Convergence issues in derivatives of Monte Carlo null-collision integral formulations: a solution

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Convergence issues in derivatives of Monte Carlo

null-collision integral formulations: a solution

Jean-Marc Tregan, Stéphane Blanco, Jérémi Dauchet, Mouna El-Hafi,

Richard Fournier, L Ibarrart, P Lapeyre, Najda Villefranque

To cite this version:

Jean-Marc Tregan, Stéphane Blanco, Jérémi Dauchet, Mouna El-Hafi, Richard Fournier, et al..

Con-vergence issues in derivatives of Monte Carlo null-collision integral formulations: a solution. Journal

of Computational Physics, Elsevier, 2020, 413, pp.1-20/109463. �10.1016/j.jcp.2020.109463�.

�hal-02546081v2�

Convergence

issues

in

derivatives

of

Monte

Carlo

null-collision

integral

formulations:

A

solution

J.-M. Tregan

a,

∗

,

S. Blanco

a

,

J. Dauchet

c

,

M. El Hafi

b

,

R. Fournier

a

,

L. Ibarrart

b

,

P. Lapeyre

d

,

N. Villefranque

e,f

a_{LAPLACE,UMR5213- UniversitéPaulSabatier,118,RoutedeNarbonne,31062ToulouseCedex,France} b_{LaboratoireRAPSODEE- UMR5302,ENSTIMAC,CampusJarlard,81013AlbiCTCedex09,France} c_{UniversitéClermontAuvergne,CNRS,SIGMAClermont,InstitutPascal,F-63000Clermont-Ferrand,France} d_{PROMES- UPRCNRS8521,7,rueduFourSolaire,66120FontRomeuOdeillo,France}

e_{CentreNationaldeRecherchesMétéorologiques(CNRM),UMR3589CNRS,MétéoFrance,Toulouse,France} f_{LaboratoirePlasmaetConversiond’Énergie(LAPLACE),UMR5213CNRS,UniversitéToulouseIII,France}

a

b

s

t

r

a

c

t

Keywords: MonteCarlomethod Directderivatives Null-collisionalgorithm Sensitivity

Integralformulation

WhenaMonteCarloalgorithmisusedtoevaluateaphysicalobservableA,itispossibleto slightlymodify the algorithmso that it evaluates simultaneously A and the derivatives ∂ςA of A with respect to each problem-parameter

ς

. The principle is the following: MonteCarloconsiders A astheexpectationofarandomvariable, thisexpectationisan integral,thisintegralcanbederivatedasfunctionoftheproblem-parametertogiveanew integral,andthisnewintegralcaninturnbeevaluatedusingMonteCarlo.ThetwoMonte Carlocomputations (of A and∂ςA)are simultaneouswhentheymake useofthesame randomsamples,i.e.whenthetwointegralshavetheexactsamestructure.Itwasproven theoretically thatthiswas alwayspossible, butnothingensuresthatthe twoestimators have the same convergence properties: even when alarge enough sample-size is used sothat A isevaluatedveryaccurately,theevaluationof∂ςA usingthesamesamplecan remaininaccurate.Wediscussheresuchapathologicalexample:null-collisionalgorithms areverysuccessfulwhendealingwithradiativetransferinheterogeneousmedia,butthey aresourcesofconvergencedifficultiesassoonassensitivity-evaluationsareconsidered.We analysetheoreticallytheseconvergencedifficultiesandproposeanalternativesolution.

1. Introduction

Whennumericallysimulatinglinear-transportphysicsusingMonteCarloalgorithms,oneofthemostrecurrent difficul-tiesisthehandlingofhighlynon-homogenousorfast-variatingmedia.Thisdifficultywasencounteredsincethebeginning ofneutron-transportandplasma-physicsmodelling.Thefirststrategyconsidersthatthemediumishomogeneousbypieces inordertomake anefficientandsimpleresolution.However, tovalidate thisapproximation,wehaveto consideralarge numberofcellswhichinvolvesanincreaseincomputationtimes,asaconsequenceofthemesh-crossingprocedure.Several techniques havebeendeveloped to accelerate thismesh-crossing asin [1]. Buta quite elegant trick was soon identified asa waytobypass thisdifficulty withoutapproximation: virtual collisionnerscan be addedwherethe truecollisionners

*

Correspondingauthor.

arescarcesothatthetotalcollisionner-densityishomogeneous.Ofcourse,inordertoensurethatthephysicalproblemis unchanged,whenaparticleinteractswithavirtualcollisionner,itsimplycontinuesitspathasifnocollisionhadoccurred [2–5].Thisisthemeaningofthedenominationnull-collisionalgorithm orfictitious-collisionalgorithm.1Thefirstpractical ben-efit isthatthenextcollisioneventcanbesampledasifthemediumwas homogenous.Thenthechoiceismadetoselect a true-collisionora virtual-collisionasfunctionoftheir localrespective-amounts andthisishow thespatial information isrecovered.Butseveralotherbenefitswererecentlyforeseenin[2] andpracticallytestedin[6–22],mainlyfor radiative-transfer applications.The mainidea isthat null-collisionalgorithms transformthe non-linearityofBeer-extinction intoa linear-recursiveproblemthatMonteCarlohandleswithoutapproximation[15].Thiswasforinstanceusedin[6] todealwith absorption-spectra ofmolecular gases combiningvery numeroustransitions: thesummation overall transitionscould be treatedbytheMonteCarloalgorithmitself,whichwaspreviously assumedimpossiblebecausethissummationwasinside theexponentialofBeer-extinction.Similarly,thevanishingoftheexponentialallowedtheextensionofimplicitMonteCarlo algorithms forinversionofabsorptionandscatteringcoefficientsfromintensitymeasurements[7].Outsideradiative trans-fer, averysimilarideawasusedtosolveElectromagneticMaxwellequationsforenergypropagationinparticle-ensembles ofstatistically-distributedshapesdespiteofthenonlinearityassociatedtothesquareoftheelectricfield[14].Againsimilar isthealgorithmproposed in[15] solvingBoltzmannequationformicro-fluidicsapplicationsdespiteofthenonlinearityof thecollisionoperator.

Back to radiative-transfer applications, the ideas suggested in [2] have motivated significant developments in the computer-graphicscommunityforthe cinemaindustry.Here thebenefitofusingnull-collisionsis thatit extendsto par-ticipating media(aerosolsorclouds)theorthogonalitybetweendata-descriptionanddata-treatmentthatwas attheheart ofthemostrecentuseofMonteCarloforrenderingcomplexscenes[8,10,9,11].Thealgorithmisindeedprocessedwithout anyknowledgeoftheexactspatial-information,anditisonlywhenacollisionoccursthataccesstothefield isrequired: the interactionbetweentheradiative-transfer algorithmandthefield-data isstrictly restrictedtothisvery moment.This allows the implementationof numerousaccelerationtechniques withlittlechanges by comparisonwiththose developed forhandlingcomplexsurfaces.Oneofthesetechniquesconsistsinthesettingofanaccelerationgrid,adjustingtheamount ofvirtualcollisionnerssothatthetotalcollisionner-densityisbothhomogeneousineachcellofthegridandcloseenough to therealdensity-field.Thisavoidsthesamplingoftoomanyuselessvirtual-collisions.Thisisoneofthestarting points ofthepresentpaper:null-collisionalgorithmsallowtheuseofanyamountofvirtual-collisionnersbutnumericalefficiency justifiesthatonetriestoreducethemtotheminimum.

However, weshow herethatreducing theamountofvirtual-collisionnerstoa minimumleads toconvergence difficul-tieswhenevaluatingsensitivities.SensitivityevaluationisaverygeneralfeatureofMonteCarlotechniques:whenaMonte Carlo algorithmis usedto evaluatea physicalobservable A, itis always possibleto modifythe algorithmin such away that itevaluatesboth A andthederivatives

∂

ς A of A withrespecttoeachproblem-parameter

ς

,andmostcommonlythe corresponding implementationisquitestraightforward[23–29].2 Butwe willshow thatevaluatingsensitivitiesusing null-collisionalgorithmsisrapidlypathological:thebetterweadjusttheaccelerationgrid,theworsethestatisticalconvergence rate. Thiswasexperiencedforinstancewithradiative transferincloudyatmospheres[30].InSec.2wewillillustratethis pathologicalbehaviour evaluatingthetransmissivityofabeamthroughanon-homogeneous column.Thenwe proposean alternativeapproachinSec.3wherethedesignofthesensitivity-evaluationalgorithmstartsfromthestandardintegral so-lutionoftheBoltzmannequation,i.e.withoutvirtual-collisionners.Theresultingsamplingrequirementsarethenaddressed withthenull-collisionapproach viewedasa simplerejection-samplingapproach.Thisintroduces thecostofsamplingan additionalrandomvariable,butatthiscosttheconvergencedifficultiesvanish.Weillustratethenumericalbehaviourofthis modifiedalgorithmisSec.4usingabenchmarkinspiredof[2].

2. Convergencedifficultieswhenevaluatingsensitivities

In thissection we designaMonte-Carlo algorithmusingthestandard null-collisionapproachfortheevaluationofthe distributionfunction f ,i.e.thesolutionofBoltzmannequation,andapply thetechniqueof[25,23] forsimultaneous evalu-ationofasensitivity

∂

ς f withrespectto

ς

,where

ς

isaparameterappearingintheabsorptionandscatteringcoefficients. TheBoltzmannequationthatweuseislinearwithconstantspeedparticles.Itmatchesthemonochromaticradiative-transfer equationexactlyandallapplicationexampleswillberestrictedtoradiativetransfer.Wechoosetomakeuseofthenotation f instead ofthe moreradiative-transferorientednotation I

=

h

ν

c f (thespecific intensity)inordertosimplifytheaccess forreadersoftheplasmaandneutronicscommunities.Themonochromaticradiative-transferequationbecomes

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

∂

tf

+

c

ω



. 

∇

f

= −(

ka

+

ks

)

c f

+

kac feq

+



4π ksc fpS

(

− 

ω



| − 

ω

)

d

ω



,

∀

x

∈ , ∀ 

ω

∈ S

2 f

(



y,

ω



₊

)

=

f∂

(

y,



ω



+

),

∀

y

∈ ∂, ∀ 

ω

+

∈ S

2₊ f

(



x,

ω



,

0

,

ς

)

=

f0

(



x,

ω



),

∀

x

∈ , ∀ 

ω

∈ S

2 (1)

1 _{Similarkeywordsarepseudo-collision,null-events,fictitious-events,null-collisions,Woodcocktracking,deltatracking andmaximumcross-section.} 2 _{Thisisnotatallstraightforwardfordomain-deformationsensitivities[25,26],butweheresticktopureparametricsensitivities.}

where f

≡

f

(



x

,

ω



,

t

,

ς

)

with



x the location,

ω



the propagationdirectionand t thetime. Forincoming scatteringin any direction

ω



oftheunitsphere

S

2_{,wewrite f}

_≡

_f

₍

_

_x

_,

_ω

_



_,

_t

_,

_ς

₎

_andp

S isthesinglescatteringphasefunction,i.e.p_S

(

− 

ω



|

−



ω

)

d

ω

 is the probability densitythat the scatteringdirection is

ω



for thisincoming direction

ω



. Theconstant particle-speedisc andthecoefficientska

≡

ka

(



x

,

t

,

ς

)

,ks

≡

ks

(



x

,

t

,

ς

)

andke

=

ka

+

ks aretheabsorptioncoefficient,thescattering coefficient andthe extinctioncoefficient respectively. feq

≡

feq

(



x

,

t

)

isthe equilibriumdistribution (following thePlanck function).



isthegeometricaldomainand

∂

itsboundaryatwhichthedistributionfunction f∂isknownforalllocations



y andalldirections

ω



₊ oftheincominghemisphere

S

2

+. f0 istheinitialcondition.

Introducingnull-collisions. Inordertodesignanullcollisionalgorithm(NCA)[2] we addafieldofvirtualcollisionnerssuch thatthetotalextinctioncoefficientispracticable,inthesensethatwecansamplethecorrespondingBeerextinction:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

∂

tf

+

c

ω



. 

∇

f

= −ˆ

kc f

+

kac feq

+



4π ksc fpS

(

− 

ω



| − 

ω

)

d

ω



+



4π knc f

δ(

ω



− 

ω



)

d

ω



,

∀

x

∈ , ∀ 

ω

∈ S

2 f

(



y,

ω



₊

)

=

f∂

(

y,



ω



+

),

∀

y

∈ ∂, ∀ 

ω

+

∈ S

2₊ f

(



x,

ω



,

0

,

ς

)

=

f0

(



x,

ω



),

∀

x

∈ , ∀ 

ω

∈ S

2 (2)

wherekn

≡

kn

(



x

,

t

,

ς

)

isthenull-collisioncoefficient,k

ˆ

=

ka

+

ks

+

kn isthetotalextinction-coefficient and

δ

isthe Dirac distribution. Equation (2) is strictly equivalent to Eq. (1) because of the Dirac distribution that ensures



_{4π k}nc f

δ(

ω



−



ω



)

d

ω



=

knc f .

When numericallyaddressing thesolution f

(



x0

,

ω



0

)

ofthistransport equationat

(



x0

,

ω



0

)

(alsosolutionofEq.(1)) using the MonteCarlomethod,one ofthe moststandard approachconsistsin asimple statisticalinterpretationthat allows to view f

(



x0

,

ω



0

)

asanaverageoverradiativepaths thataretrackedbackwardfromtheobservationlocation

(

x



0

,

ω



0

)

tothe sources [2].Inthisreading,the puretransportterm

∂

tf

+

c

ω



. 

∇

f correspondstothespatial andtemporalpropagationof f in direction

ω



atconstant-speed c.Thecollisionalterm

−ˆ

kc f correspondsto eitheran absorptionorascatteringevent (including the null-collisionevents that are forward scatteringevents). When combining it withthe transport term this leadstocollisionlocationsthataredistributedexponentiallyalongthelineofsight(Beerlaw).Trackingthepathbackward, thismeansthattheprecedingcollisionat



x1 isatadistance

λ

0 thatisarealisationofarandomvariable

0 ofprobability density p0

(λ

0

)

=

exp

(

−ˆ

k

λ

0

)

(see Fig.D.1).Once



x1 issampled,thecollisiontype issampledinturntodecidewetheran absorption,atruescatteringoranullcollisionoccurs.Inthebackwardtrackingpicture,thiscorrespondsrespectivelytothe threeremainingterms

•

withkac feq an absorption event is translated into thermalemission andthe algorithm stops with the Monte Carlo weight feq

(



x1

)

(thesourceatx



1),

•

with



_{4π k}sc fpS

(

− 

ω



|

− 

ω

)

d

ω

ascatteringeventistranslatedintothesamplingofa“previous”direction

ω



1andthe algorithmcontinuesrecursivelyasifevaluating f

(



x1

,

ω



1

)

,

•

with



_{4π k}nc f

δ(

ω



− 

ω



)

d

ω

 andits Diracfunction,a nullcollision eventistranslatedinto apure forward scattering event,i.e.the“previous”direction

ω



1 isequalto

ω



0.

Ofcoursethestatisticaltranslationincludestheboundaryconditions:whenbackwardreaching theboundaryatalocation



xianddirection

ω



i,thealgorithmstopswiththeMonteCarloweight f

(



xi

,

ω



i

)

(theincomingsourceattheboundary).The correspondingMonteCarloalgorithmisdetailedinAlgorithm1andillustratedinFig.D.1.

Integralformulation. Thisnull-collisionalgorithmbelongstothefamilyofanalogMonteCarloalgorithms,i.e.algorithmsthat canbedesignedwithoutanyformaldevelopmentbecausetheyonlynumerically-implementthewellestablishedstatistical picturesofradiationphysics.However,inthepresentcontextitisverymuchusefultoalsochooseaviewpointunderwhich thesamealgorithmappearsasastatisticalestimateoftheintegralsolutionofEq.(2).Forsakeofclarityweonlywritethis integralsolutionatthestationarylimit:

f

(

x,



ω



,

ς

)

=

exp

⎛

⎝−

λ∂



0

ˆ

k

˜

x



d

˜λ

⎞

⎠

f∂

(



y,

ω



)

+

λ∂



0 exp

⎛

⎝−

λ



0

ˆ

k

˜

x



d

˜λ

⎞

⎠

⎛

⎜

⎜

⎜

⎜

⎝

ka

(



x

,

ς

)

feq

(



x

)

+

ks

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω

f

(



x

,

ω





,

ς

)

+

kn

(

x





,

ς

)

f

(



x

,

ω



,

ς

)

⎞

⎟

⎟

⎟

⎟

⎠

d

λ

(3)

where

˜

x

= 

x

− ˜λ 

ω

,x





= 

x

− λ 

ω

,



y

= 

x

− λ

∂

ω



,with

λ

∂thedistancetothefirstboundary-intersectionstartingat



x inthe

direction

− 

ω

,i.e.

λ

∂

=

min

{

x

−

z



; z

∈

Vect−

(

x



,

ω



)

∩∂}

whereVect−

(



x

,

ω



)

= {

x

−λ



ω



;

λ



∈ R

+

}

.ThisstandardFredholm equation,typicaloftheformalsolutionoflinear-transportphysics,canbetransformedusingthefollowingproperty

exp

⎛

⎝−

λ∂



0

ˆ

k

˜

x



d

˜λ

⎞

⎠ =

+∞



λ∂

ˆ

k

(



x

)

exp

⎛

⎝−

λ



0

ˆ

k

˜

x



d

˜λ

⎞

⎠

d

λ

togive f

(



x,

ω



,

ς

)

=

+∞



λ∂

ˆ

k

(



x

)

exp

⎛

⎝−

λ



0

ˆ

k

˜

x



d

˜λ

⎞

⎠

f∂

(



y,

ω



)

d

λ

+

λ∂



0

ˆ

k

(



x

)

exp

⎛

⎝−

λ



0

ˆ

k

˜

x



d

˜λ

⎞

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

ka(x,ς) ˆ k(x) f eq

₍

_

_x

₎

+

ks(x,ς) ˆ k(x)



4π pS

(

− 

ω



| − 

ω

)

d

ω

f

(



x

,

ω





,

ς

)

+

kn(x,ς) ˆ k(x) f

(



x 

_,

_ω

_

_,

_ς

₎

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

d

λ

(4) Then

•

p_ˆ

(λ)

= ˆ

k

(



x

)

exp

−



₀λk

ˆ

˜

x



d

˜λ



canbe viewedasthe probability densityfunctionof thefree path

ˆ

(the distance untilnextcollision),

•

PA

=

k_ˆa k, PS

=

ks ˆ k andPN

=

kn ˆ

k canbeviewedastheprobabilitiesthatthecollisionisan absorption,ascatteringevent oranull-collisionrespectively,

•

and the two integrals over

[

0

,

λ

∂

[

and

[λ

∂

,

+∞[

can be gathered into a single integral over

[

0

,

+∞[

using the

Heavisidefunction

H

togive f

(



x,

ω



,

ς

)

=

+∞



0 p_ˆ

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

w∂

+

H

(λ

∂

− λ)

⎛

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)

wA

+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω

f

(



x

,

ω





,

ς

)

+

PN

(



x

,

ς

)

f

(

x





,

ω



,

ς

)

⎞

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(5)

with w∂

=

f∂

(



y

,

ω



)

and wA

=

feq

(



x

)

. Thislast equation is theintegral formulationthat we neededin orderto con-structAlgorithm1atthestationarylimit:Algorithm1isindeednothingmorethanthealgorithmic-readingofEq. (5) (and reciprocallyEq. (5) isnothingmorethantheintegraltranslationofAlgorithm1,[24]):

•



0+∞pˆ

(λ)

d

λ

standsforthesamplingofthedistanceofthecollision(accordingtothek-field),

ˆ

•

H(λ

− λ

∂

)

stands forthecasewherethe sampledcollision isoutsidetheboundary,thenthe algorithmstopsatthe

boundarywiththeMonteCarloweightw∂(thevalueof f correspondingtotheincomingradiation),

•

H(λ

∂

− λ)

stands for the case where the sampled collision is at a location



x inside the volume, andthen three

collisiontypesarepossible:

– PA

(



x

,

ς

)

standsforthecasewherethecollisionisanabsorption,thenthealgorithmstopsattheboundarywiththe MonteCarloweightwA(thevalueof feq atthecollisionlocation),

– PS

(



x

,

ς

)

stands for the case where the collision is a scattering event, then



4π pS

(

− 

ω



|

− 

ω

)

d

ω

 stands for the sampling ofa newdirection

ω



 accordingto the phase function andthealgorithm continues recursively withthe estimationof f at



x indirection

ω



,

– PN

(

x





,

ς

)

standsforthecasewherethecollisionisnull,thenthealgorithmcontinuesrecursivelywiththeestimation of f atx



intheunchangeddirection

ω



Straightforwardapplicationofsensitivity-evaluationtechniques. Nowthat wehaveconstructed theintegralformulationof Al-gorithm 1wecanapply thesensitivity-evaluation techniqueintroducedin[23,25,26].ItconsistsinderivatingEq. (5) with respectto

ς

andmultiplyinganddividingbyeachoftheprobabilitiesandprobabilitydensityfunctionsthatdependon

ς

. ThisleadstoanintegralformulationofthesensitivitythathastheverysamestructureasthatofEq. (5):

∂

ςf

(



x,

ω



,

ς

)

=

+∞



0 p_ˆ

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

wς_∂

+

H

(λ

∂

− λ)

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)

wς_A

+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎛

⎝

∂ς ks(x _,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(



x

,

ω





,

ς

)

⎞

⎠

+

PN

(

x





,

ς

)

⎛

⎝

∂ς kn(x _,ς) kn(x,ς) f

(



x 

_,

_ω

_

_,

_ς

₎

+ ∂

ςf

(



x

,

ω



,

ς

)

⎞

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(6) withwς_∂

=

0 and wς_A

=

∂ςka(x,ς) ka(x,ς) f

eq

₍

_x

_



₎

_{becausek is}

ˆ

_{independentof}

_ς

_{.Becauseoftheiridenticalstructure,wecangather} Eq. (5) and(6) intooneusingthevectorialnotation



w

;

wς



:



f

(



x,

ω



,

ς

)

; ∂

ςf

(

x,



ω



,

ς

)



=

+∞



0 p_ˆ

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)



w∂

;

wς∂



+

H

(λ

∂

− λ)

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)



wA

;

wς_A



+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎧

⎨

⎩

f

(



x

,

ω





,

ς

)

;

⎛

⎝

∂ς ks(x _,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(



x

,

ω





,

ς

)

⎞

⎠

⎫

⎬

⎭

+

PN

(

x





,

ς

)

⎧

⎨

⎩

f

(



x

,

ω



,

ς

)

;

⎛

⎝

∂ς kn(x _,ς) kn(x,ς) f

(



x 

_,

_ω

_

_,

_ς

₎

+ ∂

ςf

(



x

,

ω



,

ς

)

⎞

⎠

⎫

⎬

⎭

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(7)

Thealgorithmic-readingof(7) leadstoAlgorithm2thatevaluates simultaneously f and

∂

ς f . Therecursivenatureofthis algorithmcomes fromthe factthat thefinalbracketsin thescatteringandnull-collisiontermscontain f and

∂

ς f atthe samelocationinthesamedirection.Thefactthattheirsensitivitypartincludesasummationistranslatedintoanalgorithm incrementingtheMonteCarloweightasexplainedinAppendixA.

Simulationexamples. Atthisstage,we designeda null-collisionalgorithm,constructedthecorresponding integral formula-tionandappliedthepropositionof[23,25,26] inastraightforwardmannersothatthealgorithmalsoevaluatessensitivities. We now test thissimulation strategy by evaluating thetransmissivity ofa non-diffusive heterogeneous columnandalso evaluating the sensitivityofthistransmissivity w.r.t.

ς

, a parameterinfluencing the absorptioncoefficient. Hereafter this configurationis calledheterogeneous-slab (see Fig.D.2):



is acolumnoflength L withe



x the normalincomingat

loca-tion y

=

0.The equilibriumdistribution isnull (coldmedium, feq

_≡

_{0). Theboundary conditionsare f}∂

₍

₀

_,

_e

_

x

)

=

0 and

f∂

(

L

,

− 

ex

)

=

finc.Theabsorptionandscatteringcoefficientsare ka

(

x

,

ς

)

= (

ς

−

γ

)

−

atan(α(x−β))+π

2

π/2

+

γ

andks

≡

0. Algo-rithm2isusedtoevaluateboth f

(

0

,

e



x

,

ς

)

and

∂

ς f

(

0

,

e



x

,

ς

)

thatcorrespondtothetransmissivityT anditsderivative

∂

ς T respectively:T

=

f

(

0

,

e



x

,

ς

)/

finc and

∂

ς T

= ∂

ς f

(

0

,

e



x

,

ς

)/

finc.Wechosethisparticularprofileofka,becauseitispossible tocalculateT and

∂

ς T analytically(seethecaptionofFig.D.2).ExampleMonteCarloresults,usingN

=

10000 samples,are comparedtotheanalyticalsolutioninTablesD.5a andD.5b.Thestatisticaluncertaintyisnoted

σ

(the standarddeviation oftheMonteCarloestimator).InTableD.5bwealsoprovidethenumberofsamplesN1%requiredtoachievea1% accuracy. Thesimulationswere madeusingfivedifferentk-profiles

ˆ

(eachoverestimatingka atalllocations),withacceleration-grids,

ˆ

k beinguniformwithineachmesh(seeFig.D.2):

•

fork20%

ˆ

nogridisused:theprofileofk is

ˆ

uniform,equalto1

.

2 timethemaximumka-value.

•

fork1

ˆ

nogridisused:theprofileofk is

ˆ

againuniform,exactlyequaltothemaximumka-value.

•

fork

ˆ

10 thegrid is constructed insuch a way that acrosseach meshthe variations ofka are 1

/

10 of the maximum ka-value,andtheprofileofk is

ˆ

uniformwithineachmesh,exactlyequaltothemaximumka-valueinsidethemesh.

•

fork100

ˆ

andk1000

ˆ

thegridisconstructedthesamewaywith1

/

100 and1

/

1000 variationrespectively.

The transmissivity results ofTable D.5a confirm that the estimation of T is insensitive to the adjustment of thek-field

ˆ

(only thecomputation timeis affected).Butthe sensitivityresultsofTableD.5bclearly showtheopposite:the statistical convergence isworse whenk is

ˆ

closeto k and thenumber of samplesrequired to reach a givenaccuracy level can be

risen up toinfinity whenmatchingk to

ˆ

k exactly. Thisisthe pathologicalbehaviour thatwe announcedin introduction: sensitivitiescannotbeevaluatedaccuratelywhenusingaccelerationgridsreducingthenumberofvirtualcollisions. Thevarianceofthesensitivityestimate. Forabetterunderstandingofthisbehaviour,westudiedahomogeneous-slab forwhich the variance oftheMonte Carloestimatecan be calculatedanalytically. Thiscaseisidentical tothe previous one (trans-missivityofa purelyabsorbingcolumn) butnowk

=

ka isuniform:ka

(

ς

)

≡

ς

,T

=

exp

(

−ς

L

)

and

∂

ς T

= −

LT .Ofcourse, there isno needtomake useofa null-collisionalgorithmassoonask is uniform. Weonly doitfortheoretical reasons (with k

ˆ

>

k uniform).Thisallowsusto fullyidentifythereasonswhythe varianceof thesensitivityestimate riseswhen reducingkn

= ˆ

k

−

k.Thismaysoundtrivialassoonaswhenencounteringanull-collisionevent,theMonteCarloweightof the sensitivityalgorithmincludes afactor ∂ςkn(x,ς)

kn(x,ς)

=

1

/

kn (see Eq. (7)),butreducingkn alsoreducesthenumberofsuch

null-collisionoccurrences.Thismayleadtoacompensation,maintainingthevarianceatafinitevalue.Thedevelopmentsof AppendixB.1showtheopposite:thestatisticaluncertaintyisindeed

σ

∂ς T

=



L2_e−kaL

kn+1/L kn



−

L2_e−2kaL

√

N (8)

Fig.D.3illustrates themeaningofthisdependenceof

σ

∂ςT withtheproblemparameters.Inthisidealisedcase,lookingat the behaviour of such an algorithm applyingsensitivity-evaluation techniquesin a straightforward manner,the difficulty is well identified: when kn

ˆ

k approacheszero,the number ofsamples requiredfor a 1% accurateevaluation ofthe sensi-tivitytends toinfinity(seeFig.D.3c).Thisfigurealsodisplaysthebehaviour ofan algorithmimplementingtheverysame sensitivity-evaluationtechnique,butwithouttheuseofnull-collisions(whichispossiblehereinthisidealiseduniformcase). Withoutnull-collisions,therelativevalueofthestandarddeviationofthesensitivity-estimate(Fig.D.3b)isidenticaltothat ofthemainquantity(thetransmissivity-estimate,Fig.D.3a).Thisisanidealbehaviour:thesensitivityisestimatedwiththe samerelativeaccuracyasthatofthemainquantity.Altogetherinthissimpleexample,weseethatevaluatingsensitivities canbeperfectlycostlessbefore usingnull-collisionsandmaybecomepathologicalwhennull-collisionsareintroduced.

Notethat inthegeneralcase, evenwithoutnull-collisions,evaluatingsensitivities canbe truly difficult.Understanding the relativevariance ofsensitivityestimatesandcomparing themtotherelative varianceofthe algorithmestimatingthe mainquantitywasindeedoneofthemainconcernsoftheinitialworkofDe Lataillade[23].Essentially,seriousdifficulties arise assoon as thescattering optical-thickness ishigh. The objectiveof the presentpaper is not at all to address this specificissue:attheendofthefollowingsection,whenanalternativesolutionwillbeproposedforevaluatingsensitivities innullcollisionsalgorithms,theproblemsassociatedtohighlyscatteringmediawillremainunsolved.

3. Analternativeapproach

The precedingsection identifiesconvergencedifficultieswhenevaluating sensitivitiesusingnull-collisions.Theses diffi-cultiesarenotassociatedtothestandardsensitivity-evaluationalgorithmitself:consideringslabtransmission,wehaveseen that when wedo notmake useofnull-collisions, thesensitivity-evaluationalgorithm convergesaswell asthealgorithm evaluating themainquantity.Sotheobserveddifficultiesareonlytheconsequencesofintroducingvirtual-collisionners.At thisstage,null-collisionalgorithmsappearthereforeasperfecttoolsforhandlingheterogeneousfields,butareincompatible withthesimultaneousevaluationofsensitivities.

We haveseen thatthisproblemisrelatedtotheterm _k1

n appearingintheMonteCarloweightofthesensitivity algo-rithm.Atwhichstagedidthistermappearandcanwebypassthisstep?Clearly, _k1

n appearedwhenderivatingwith

ς

the null-collisionprobability PN

(

ς

)

=

1

−

PA

(

ς

)

−

PS

(

ς

)

,withPA

(

ς

)

=

ka

(

ς

)/ˆ

k andPS

=

ks

(

ς

)/ˆ

k.Afirstwaytosuppressthis

1

kn termconsistsinmakingk dependent

ˆ

on

ς

.Thisisalwayspossiblebecausek is

ˆ

afree parameterandwecantherefore adjustit tothevariationofka

(

ς

)

+

ks

(

ς

)

sothat PN doesnotdependon

ς

anymore.We firsttestedthissolutionandit proved itselfalreadyquitepractical:thecorrespondingdetailsareprovidedinAppendixC.Butwefinallyretainedanother algorithm more efficientin all thecases that were studied,starting from theintegral solution ofthe original Boltzmann Eq. (1),i.e.priortotheintroductionofvirtual-collisionners.Theideaconsistsinfirstdesigningan algorithmevaluating si-multaneously f and

∂

ς f asiftheheterogeneityofthefieldcouldbehandledwithoutdifficulty andonlyintroducenull-collisions inasecondstage.Forthis,wecansimplyrewriteEq. (5) withkn

=

0 (novirtualcollisionners):

f

(



x,

ω



,

ς

)

=

+∞



0 p

(λ)

d

λ

⎛

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

w∂

+

H

(λ

∂

− λ)

⎛

⎜

⎝

PA

(



x

,

ς

)

wA

+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω

f

(

x





,

ω





,

ς

)

⎞

⎟

⎠

⎞

⎟

⎟

⎟

⎠

(9)

TheonlydifferenceswithEq. (5) arethat

•

PN

=

0,

•

therandomvariable

ˆ

ofprobabilitydensityp_ˆ

(λ)

= ˆ

k

(



x

)

exp

−



λ 0 k

ˆ

˜

x



d

˜λ



(thefreepathinthek-field)

ˆ

isreplaced withtherandomvariable

ofprobabilitydensityp

(λ)

=

ke

(



x

,

ς

)

exp

−



λ

0ke

(

x

˜

,

ς

)

d

˜λ



(thefreepathintheoriginal ke-field).

Thisequationcan thenbederivatedwithrespectto

ς

andmultiplied/dividedby eachoftheprobabilitiesandprobability densityfunctionsthatdependon

ς

(exactlythesamewayEq. (7) wasconstructedfromEq. (5))togive

∂

ςf

(



x,

ω



,

ς

)

=

+∞



0 p

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

⎛

⎝−

w∂ λ∂



0

∂

ςke

(x

l

,

ς

)

dl

+

wς∂

⎞

⎠

+

H

(λ

∂

− λ)

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)

⎛

⎝−

wA λ



0

∂

ςke

(x

l

,

ς

)

dl

+

wςA

⎞

⎠

+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

−

λ



0

∂

ςke

(x

l

,

ς

)

dlf

(



x

,

ω





,

ς

)

+

∂ς ks(x,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(

x





,

ω





,

ς

)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(10)

Amainpoint ofthe presentpaperisthat thisintegral equation,althoughit wasderived thesamewayasEq. (7),cannot be interpretedinalgorithmic terms: theintegral pattern



∂

ς kedl isnotyettransformed intoa statisticalexpectation. An additionalrandomgenerationwillberequired.Atthisstageletusintroducean arbitraryrandomvariable

L

ofprobability densityfunction p_L andwrite



∂

ς kedl

=



p_L

(

l

)

dl∂ςke

p_L(l).ReportingthisintoEq. (10) andusing



p_L

(

l

)

dl

=

1 leadsto

∂

ςf

(



x,

ω



,

ς

)

=

+∞



0 p

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

λ∂



0 p_L

(

l

|λ

∂

)

dl



−

w∂

∂

ςke

(x

l

,

ς

)

p_L

(

l

|λ

∂

)

+

wς_∂



+

H

(λ

∂

− λ)

λ



0 p_L

(

l

|λ)

dl

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)

−

wA∂ς k_pLe(_(lx_|λ)l,ς)

+

wς_A



+

PS

(

x





,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎛

⎜

⎜

⎜

⎝

−

∂ς ke(xl,ς) p_L(l|λ) f

(

x





,

ω





,

ς

)

+

∂ς ks(x,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(



x

,

ω





,

ς

)

⎞

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(11)

Atthisstage,null-collisionshavenotbeenintroduced.Therefore,thealgorithmicreadingofEq. (11) wouldnotbepractical assoonastheke-field isheterogeneous:the difficultywouldcome fromthe samplingof

.The objectiveofintroducing null collisions will therefore be to replace

with another path-length

ˆ

, shorter in average but easy to sample, and compensatethetoomanycollisionsbythefactthatsomeofthemarenull.However,thisnotastrivialasinthealgorithm forthemainquantitybecauseofthenewrandomvariable

L

thatweneededtointroducewhentransformingEq. (10) into

Eq. (11) (transformingitintoanexpectation).Indeed

∂

ς ke needstobeintegratedalongthewholepath,nowincludingnull collisions. Instatisticalterms,thismeansthat therecursivityofthepath-sampling algorithmisonlyinsurediftheMonte Carloweightassociatedtonullcollisionsincludestheterm

−∂

ς ke

(

xl

,

ς

)/

pL

(

l

|λ)

f

(



x

,

ω



,

ς

)

,exactlylikefortruescattering

events:

∂

ςf

(

x,



ω



,

ς

)

=

+∞



0 p_ˆ

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

λ∂



0 p_L

(

l

|λ

∂

)

dl



−

w∂

∂

ςke

(x

l

,

ς

)

p_L

(

l

|λ

∂

)

+

wς_∂



+

H

(λ

∂

− λ)

λ



0 p_L

(

l

|λ)

dl

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)

−

wA∂ς k_pLe(₍x_l|λ)l,ς)

+

wςA



+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎛

⎜

⎜

⎜

⎝

−

∂ς ke(xl,ς) p_L(l|λ) f

(

x





,

ω





,

ς

)

+

∂ς ks(x,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(



x

,

ω





,

ς

)

⎞

⎟

⎟

⎟

⎠

+

PN

(



x

,

ς

)



−

∂ς ke(xl,ς) p_L(l|λ) f

(



x

,

ω



,

ς

)

+ ∂

ςf

(



x

,

ω



,

ς

)



⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(12)

Equations (5) and(12) havenowasimilarstructure:allthesamplesusedtoevaluate f canalsobeusedfortheevaluation of

∂

ς f .Butinordertocompletetheevaluationofsensitivity,we mustaddonesample (of

L

)percollision.Thankstothis similar structure,we cangather them intoasingle vectorialwriting (exactly thesamewayEq. (7) was constructedfrom Eq. (5) and(6)):



f

(

x,



ω



,

ς

)

; ∂

ςf

(



x,

ω



,

ς

)



=

+∞



0 p_ˆ

(λ)

d

λ

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

H

(λ

− λ

∂

)

λ∂



0 p_L

(

l

|λ

∂

)

dl



w∂

; −

w∂

∂

ςke

(x

l

,

ς

)

p_L

(

l

|λ

∂

)

+

wς_∂



+

H

(λ

∂

− λ)

λ



0 p_L

(

l

|λ)

dl

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

PA

(



x

,

ς

)



wA

; −

wA∂ς k_pe(xl,ς) L(l|λ)

+

wςA



+

PS

(



x

,

ς

)



4π pS

(

− 

ω



| − 

ω

)

d

ω



⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

f

(



x

,

ω





,

ς

)

;

⎛

⎜

⎜

⎜

⎝

−

∂ς ke(xl,ς) p_L(l|λ) f

(

x





,

ω





,

ς

)

+

∂ς ks(x,ς) ks(x,ς) f

(



x 

_,

_ω

_



_,

_ς

₎

+ ∂

ςf

(



x

,

ω





,

ς

)

⎞

⎟

⎟

⎟

⎠

⎫

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎭

+

PN

(



x

,

ς

)



f

(



x

,

ω



,

ς

)

;



−

∂ς ke(xl,ς) p_L(l|λ) f

(



x

,

ω



,

ς

)

+ ∂

ςf

(

x





,

ω



,

ς

)



⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(13)

The algorithmic-readingof(13) leadstoAlgorithm3that isanalternativetoAlgorithm2forevaluatingsimultaneously f and

∂

ς f .AsexplainedinthealgorithmicreadingofEq. (7),therecursivenatureofthisalgorithmcomesfromthefactthat thefinal bracketsinthescatteringandnull-collisiontermscontain f and

∂

ς f atthesamelocationinthesamedirection. The fact that their sensitivitypart includes a summation is translated into an algorithm incrementing the Monte Carlo weightasexplainedinAppendixA.Itisinterestingtonotethattheintegral



p_L

(

l

)

dl

∂

ς ke

(

xl

,

ς

)/

pL

(

l

)

canbemoreorless

difficultto evaluatedependingontheprofileof

∂

ς ke.Butthiscanbe easilyhandledusingimportancesamplingbasedon thek-adjustment

ˆ

grid,asexplainedinAppendixD.

4. Simulationsusingthealternativeapproach 4.1. Transmissivityofapurelyabsorbingcolumn

ApplyingthealternativeapproachofSec.3totheevaluationofcolumn-transmissivitiesleadstothecontentofFig.D.3d andTableD.5c.Forthe homogeneous-slab configuration,Fig. D.3showsthat notonly thepathologicalbehaviour ofSec.2

isremoved, butthesensitivityisestimatedwithastatisticaluncertaintythatisperfectforasimultaneous evaluation:its dependenceontheparametersoftheproblemisidenticaltothatofthemainquantity.Asabove,inthisverysimplecase, thisuncertaintycanbeexpressedanalytically(seeAppendixB.2)andindeed

σ

∂ς T

∂

ςT

=

σ

T T

=

!

1

−

e−kaL

√

N

Fortheheterogeneous-slab configuration,theuncertaintycannotbepredictedtheoretically,buttheconclusionsofFig.D.5are identicaltothoseofFig.D.3:intermsofrelativeaccuracy,theconvergencerateisequaltothatofthealgorithmevaluating themainquantity.Itisthereforestrictlyindependentoftheadjustmentofk to

ˆ

k.Theuseofanaccelerationgriddoesonly whatweexpect:itreducesthenumberofnull-collisionsbutdoesnotimpactthevarianceanymore.

4.2. Fullradiativetransferina3Dconfiguration

In[2],a cubicbenchmark configurationwas usedtotest null-collisionalgorithms whendealing withthree-dimension highly-heterogeneous fields for all ranges of optical thickness and single-scattering albedo. We here make use of the sameconfiguration, namedheterogeneous-cube hereafter, in orderto test ouralternative approach with3D radiation(see Fig.D.4):

•

radiationismonochromatic;

•

thecubeisofside2L,with0K blackfaces;

•

theinside-temperaturefieldissuchthat feq variesfrom feq

=

fmaxeq (atthecentre ofthefaceatx

= −

L)to feq

=

0 (at x

=

L and

(

y

= ±

L

,

z

= ±

L

)

)andmimicstheshapeofflame: feq

(

x

,

y

,

z

)

=

η

(

x

,

y

,

z

)

fmaxeq (seethe

η

profileinFig.D.4);

•

thefieldsofabsorptionandscatteringcoefficientsfollowsthesamespatialdependence:ka

(

x

,

y

,

z

)

=

η

(

x

,

y

,

z

)

ka,maxand

ka

(

x

,

y

,

z

)

=

η

(

x

,

y

,

z

)

ks,max;

•

Thesingle-scatteringphasefunctionisthatofHenyey-Greensteinwithauniformvalueoftheasymmetryparameterg;

• ˆ

k isadjustedtok usingaregular cubic-grid(k uniform

ˆ

within eachmesh):theonly parameterfork is

ˆ

thereforethe numberofmeshperdirection.

Theevaluated quantity A

(

x

,

y

,

z

)

isthestationarynet-powerdensityandthefreephysicalparameters areka,maxL,ks,maxL and g. In TableD.7 we reproducethe computations ofTable 1 in[2], i.e.testingwide ranges of optical thicknesses but fixing g

=

0 (isotropicscattering).Inthesametablewealsoprovidetwosensitivities,

∂

ka,max and

∂

ks,max,that weevaluated simultaneously with A.As in[2], althoughthey are notdisplayed, we checkedthat simulationresultswithnon-isotropic scatteringleadtotheexactsameconclusions.

Theseconclusions arevery similartothose reachedon theslab-transmissivityexample:TableD.8 highlightsthesame featuresasFig.D.5,thestandarddeviationofAlgorithm3beingindependentofk,

ˆ

whereasitincreaseswhenk gets

ˆ

close toke forAlgorithm2.

5. Conclusion

The simulation examplesof the precedingsection show that the solutionproposed inSec. 3 ispractical. Sensitivities can be accurately evaluated even when using null-collision algorithms. We still face convergence difficulties for highly-scatteringmedia, butthisisan open question identifiedinall linear-transport physics,independentlyof theuseof null-collisions.

Thewaywebypassedtheconvergencedifficultiesspecifictonull-collisionsisinrupturewiththeprincipleofevaluating sensitivities

∂

ς A simultaneouslywiththemainquantity A:inallprevious works,thetwoevaluationsweretruly simulta-neousinthesensethattheverysamesampleswereusedinthe A and

∂

ς A MonteCarloalgorithms.Here,allthesamples usedtoevaluate A arealsousedfortheevaluationof

∂

ς A,butinordertocompletetheevaluationof

∂

ς A,someadditional randomvariablesmustbesampled.Thesensitivityevaluationhasthereforeaspecificcomputation-cost.Inallourtest-cases, thiscostwasvery smallcomparedtothetotalcomputation cost,andconsidering thewiderangeof testedopticaldepths