samples.

equation. Notice thatanother advantageofthe followingcluster schedulingapproachis toavoidwasting resourceswhen theprocesseshavedifferenttimeexecution.

4. Numericalexperiments

The numericalexperimentswere realizedusingSLOTH[32] aUQ pythonlibrarydevelopedattheInstitute of Compu-tational Science (ICS)ofLugano.For thiswork,we extended itto themonodomain equation(and ingeneralto all types of3D

+

1PDEs)byemployingUtopia[38] fortheﬁniteelementformulation.Theexperimentshavebeenperformedonthe multicorepartitionofthesupercomputerPizDaintoftheSwissNationalSupercomputingCentre.

Parametersfortheexperiments Theparametersofmonodomainequation(1) usedforthenumericaltestsare:

•

G0

(

x

)

=

3

.

325

·

10−3cm2mV ms−1 from(7) isthemeandiffusionﬁeld.

• α

=

1

.

4

·

10−3 _mV−2_ms−1_,_u

rest

=

0 mV, uth

=

28 mV,andupeak

=

115 mVare thevaluesforthe ionchannel model Iion

(

u

)

in(2).

(10)

Fig. 4. Stimulus locations for the cube and the ventricle.

Fig. 5. KL eigenvalues decay for the cube (left) and the idealized ventricle (right).

•

Iapp

(

x

,

t

)

=

urest

+

upeakexp

−

(

x

−

x0

)

2

σ

2

χ

[0,t1)

(

t

)

where t1

=

t

=

0

.

005 ms is the function we rely on for the

appliedstimulus. Forthecube,

σ

=

0

.

5 cmwhereas fortheventricle itwas set

σ

=

1 cm.Here,x0 isthelocation of stimulusandareshownforbothgeometriesinFig.4.

ParametersforKLexpansion Weusethecorrelationkernel

k

(

x

,

y

)

=

e−

d(x,y)2

σ_KL

_,

for

σ

KL

=

0

.

25 forthecubeand

σ

KL

=

0

.

5,whiled

(

x

,

y

)

istheEuclidean distancebetweenx andy.Thedumpingfactor s in (7) forenforcinguniformellipticityissetto s

=

0

.

3 for bothgeometries.Thelowrankpivoted Choleskydecompositionon thecovariancematricesinducedbythecorrelationkernelsaboveyieldedthestochasticdimensionM

=

66 forthecubeand

M

=

87 fortheidealizedventricle.WefurthermorereporttheKLeigenvaluesdecayinFig.5.

4.1. Numericalexperimentsin1D

+

1

We intend to verifythe convergencerate ofboth MC and MLMCasdeveloped inSection 3.We consider an interval anduseahierarchynestedmeshes,obtainedbyuniformcoarseninginspaceandtime.InTable1,wereportallthedetails regarding theconsidered mesh hierarchy,i.e.the numberof elements,the meshsize, the numberoftime steps andthe timestepsize.

Wecreateareferencesolution,denotedasvre f,bycalculatingthemeanover20,000 samplesontheﬁnestlevelL

=

5. For the purpose ofverifying the MC convergencerate, we run several MCs on each level l with Nl

∼

24l, andcompute thedifferencebetweentheexpectationcalculatedateachlevel,denotedasvl,andthereferencesolution.Weevaluatethe rootedmeansquareRMSE

= E[

el

2

]

1/2,whereel

= |

vl

−

vref

|

and

el

2

=

[0,T]

el

(

·,

t

)

2_L2₍_D₎dt

=

[0,T]

D e2_l

(

x

,

t

)

dx dt

.

(11)

Table 1

DetailsabouttheconsideredMeshhierarchy.t andh arethetimeandspacediscretizationsteps,m andn are thenumberofsubdivisionsfortimeandspaceintervals.

l 0 1 2 3 4 5

n 31 62 125 250 500 1000

h 0.032 0.016 0.008 0.004 0.002 0.001

m 4 8 16 32 64 128

t 0.16 0.08 0.04 0.02 0.01 0.005

Fig. 6. Controlled convergence graphs for MC and MLMC for the 1D+1 example.

Table 2

Detailsaboutthemeshhierarchyforthecubegeometry.

l 0 1 2 3

Dof’s 5400 65,219 898,317 13,301,753

h 0.2 0.1 0.05 0.025

t 0.04 0.02 0.01 0.005

We plot the rooted mean square error versus the level, in what we will refer to as the controlled convergence graph. We usethisterminologyasitindicates that we arecontrolling thenumberofsamples inorderto haveastochastic and discretizationerrorsofthesameorder.TheplotisshowninFig.6.

In the same plot, we show the controlled convergence for the MLMC. For each level L, we perform an MLMC run involving as many levels by setting the number of samples on the coarse level to 24L. The number of samples for the following levelsis then calculatedby dividingthisnumber withtheidealsamplingratio,which corresponds inour case to 16.Thisnumbercanbe deducedby enforcingthath2

lN −1/2 l

=

h2l+1N

−1/2

l+1 which representstheerrorcontributionfrom levelsl andl

+

1.

4.2. Numericalexperimentsin3D

+

1

Weintendinthenextsubsectionstoestimateandverifytheconvergencebehavior fortheMCandtheMLMCestimators, as done for the 3D

+

1 case above. Furthermore, a comparison in terms of work to accuracy for both methods will be conducted.

Werely onthecubeandtheventriclegeometries,introduced inSection 3(cf.Fig.1).As opposedtoa classicalMLMC approach,whereone wouldreﬁne aninitiallycoarse meshinordertoreduce thevariance,wehereconsidertherealistic setting ofan application forwhichonly one mesh ofthe givengeometry isprovided. Thismesh is providedat theﬁne discretizationlevel,thereforethegoalfortheMLMCistoreducetheworkloadbyconsideringcoarsermeshes.Werelyon anestedmeshhierarchyof3and4levelsrespectivelyfortheventricleandthecubegeometries.Thenumberofspace-time degreesoffreedom(Dof’s),thespaceandtimediscretizationstepsofalltheconsideredlevelsforthecubeandtheventricle arerespectivelyreportedinTables2and

3

.

(12)

Table 3

Detailsaboutthemeshhierarchyfortheventriclegeometry.

l 0 1 2

Dof’s 154,546 2,120,420 31,184,747

h 0.1 0.05 0.025

t 0.02 0.01 0.005

Fig. 7. Convergence rate for the MC estimator for the cube and the ventricle.

Fig. 8. Mean (left) and variance (right) at the ﬁnal time state for the cube.

4.2.1. Monte-Carloin3D

+

1

Weconduct aMonte-Carlostudy of N

=

10000 sampleson theﬁnediscretizationlevelsinordertoobtain areference solution. In Figs. 8 and 9, we show the mean and the variance at the ﬁnal time state for the cube and the ventricle, respectively.

InSection3,wepresentthetheoreticalsemi-linearconvergenceoftheMCmethod(cf.Equation(16)).Weverifythisin Fig.7inwhichwe plottherootedmeansquareerror(RMSE)against thenumberofsamples(N) usedfortheestimation process. Itcanbe observedhow theRMSEindeedbehavesasRMSE

=

O(

N−0.5

₎

_._Notice_that _these_plots_rely_on_averaged multipleruns.

4.2.2. MutilevelMonte-Carloin3D

+

1

ThesettingfortheMLMCisvery similartothatofthe1D

+

1case.We underlinetheimportanceoftheidealsampling ratioasdetailedatthe endofSection 3.We provideinFigs. 10and

11

a visualization oftheMLMCestimator(mean at coarselevelandcorrectionsbetweensuccessivelevels)forthecubeandtheventriclerespectively.

Fig.12presentsthecontrolledconvergencegraphofMCandMLMCforboththecubeandtheventricle.Itdemonstrates that thegeneralconvergencerateobtainedforthesetwomethodsisasexpected. MCandMLMChavethereforeasimilar controlledconvergencerate.

Workbehavior foraﬁxedMLMCsetting: Asmentionedintheprevious paragraph,theplotsforcontrollederrorinFig.12

(13)

Fig. 9. Mean (left) and variance (right) at the ﬁnal time state for the ventricle.

(14)

Fig. 11. Mean at coarse level and corrections between successive levels for the ventricle.

Fig. 12. Controlled convergence (MC and MLMC) graphs for the cube (left) and the ventricle (right).

underlinethisaspect,weattemptheretoevaluatethisdifferenceofproducedworkforbothmethods,whenevertheMLMC setting(ﬁnelevel,numberoflevels,meshhierarchy,...)hadbeenpreviouslyset.LetusdenotewithW_ltheexpectedwork forsolvingasampleatlevell,

∀

l

=

0

,

· · · ,

L.WealsointroduceWl tobedeﬁnedas

Wl

=

Wl−1

+

Wl

,

∀

l

=

0

,

· · · ,

L

withW₋₁

≡

0.Wecanthereforeformulatethetotalworkexpressionforbothmethodsinthefollowingway W_MC,L

=

WLNL

,

WMLMC,L

=

L

l=0

WlNl

,

(28)

where Nl is thenumberofsamples computedoneach level l,

∀

l

=

0

,

· · · ,

L.We furthermorecan usethe samplingratio betweenlevelsforMLMC,whichwewilldenoteherewith

β

,torewritetheMLMCtotalworkinfunctionofthenumberof samplesattheﬁnelevelexclusively

W_MLMC,L

=

_L

l=0

β

L−lWl

NL

.

ThelatterexpressionallowsustoselectdifferentvaluesofthenumberofsamplesNL atthefinelevel,thereforeevaluating theworkfordifferentsamplings.InFig.13,weplottheRMSEversusthetotalamountofworkforbothmethods.Wehere define theexpectedwork W_l forcomputingasample atlevell to be thetotalrunning timeforsolving a sample atthe samelevel,averagedthroughsolvingahundredofsamples.Furthermore,wenormalizewithrespecttothetimeforsolving asampleonthefinelevel.

Asymptoticalworkbehavior ofMC/MLMC: Equivalentlytothecontrolledconvergenceconcept,wewouldlikenowtostudy theworkinthecontextofthecontrollederror.Followingthetheory,inordertoachieveanerroroforder2−2l_,_{the cost}_for

(15)

Fig. 13. WorkcomparisonbetweenMCandMLMCforthecube(left)andtheventricle(right).Works(WMC,L andWMLMC,L asdeﬁnedin(28))are

measuredonthebasisofCPUtime.

solvingasamplewithlinearFEisgivenby CFE,l

=

2γdl

,

where

γ

isthecomplexityofthesolutionmethodusedandd isthedimensionforthePDEproblemconsidered(equalto4 inourcase).Thenumberofsamplestoexecuteonalevell forgettinganerroroforder2−2l isgivenby

N_MC,l

=

24l

.

Fromthese,onecanrecoverthetotalamountofworkrequiredbybothmethodsforaﬁnediscretizationlevelL

W_MC,L

=

CFE,LNMC,L

=

2γdL24L

=

2(γd+4)L (29) and W_MLMC,L

=

L

l=0 CFE,LNMC,L−l

=

L

l=0 2γdl24(L−l)

=

24L L

l=0 2(γd−4)l (30)

Thiscanberewrittenas

W_MLMC,L

=

⎧

⎨

⎩

L24Lif

γ

d

=

4

,

2γdL

−

24L 2γd−4

₋

₁ if

γ

d

=

4

.

(31)

Therefore, incasedimension d and complexity

γ

areknown, equations(29) and(31) givea theoretical estimate on the asymptotical behavior forthe work produced by both MC andMLMC. In our case, since we are measuring the work in termsoftimetosolution(CPUtime),wecanestimatetheproductofthosevaluesas

γ

d

=

log₂

(

W_l₊₁

/

W_l

).

(32)

Thefollowingestimatesyield

• γ

d

≈

4 forthecube

• γ

d

≈

2 fortheventricle

TheMCandMLMCproducedworkforthecubeexperimentsarethereforegivenby

W_MC,L

=

28L

,

WMLMC,L

=

L24L

.

(33)

Fortheventricleexperiments,weobtain

W_MC,L

=

26L

,

WMLMC,L

=

3

(

24L

−

22L

)

4

.

(34)

Thisdemonstrates veryclearly that theasymptoticbehavior ofMLMCintermsof computationalwork issigniﬁcantly re-ducedincomparisonwithstandardMC.Weillustrate thisinFig.14.Werecallthatthecontrolled RMSEel atlevell isof

(16)

Fig. 14. MCandMLMCasymptoticalworkforthecube(left)andtheventricle(right).The“dimensionalcomplexity”(γd asdeﬁnedin(32))ismeasured onthebasisofCPUtime.

order2−2l_._From_this,_we_can_deduce_that_e−0.5

l

∼

2l.Puttingthistogetherwithequations(33) and(34),showsthatforthe cubewehave

el

=

O

(

W_MC−1/4

),

whereasfortheventricle

el

=

O

(

W_MC−1/3

).

ThisisillustratedinFig.14. 5. Conclusions

We presented a novel approach aimed at performing uncertainty quantiﬁcation fortime-dependent problems, in the presence of high-dimensional input uncertainties andoutput quantities-of-interest.We assessed the performance ofthe proposed frameworkbyshowingindetailsitsapplicationtocardiacelectrophysiology.Theresultsshowthattheproposed methodattainstheidealconvergenceratespredictedbythetheory.Ourmethodologyreliesonacloseintegrationof multi-levelMonteCarlomethods,paralleliterativesolvers,andaspace-timediscretization.Thisallowstoexploittheir synergies, suchasfortheinitializationofNewton’s methodusingthesolutionofpastsampleswhentheybecomeavailable.The pro-posedmethodcanalsobeextendedtospace-timeadaptivityandtime-varyingdomains,bothofwhichareveryrelevantin thecontextofcardiacelectrophysiologyandwillbeaddressedinfutureworks.

CRediTauthorshipcontributionstatement

S.Ben Bader: Conceptualization, Investigation,Methodology,Software,Visualization, Writing –originaldraft, Writing – review&editing. P.Benedusi: Software,Writing–originaldraft,Writing–review&editing. A.Quaglino: Conceptualization, Methodology, Software,Writing – originaldraft, Writing – review &editing. P.Zulian: Software,Writing –original draft, Writing–review&editing. R.Krause: Conceptualization,Fundingacquisition,Methodology,Resources,Supervision. Declarationofcompetinginterest

The authors declare that they haveno known competingﬁnancial interests or personal relationships that could have appearedtoinﬂuencetheworkreportedinthispaper.

Acknowledgements

The authors would like to thankthe Swiss NationalScience Foundation (SNSF) for their support through the project “Multilevel Methods and Uncertainty Quantificationin Cardiac Electrophysiology” in collaboration with theUniversity of Basel(grantagreementSNSF-205321_169599),butalsothroughtheproject“HEARTFUSION:Imaging-drivenPatient-specific CardiacSimulation”incollaboration withtheUniversityofBern(grantagreementSNSF-169239).Furthermore,theauthors wouldliketo thanktheDeutscheForschungsgemeinschaft(DFG) whichhassupportedparts ofthesoftwareusedherein theSPPEXA program“EXASOLVERS –ExtremeScaleSolvers forCoupledProblems”andSNSFunderthelead agencygrant agreement SNSF-162199.Part of thesoftware toolsutilized inthispaper were developedas partofthe activities of the Swiss Centre forCompetence in Energy Research on the Future Swiss Electrical Infrastructure (SCCER-FURIES), which is financially supported by the SwissInnovation Agency (Innosuisse –SCCER program). Theauthors gratefullyacknowledge financial support bythe Theo Rossidi MonteleraFoundation, theMantegazza Foundation, andFIDINAM tothe Center of ComputationalMedicineinCardiology.

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