Monte-Carlo and sensitivity transport models for domain deformation

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Monte-Carlo and sensitivity transport models for

domain deformation

P. Lapeyre, Stéphane Blanco, Cyril Caliot, J. Dauchet, Mouna El-Hafi,

Richard Fournier, Olivier Farges, Jacques Gautrais, Maxime Roger

To cite this version:

P. Lapeyre, Stéphane Blanco, Cyril Caliot, J. Dauchet, Mouna El-Hafi, et al.. Monte-Carlo and

sensi-tivity transport models for domain deformation. Journal of Quantitative Spectroscopy and Radiative

Transfer, Elsevier, 2020, 251, pp.1-13/107022. �10.1016/j.jqsrt.2020.107022�. �hal-02639249�

Monte-Carlo

and

sensitivity

transport

models

for

domain

deformation

P.

Lapeyre

a,∗

_,

_S.

_Blanco

b

_,

_C.

_Caliot

a

_,

_J.

_Dauchet

c

_,

_M.

_El

_Hafi

d

_,

_R.

_Fournier

b

_,

_O.

_Farges

e

_,

J.

Gautrais

g

_,

_M.

_Roger

f

a PROMES CNRS, Université Perpignan Via Domitia - 7, rue du Four Solaire, Font Romeu Odeillo 66120, France b LAPLACE, UMR 5213 - Université Paul Sabatier, 118, Route de Narbonne, Toulouse Cedex 31062, France c Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, Clermont-Ferrand F-630 0 0, France

d Université Fédérale de Toulouse Midi-Pyrénées, Mines Albi, UMR CNRS 5302, Centre RAPSODEE, Campus Jarlard, Albi CT Cedex, F-81013, France e LEMTA - UMR 7563 - Université de Lorraine, Vandœuvre-lès-Nancy, France

f CETHIL, UMR 5008, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, Villeurbanne F-69621, France

g Centre de Recherches sur la Cognition Animale (CRCA), Centre de Biologie Intégrative (CBI), Université de Toulouse, CNRS, UPS, France

Keywords:

Monte Carlo Domain deformation Sensitivity transport models Sensitivity estimation Shape sensitivities

a

b

s

t

r

a

c

t

Weaddressthequestionofevaluatingshapederivativesofobjectivefunctionsforradiative-transfer engi-neeringinvolvingsemi-transparentmedia.AfterrecallingthestandardMonte-Carloapproachto sensitiv-ityestimationanditscurrentlimitations,anewmethodispresentedforthespecificcaseofgeometrical sensitivities.Thismethodisthentestedonconfigurationswithmultiple-scatteringandabsorbing (non-emitting)semi-transparentmedium.Anewgeometricalsensitivityalgorithmispresentedwithfulldetails inordertoextend,onseveralexamples,itsimplementationincomplexgeometries.

1. Introduction

Theoptimizationofengineeringprocessesinvolvesanobjective function drivenby physicalmechanisms.Inthefield of geometri-caldesign,shapeoptimizationmodelsarerequiredtofindthe ex-tremum ofan objectivefunction, denoted hereas J

(

π



)

_, that de-pends on a vector of design parameters

π

. Among optimization methods, the gradient descent method can be usedto find a lo-calextremum,andstochasticmethods(geneticalgorithms,particle swarm optimization)can be usedto look fora global extremum

[7,9,20].1_{Inanycase,thederivativeofJ}

₍

_π

_

₎

_{withrespectto}

_π

_{ isa}

valuable piece ofinformationforthe optimizationofengineering processes[2,5,15,22].

Our work addressprocess configurations that havea complex geometryandwheretheradiativetransfersareamajorcomponent of heat transfer models, and therefore of the objective function

[8].Classically, the Monte-Carlomethod is regardedasa method of reference in such cases because it remains unbiased even for configurations withhigh level of complexity in radiative

proper-∗ _{Corresponding author.}

E-mail address: [email protected] (P. Lapeyre).

1 Some of the references are PHD theses in french. However, every information

that is essential to this work is fully reported in the present text.

ties andgeometry [6,8,10,27]. To carry out the computation, the models are statedin an integral formulation, whichis then con-sidered as an expectation, which in turn is estimated by sam-pling,yieldingan unbiasedestimateoftheexpectation aswellas itsconfidenceinterval.Consideringsensitivities,awell-known ad-vantageoftheMonte-Carlomethod isitsabilityto estimatesuch expectation andits derivatives by usingthe very samesampling, avoiding additional computation time [1,11,14,17–19,26]. This ad-vantage has however some limitations when sensitivity to geo-metricalparametersisconsideredbecausetheintegralformulation ofthe modelraises formalization andimplementationdifficulties

[25]. Those limitations led us to consider the geometrical sensi-tivitiesrestartingfromthelocaldifferentialequations,namelynot formulatingfirstthephysicalmodelsasintegrals.

Several studies have addressed the question of finding shape derivatives for linear physical models through shape sensitivity analysis[24]. Inthe field ofdeterministic approach, [21] propose ageneralization ofthesensitivitymodelstotheradiative transfer equation withspecular anddiffuseradiative boundary conditions inordertosolvetheshapeoptimizationproblems.Inthepresent work,weextendthequestionofevaluatingshapederivativeswhen usingtheMonte-Carlomethodasa methodofreference. Towards thisgoal,weproposeaconstructionoftheshapesensitivity mod-els that overcomethe limitations mentioned above, still

preserv-ingtheintuitive aspectofthemethodanditsunbiasedprediction power.Inshort,weconsiderthegeometricalsensitivityasa quan-titythatistransported.

We show howtoformulate itstransport model[3], aswell as how to translate the boundary conditions of the physical prob-lemintothecorrespondingboundaryconditionsforthesensitivity transportmodel. Then andonly then, theMC methodis usedto drawestimatesfromthisnewtransportmodel.Asaresult,wegain thatgeometrical sensitivities canbe solved inconfigurations that were previously unsolvable by classical differentiation of Monte-Carlointegrals.Asaside-effect,welosethewell-knownadvantage ofestimating sensitivities using the same sampling— yet atthe boundariesonly.

In Section 2, we will briefly review the existing Monte-Carlo methodusedtoestimateanobjectivefunctionJ(

π

)andits deriva-tive

∂

_πJ(

π

)andthedifficultiesraisedbygeometrydeformation.In nextsection,wewillintroduceournewsensitivitymodel.Wewill usethereanacademicconfigurationwithanuni-dimensional geo-metricalparameter

π

inordertoexposeitinfulldetails.

Finally,thegeometricsensitivitymodelwillbeappliedonfour differentconfigurations,presentingthescope ofourdevelopment so far for complex geometry, reflective boundary conditions and curvesurfaces.

2. ClassicalapproachforsensitivityinMonteCarlomethods In this section, we briefly summarize the sensitivity calcula-tion as described in [12] and [17]. Any radiative quantity J ex-pressed as a linear integral function can be estimated by the Monte Carlo method as long as it is stated as an expectation (Eq. (1) of the first insert). Formulated as such, the objective function might depend on the parameter

π

through the den-sity probability functions, the Monte Carlo weight function and the integration domain. One of the advantages of the Monte Carlo method is that the sensitivities of the objective function canalways bedefinedfromthesame randomvariable(same do-main definition and same probability density functions) as the objective function (Eq. (2) of the first insert). Numerically, this means that the objective function and its sensitivities are esti-matedsimultaneously from the sameset of the random variable samples.

However, when theintegrationdomain isparameterized by

π

(e.g. due to geometry deformation), two major difficulties have beenpointed out in previous works [17,25]. The first one lies in constructingtheformalexpressionofthedeformationvelocity vec-tor inside the domain (see Eq. (4) in the first insert). The sec-ond one arises from the implementation of this formal expres-sion when dealing with multiple scattering and multiple reflec-tion.These two difficulties jeopardizethe extension of the exist-ing method to configurations withcomplex geometry. Therefore, wepresentan alternativeproposalwherethegeometric sensitivi-tiesareobtainedthroughthedifferentiationoftheradiative trans-fermodelitself.

As a last point, we recall that the typical objective functions forgeometrical designof(stationary)radiativetransfer, when ad-dressedwithMonteCarlo,areintegralsoverfrequency,over posi-tionsxandoverdirectionsuoflinearfunctionsofthe monochro-maticintensityI

(

x,u

)

.TheMonteCarlo algorithmsthereforestart bysamplingthecorrespondingspacesandthealgorithmcompletes byestimatingI

(

x,u

)

atthesampledfrequency.Thefourexamples that willbe presentedbelow willinclude such preliminary sam-pling,but all ourformal developmentswill focus on the expres-sionsforIand

∂

_πI.

ClassicalapproachtoestimatesensitivitiesinsideaMonte Carloalgorithm

The random variable



, established over a multiple di-mensiondomainDaccordingtoaprobabilitydensity func-tionp_

(

γ

)

,isidentifiedsuchas:

J

(

π

)

= 

D(π )

p_

(

γ

,

π

)

d

γ

wˆ

(

γ

,

π

)

=E



wˆ

(

,

π

)



(1)

where wˆ isa randomvariable functioncalledthe Monte Carloweightfunction.Whentheintegrationdomainis inde-pendentof

π

itisshownin[12]thatthesensitivityofJ

(

π

)

withrespectto

π

is:

∂

πJ

(π )

=  D p _

(γ

,

π )

d

γ



∂

πp _

(γ

,

π )

p _

(γ

,

π )

wˆ

(γ

,

π )

+

∂

πwˆ

(γ

,

π )



(2)

∂

πJ

(

π

)

isan expectationofanewrandom variable func-tionwˆ_π ofthesamerandomvariable



.

ˆ w_π

(

γ

,

π

)

=



∂

πp_

(

γ

,

π

)

p_

(

γ

,

π

)

wˆ

(

γ

,

π

)

+

∂

πwˆ

(

γ

,

π

)



(3)

When the integration domain does depend on

π,

it is shownin [16,17]that thesensitivityofJ

(

π

)

withrespectto

π

canbewrittenas:

∂

πJ

(

π

)

=  D(π ) p_

(

γ

,

π

)

d

γ

×



∂

πwˆ

(

γ

,

π

)

+wˆ

(

γ

,

π

)

∂

π_pp

(

γ

,

π

)



(

γ

,

π

)

+

∇

.



ˆ w

(

γ

,

π

)

p_

(

γ

,

π

)



v

π



p_

(

γ

,

π

)



(4)

where

v

π denotethedeformationvelocity.Thisvelocityis constrainedonly atthe integration domain boundaries, and mustbeextendedarbitrarilyalloverthedomain.

3. Analternativeproposal:Thesensitivitytransportmodel In this section, the intensity geometric sensitivity

∂

_πI is in-troducedatthemodelinglevel.Thatis,insteadofderivatingI(

π

) stated asan integral, we view the sensitivityas a quantity with itsowntransportmodel.Thismodelisobtainedbyderivatingthe radiativetransferequation(RTE).Inthepresentwork,weconsider the caseofa scattering, absorbingnon-emissive medium (adding mediumemissionwouldnotconveyanyadditionalidea).

Denotingthedomain



,andtheboundary

∂



(

π

),theRTEthen reads:



 u.

_∇

_I

₍

_x,u,

_π

₎

₌₋

₍

_ka+ks

₎

_I

₍

_x,u,

_π

₎

+ks 4πp

(

u

|

u

)

duI

(

x,u,

π

)

forx∈



I

(

x,u,

π

)

=I_{∂(π )}

(

x,u,

π

)

for x∈

∂

(

π

)

and u.n>0 (5)

TheRTEdifferentiationleadsto



 u.

_∇

_s

₍

_x,u,

_π

₎

₌₋

₍

_ka+ks

₎

_s

₍

_x,u,

_π

₎

+ks 4πp

(

u

|

u

)

dus

(

x,u,

π

)

forx∈



s

(

x,u,

π

)

=s_{∂(π )}

(

x,u,

π

)

for x∈

∂

(

π

)

and u.n>0 (6)

wheres

(

x,u,

π

)

=

∂

πI

(

x,u,

π

)

.Sincethegeometricalparameter

π

only appears at the boundary conditions, it does not affect the operators of Eq. (5). We can therefore notice that the sensitivity transport equation isidenticaltothe intensitytransportequation in the domain



. It implies that, in the domain, the algorithms usedto solvethe RTEcanbe usedunchangedto solvethe

sensi-tivity transport equation. Thesensitivity transport andthe inten-sitytransportonlydifferoverboundaryconditions.Therefore sen-sitivity can be regarded asa quantity that is transported by the sametransport modelasthe intensity,onlychangingthe sources atboundaries.Thequestionthatremainsisthentodeterminethe sourcesofsensitivityatboundaries.Weshowbelowthat sensitiv-ityboundaryconditionsinvolveacouplingwithintensityand de-pendontheradiativepropertiesoftheboundary(Eq.(9)) sothat thesensitivitytransportcannotbe solvedonits own.Inthe con-textofMonteCarloformulation,thiswillresultinanintegral for-mulationofthesensitivitywhichincludesEqs.(5)and(6)(details inSection4),whichrequiresinturntosolvesystems(5)and(6)in parallel.

Generalexpressionforsourcesofsensitivityatboundary

Let us denote I_B the field of the incoming radiative intensity at the boundary. All Monte Carlo approachesrely on an integral formulationoftheRTEsolutionoftheform

I

(

x,u,

π

)

= 

D(π )

p_

(

γ |

x,u

)

d

γ

wˆ

(

γ

,I_B

)

(7)

where p_

(

γ |

x,u

)

istheprobability densityofpath

γ

propagating the radiativesources fromthe boundarythrough themedium up toxindirectionu.

Identically,letusdenotes_Bthefieldoftheincomingsensitivity attheboundary.The MonteCarlo algorithmevaluating the inten-sity is straightforwardly translated into a Monte Carlo algorithm evaluatingthesensitivitybasedonthefollowingintegral formula-tioninwhichonlytheboundarysourcesarechanged:

s

(

x,u,

π

)

= 

Dπ(π )

p_π

(

γ

_π

|

x,u

)

d

γ

_π wˆ

(

γ

_π,s_B

)

(8)

The absorbing and scattering medium properties are preserved during theprocess andtherefore the spaceof optical paths



is identicalto thatofthesensitivitypaths



_π.The mainpoint here isthat thesensitivityboundaryconditionsare transportedbythe samepathsastheintensityboundaryconditionsandwecan there-foreusethesamesetofsampledpathstoestimatesimultaneously bothquantities.

Howeveratthispointtheboundaryconditionsofthesensitivity modelareunknown.Themainobjectiveofconstructingthe sensi-tivitymodelistoformulatethoseboundaryconditions:the sensi-tivitiesassociated to thegeometrical perturbations forall incom-ingdirectionsattheboundary.Preservingthecompletegenerality of the emission, absorption,reflection andscattering phenomena compatible with the radiative transfers equation, theseincoming sensitivities canbewrittenundertheformofalinearapplication

Lof

• the sensitivitiess

(

x_,u_,

π

)

inall outgoing directionsu (as for any surface reflection problem, only hereit is the sensitivity thatisreflected),

• theintensitiesI

(

x_,u_,

π

)

alsoinalloutgoingdirectionu,

• and the black-body intensity at the local temperature

Ib

(

T

(

x,

π

))

.

s _∂(π)

(

x,u,

π

)

=L

s

(

x,u  ,

π

)

,I

(

x,u  ,

π

)

,Ib

(

T

(

x,

π

))



withu.n> 0

(9)

Themainpoint hereisthefactthat Iappearinthislinear ap-plication. This impliesthat via the boundary condition, the sen-sitivity model is coupled with the intensity model: the sources of sensitivity have to be evaluated, from the outgoing intensity (Fig.1),beforebeingpropagatedintothedomain.AppendixAand

Appendix B provide practical examples of these coupling at the

Fig. 1. Schema corresponding to the sensitivity boundary condition ( Eq. (9) ).

boundary. These examples correspond to the four configurations studiedinnextsections.

AroadmaptothesensitivitytransportmodelintheMonte Carlocontext

• The sensitivity transport equation is well established andfamiliar:itrepresentsthesametransportphenomenaas fortheintensity(Section3).

• Regardingtheboundaryconditions,asfarasour config-urationsareconcerned:

sources of sensitivities appear at geometrically modified boundaries(Section5):

theimpactofthevolumeofsemi-transparentmedium added orwithdrawnwhen modifying the surface; thetermu_.

_∇

_{I ofEq.(18)isdirectlyderivedfromthe}

radiativetransferequation(Section5,AppendixA). sources of sensitivity appear as Dirac distributions

where intensity displays discontinuities at the boundary(Section5,AppendixA)

everywhere the boundary condition is reflecting for the intensity,itisaswellreflectingforthesensitivity • Fromanalgorithmicpointofview:

well-identified difficulties arise from sources expressed as Dirac distribution in location and/or direction (Section5),

the coupling between sensitivity transport equation and intensity transport equation through the boundary conditionsisdealtwithstandarddoublerandomization technique(Section4).

Inconfiguration 1,the sourceof radiationis ablack cylinder inside a square cavity filled withabsorbing andscattering mate-rial.The objectivefunction isthelocal absorptionpower andthe geometricparameter

π

actson thesizeofthesquare cavity.This configuration isdiscussed infull details in Section 3 in order to illustratethe entireprocess ofdesigning ashape sensitivity algo-rithm: expressing the sensitivity sources (as function of the in-coming intensities); choosing a Monte Carlo algorithm propagat-ing thesesources asif they were known (according to the stan-dard practice of Monte Carlo in radiative transfer); coupling it withaMonteCarloalgorithmevaluatingI.Inthefollowingsection

(5)three other configurations are successively discussed. Config-uration2 is verysimilar to configuration1 butnow thesquare cavityisemitting witha temperaturediscontinuity. Thisleads to thedefinition ofa new type ofsensitivity sources:Diracsources atthediscontinuitylocation.Configuration3dealswiththesame familyofsensitivitysourcesbuttheyarenowrelatedtoreflection atgeometricallymodifiedboundariesandthe techniqueisscaled uptorealisticthermalsolarplants.TheDiracsourceswillherebe attheedgeofheliostatsofincreasingsizes. Withconfiguration4 wegobacktosemi-transparentmedia,still withcomplex geome-triesbutnowconcentratingonthequestionofevaluatingsources atalteredcurvedsurfaces.

Fig. 2. On the left, the square cavity geometry filled by a semi-transparent medium, lightened by an emissive cylinder at its center. On the right, the dashed line represents the uniform scaling of the cavity. The solid multiple-scattering path is a typical realization of a radiative path use in a Monte Carlo algorithm evaluating the local absorption power at  x0 . The same path is used to evaluate the sensitivity. The dotted multiple-scattering path illustrates how we translate in Monte Carlo terms the coupling of the

sensitivity model with the initial radiative transfer model: the incoming sensitivity at  xγ requires the evaluation of the outgoing intensity at the same location and thanks to double randomization, only one single sample of a standard Monte Carlo path is required to complete the sensitivity realization.

4. Foundationalalgorithm(configuration1)

Here, weexpose thealgorithmic principlesofthemethod, us-ing,forthesakeofclarity,afairlysimpleconfigurationwitha sim-plegeometricalalteration andhomogeneous mediumand bound-aries.

Let us consider a square cavity containing a gray semi-transparentabsorbing, scattering,butnon-emitting medium. Cav-itydimensionsaresetalongthexandyaxesandareinfinitealong thezaxis.Atthecenterofthesquareaninfiniteisothermal black-bodycylinderattemperatureT.Thesquarewallsarealsoblackbut emitnoradiation(Fig.2).Thesemi-transparentdomainisdenoted as



,anditsboundary

∂



(

π

)isconstitutedbythecavity bound-aryR

(

π

)

andthecylinderemittingboundaryF.Thegeometrical parameter

π

issettingthecavityboundaryR

(

π

)

:thedeformation isa colinear translationof R

(

π

)

according to the normalvector



n_{R(π )} resulting in an uniformscaling of the cavity. The objective functionisthelocalabsorptionpowerJ

(

x0,

π

)

atalocationx0and

we aimatevaluating its derivative

∂

_πJ

(

x0,

π

)

withrespect to

π

.

BothareintegralsovertheunitsphereofIands:

J

(

x0,

π

)

=  4π ka I

(

x0,u0,

π

)

du0 (10)

∂

πJ

(

x0,

π

)

=  4π kas

(

x0,u0,

π

)

du0 (11)

In the Monte Carlo context, these integrals will be evaluated by firstsamplinga directionu0 andthen evaluating I

(

x0,u0,

π

)

and

s

(

x0,u0,

π

)

.

The radiative model is the RTE of Eq. (5) withthe boundary conditions:



I

(

xR,u,

π

)

=0

I

(

x_F,u

)

=Ib

(

T

)

(12)

4.1.Thesensitivitymodel

With regard tothe square cavitydeformation andconsidering theboundaryradiativeproperties,thelinearapplicationL describ-ingsensitivityboundaryconditionsishere

s_{∂(π )}

(

xF,u

)

=0 (13)

because_{F is}unaffectedwhenchanging

π

,and

s_{∂(π )}

(

x_{R(π )},u,

π

)

=− ks  u.n_R  4π p

(

u

|

u

)

duI

(

x,u,

π

)

(14)

Wewill discussthe expressionsofsuch sensitivitiesforincoming directionsatgeometricallymodifiedboundarieswithmore gener-alityinSection5andAppendixA.Themainpointisthatthey in-cludethe intensityin outgoingdirections.Here thereason isthe following:whenenlargingthecavity(forinstance),anewlayerof semi-transparentmaterialisintroducedandthislayercaninteract withoutgoingradiationandscatteritbackintothecavity.

4.2. PropagatingthesensitivitysourceswithastandardMonteCarlo approach

The sensitivityboundaryconditionis coupledwiththe outgo-ing intensity at all pointson the geometrically modified bound-aries.Thismeansthatwehavetoevaluatethisintensitytosetthe boundary condition.Letus temporarilyassume that theintensity is known andthereforethat thesensitivity is alsoknown for all incomingdirections.We can thenconsider theseincoming sensi-tivitiesassources,asinastandardradiativetransferproblem,and thinkofMonteCarloalgorithmspropagating them.Herewewant toevaluatesatalocation x0 anddirectionu0.Weknowthatsis

solutionofthevery sameRTEasIandthestandardMonte Carlo approachwouldthereforebetouseareversealgorithm,exactlyas forthe estimationof I

(

x0,u0,

π

)

. Thismeans that the algorithms

evaluating I

(

x0,u0,

π

)

and s

(

x0,u0,

π

)

start with the very same

steps,andthereforecansharethesamesampledpaths:

• Amultiplescatteringpaths

γ

issampled,startingatx0in

direc-tion−u0,findingsuccessivescatteringlocationsxiaccordingto

Beerlawfor pure scattering, scatteringdirections_−u_i accord-ing tothe singlescatteringphase function,until reaching one ofthetwoabsorbingsolidsatlocationx_γ indirection−u_γ.

• TwoMonte Carlo weights are then computed usingthe same path:

• one for the intensity problem that is simply the value of the incomming intensity at x_γ in direction u_γ attenu-atedbycontinuousabsorptionalongthepathlengthl_γ,i.e.

I

(

x_γ,u_γ,

π

)

exp

(

−kalγ

)

,

• one for the sensitivity problem constructed the very same way but with the incomming sensitivity, i.e.

Fig. 3. Absorbed radiative intensity density J ( π) and its sensitivity ∂πJ ( π) to the uniform scaling of R (π) boundary. The phase function in the semi-transparent medium is isotropic p U(  ui) = 41π and the probability density function of scatter-

ing extinction is p (li) = k s exp (−k s l i) . Cavity dimensions are set by a variation δL of L influencing the optical thickness k e L . Single scattering albedo is uniform on the cavity ks

ke = 0 . 5 . Estimations of the absorbed radiative intensity density and its

sensitivity are obtained from 2.10 6 realizations of the corresponding Monte-Carlo

weight function.

4.3. CouplingthesensitivityMonteCarloalgorithmwithanestimate oftheoutgoingintensity

Sincethesensitivitysourcesareunknownatthisstage, the al-gorithm describedaboveisincompleteforits sensitivitypart:we would need to evaluate the intensity at each location x_γ where pathsmeetboundaries,andinalldirectionsu.Wecouldembeda full newMonteCarlo computationofIforsucheach location x_γ, butthisisunnecessaryduetoonefundamentalpropertyof Monte-Carlo methods: thedouble randomization.Since thesensitivityis expressed as an expectation of a linear function of the intensity that isitselfexpressedasan expectation,bothcanbe nestedinto one expectation [4].Hence, thesensitivitycan be estimatedwith nobiasusingonesingleMonteCarloalgorithmtoestimatethe lat-ter,combiningsensitivityandintensity.Thisisthesameideaasin reversemultiplescatteringormultiplerefectionMonteCarlo algo-rithms.Insuchverystandardalgorithms,whenacollisionlocation hasbeensampled,atasurfaceorinthe volume,fullycomputing thereflectionorscatteringsourceswouldrequiretostartaMonte Carloalgorithmwithalargenumberofsampledpathsforeach un-knownintensityineach incomingdirection.Butinstead,onlyone reflectionorscatteringdirectionissampledandamultiple scatter-ingpathisinitiatedforthisdirectiononly.Heretheprocessisless intuitive becauseitcombinessensitivitiesandintensities,butitis strictlysimilar:

• a path

γ

is sampled, as described above, from

(

x0,u0

)

to

(

x_γ,−u_γ

)

;

• therequiredsensitivitys

(

x_γ,u_γ,

π

)

isdefinedinEq.(14)asan integraloverdirectionsu andthealgorithmthereforesamples



u(justassamplingthephasefunctioninamultiplescattering algorithm);

• a new path

γ

 is started in direction _−u as if evaluating

I

(

x_γ,u,

π

)

withareverseMonteCarloalgorithm.

The resultingalgorithmis describedin Fig.C.16.The sampling of u is isotropic. If u is oriented toward the inside of the cav-ity (scalar product u.n_{R(π )}>0), the radiative intensity value is known: itisnullsince thesquare surfaceisneither emittingnor reflecting (box BinFig. C.16).So thepath

γ

 evaluating intensity isonly requiredwhenu isan outgoingdirection(dotted pathin

Fig. 4. Sensor’s response J ( π) and its sensitivity ∂πJ ( π) ( Eqs. (15) and (16) ) to a uniform scaling of R (π) . By comparison with previous simulation examples, the intensity is here integrated over the sensor’s surface and its response frequency range. The sensitivity results show that the additional integration process has no influence on the sensitivity estimation. The sensor of square surface S sensor is lo- cated at location  x0 ( Fig. 2 ) and oriented toward the − ex direction. The geometrical change of the cavity is the same as in Fig. 3 . The wavelength range is [ λmax

2 ; 2 λmax ]

where λmax(T) is the wavelength of the Plank function maximum at the cylinder

temperature T = 10 0 0 K. The dependencies with wavelength of the absorption and scattering coefficients are k a(λ) = k s(λ) = (1 /λmax)λ.

Fig. 2). Box C in Fig. C.16 displays an example of the only cases where the sensitivity Monte-Carlo weight has a non null value: when

γ

hitsthesquare,uisoutgoing,and

γ

hitsthecylinder.

Fig.3 displaysthe simulation results andillustrate the stabil-ityofsensitivities estimatedbythe model.Forcomparison,finite differenceswerecalculatedfromMonte-Carloestimationofthe ab-sorbedradiativepowerdensityfordifferentpositionsoftheupper surface.Thelatterresultsillustratethetypicaldifficulties encoun-teredwhenevaluatingthesensitivitiesfromfinitedifferences,that isthedifficultytoobtainconvergedresults.Inthisregard,if

δ

Lis chosen toosmallwithrespectto L,the varianceof thefinite dif-ferenceestimatebecomestoolargeandtheresultsareinaccurate. Onthe contrary,if

δ

Listoo large,the finitedifference converges buttowardavaluethat departsfromthetruesensitivity.This dif-ficultyiswellknownwhengradientsareestimatedby differentia-tion(itisoutlinedinthegradient-basedKiefer-Wolfowitzmethod presentedin[7]). Thefigure alsoshowsthat thelocal absorption power becomeslesssensitive to a variation of δL

L when the size

ofthecavityincreases.Table1summarizestheresultsofthelocal absorptionpowerandits shapesensitivityfordifferentparameter sets.TheclosertheprobelocationgetstotheR

(

π

)

boundarythe higherthe sensitivity. It alsoshows a practical limit to the eval-uation ofthe sensitivities inthe caseofhigh optical thicknesses. Apartfromthislimit,therearenoconvergenceissues.

4.4.Surfaceandspectralintegration

Themethodologycanbeextendedtoaspectralandsurface ob-jective function,for exampleto simulate the responsemeasured, atthepositionx0 withinthe cavity,bya sensorofsurfaceS,and

fora givenfrequency range.As long asthe measure settings are independentofthedeformationparameter,thequestionof evalu-atingthesensitivityaspresentedinSection3andthe correspond-ingmethodology(Section4)arenotimpacted.Intermsof Monte-Carlo,evaluating thesensorresponsesensitivitytoa perturbation ofthesquaresurfaceisequivalenttosampleapositiononthe sen-sorsurface,a wavelengthwithin thespecificfrequencyrangeand a directionwithin the measured hemisphere prior tosample the

Table 1

Absorbed radiative intensity density and its sensitivity results for a fixed value of L . The probability den- sity function for isotropic scattering is p U(  ui) = 41π and probability density function of scattering extinction

is p (li) = k s exp (−k s l i) . Location  x0 on the y axis is determined by yL0 = 0 . 5 and the ratio between the cylinder radius and L is set at r

L = 0 . 125 . The number of realizations of the Monte-Carlo weight functions is N = 2 × 10 6 ,

(%) represents the relative error of ∂πJ ( π) and J ( π).

ka L ks L xL0 ∂π J( x0,π) L 4πkaIbmax σL 4πkaIbmax ∂πJ(% ) J( x0,π) 4πkaIbmax l σ 4πkaIbmax J (%)

1 1 0.6375 1.47e −02 6.221e −05 0.42 3.89e −01 3.297e −04 0.09 1 10 0.6375 1.88e −02 2.299e −04 0.01 5.54e −01 3.037e −04 0.06 1 50 0.6375 4.06e −04 2.765e −05 6.80 6.64e −01 2.665e −04 0.04 1 100 0.6375 1.08e −05 2.31e −06 21.31 6.66e −01 2.602e −04 0.04 1 0.001 0.6375 1.85e −05 6.243e −08 0.34 3.53e −01 3.308e −04 0.09 10 1 0.6375 1.34e −06 1.198e −08 0.89 2.86e −01 2.669e −04 0.09 50 1 0.6375 1.16e −20 3.874e −22 3.33 1.19e −01 1.292e −04 0.01 0.001 1 0.6375 6.50e −02 2.878e −04 0.44 4.23e −01 3.493e −04 0.08 1 1 0.75 2.24e −02 9.048e −05 0.41 1.54e −01 2.199e −04 0.14 1 1 0.875 3.27e −02 1.255e −04 0.38 8.10e −02 1.531e −04 0.19 1 1 0.9875 7.70e −02 3.644e −04 0.47 4.73e −02 1.123e −04 0.24 1 1 0.9875 = y0

L 5.22e −02 3.287e −04 0.63 2.05e −02 6.766e −05 0.33

Fig. 5. The two different effects of an infinitesimal modification of the boundary on the incoming intensity at position  x∂(π) and direction  u .

sensitivitypath(Eqs.(15)and(16)).

J

(

x0,

π

)

=  S dS  _λmax λmin d

λ

 2π ka

(

λ

)

I

(

x0,u0,

λ

,

π

)

du0 (15)

∂

πJ

(

x0,

π

)

=  S dS _λmax λmin d

λ

 2π ka

(

λ

)

s

(

x0,u0,

λ

,

π

)

du0 (16)

The typical integrated radiative objectivefunctions do not intro-duceadditionaldifficultiesregardingthesensitivity.Fig.4displays thecorrespondingsimulationresults.

5. Thedifferenttypesofsensitivitysources

In this section we focus onthe sensitivity sources. So far we haveseenthatbesidethecouplingattheboundarycondition,the sensitivitymodelcanbe estimatedinthedomainexactlylikethe intensitymodel,usingstandard MonteCarloapproachesto radia-tive transfer. The following examples will all make use of most simpleforwardMonteCarloalgorithms.Moreattentionwillbe de-votedto the boundary condition definitions, in particular to the Diracsourcesduetoboundarydiscontinuitiesandtosources asso-ciatedtocurvedsurfaces.

We will see that depending on the configurations, the sensi-tivitysources can take diverseforms, namely that the linear ap-plicationexpressingtheboundaryconditionscan becomposed of distinctlinearoperators. InAppendixA,formal developmentsare providedandreportedherebyEq.(17)foratranslationcolinearto thenormalvector:

s

(

x_{∂(π )},u,

π

)

=−

∂

yI

(

x_{∂(π )},u,

π

)

v

+

∂

πIp

(

x,z,u,

π

)

(17)

Thisformulationestablishesa relationbetweentheshape deriva-tive s

(

x_{∂(π )},u,

π

)

and the material derivative

∂

_πIp

(

x,z,u,

π

)

in

this particular transformation [24]. Using this relation for black-bodyradiativeboundaryconditions,weidentifytwotypesof con-tributionstothesensitivityinincomingdirections:

s

(

x_{∂(π )},u,

π

)

=−u.

∇



(

I

(

x∂(π ),u,

π

)

 u.n

v

+u.

∇

TI

(

x∂(π ),u,

π

)

 u.n

v

withu.n>0 (18)

The first contributiontype (first termin Eq.(18)) corresponds to theeffectofmedium diffusionandmedium absorptioninthe in-finitesimallayer along the surfaceadded or suppressedafter the surfacedeformation(cf Fig.5a). Ifasurfaceisshifted towardthe outside, its displacement creates a new layer in its vicinity (the interiorvolume isincreased). The intensityexitingthe surfacein directionuwill be attenuatedby thislayer(absorptionand scat-tering) andfurthermore theintensityincomingonthisnewlayer willhaveaprobabilitytobescatteredindirectionu.When trans-lated into sensitivities, these decreases and increases of the in-comingintensityatthe boundarylead toexactlythe sameterms as the collision terms in the radiative transfer equation (−

(

ka+

ks

)

I

(

x,u,

π

)

+ks 4πp

(

u

|

u

)

duI

(

x,u,

π

)

),whichallowsto

summa-rize them using the transport operator, which explains the term



u.

_∇

_{I inEq.(18)(transportequalscollisioninthestationaryRTE).}

In configuration 1, only this first contribution term was en-countered.Thesecond termofEq.(18)corresponds totheeffects ofnonhomogeneousradiationinincomingdirectionsata geomet-ricallymodifiedboundary.Weusethenotation

_∇

_{TI forthesurface}

gradient,alsocalledtangentgradient,oftheincomingintensityat theboundary.Fig.5billustratesthiseffectontheintensityat loca-tionx_{∂(π )}indirectionu.Letusassumethatthemediumis trans-parent:beforethedeformation,theintensityindirectionuwas ex-actlytheintensityemittedorreflectedbythesurfaceatthatpoint.

Fig. 6. The two types of discontinuities generating sources of sensitivity. The image 6 a represent the discontinuity induced by the edges connecting two plane surface. The image 6 b represent the discontinuity induced by the shading. Finally the image 6 c represent the combination of both types of discontinuities according to the radiative intensity direction.

Fig. 7. Configuration 2, on the left the square cavity geometry filled by semi-transparent medium and the deformation of the upper wall. On the right the locations of the sensitvity sources.

After deformation, this intensitycomes fromanother location on thesurfacewhichexplainsthetangentialgradientinEq.(18).This tangential gradient plays a very specific role when dealing with surface discontinuities, for instance when two adjacent surfaces havedifferenttemperatures (Fig. 6a),ordifferent reflection prop-erties,orevenwhentheyhavethesamereflectionpropertiesbut their normalsarenot thesame.Inallthreecases,theintensityin anyincomingdirectionisdiscontinuouswhenmodifiedalongthe surfaceandthecorresponding Heavisidefunction leadstoaDirac when applying the tangential gradient. This creates sources that arenotcontinuouslydistributedalongthesurfacebutconcentrated onthediscontinuitylines:wethenhavetodealwithDiracsources ofsensitivity.

The same idea of a discontinuity of intensities at modified boundaries allows to think the sensitivity sources associated to shadingeffects.ThisisillustratedinFig.6bwithacurvedsurface: at point psome directions are blocked by thesurface andifthe point is displaced along the y axes, it will block newdirections and consequently become a local source ofsensitivity. The same shading effect occurswith adjacentplane surfaces withdifferent normalorientationsassoonasthediscontinuitypisgeometrically modified(see6c).

Forconfiguration2,wehandleformallythisspecificfamillyof sensitivitysources,startingfromEq.(18)andwegiverigorous def-initionsofDiracsourcesofsensitivity.Forfurthercases,theformal statement of those sensitivity sources at discontinuitiesis still a work inprogress. Wedo not addressthem yet usingthe general sensitivityboundary conditionstatedin Eq.(18)butthey are es-timated onan adhocbasis byanalyzing geometricallythe differ-ential implications of surfaces deformations. This is the case for

thecurvedandreflectivesurfaceexamplesofconfiguration3and configuration4(seeAppendixB).

In the following ofthis section, configuration 2 gives an ex-ampleofanedgeboundarydiscontinuitytreatedfromthegeneral sensitivityboundary conditionstatementEq.(18)in an academic case:thesamesquare cavityasinpart4butwithadifferent de-formation.In configuration3, theintensityedge discontinuity is extendedonathermalsolarconfigurationwithreflectivesurfaces. Itisshowedthatthisparticulartypeofsensitivitysourcescanhave a heavy impact on the algorithm when considering complex ge-ometry. We recognize here similar difficulties as reportedin an-otherworkaboutshapesensitivities[13]whereoptimizingthe al-gorithmprocessdemandedgreateffortsinthecaseofimage syn-thesistransparentmediumconfigurations.Theboundarycondition statementofconfiguration3hasbeendoneonanadhocbasisin

AppendixB.Finallyconfiguration4givesanexampleofashading discontinuityinthecaseofcurvedsurface.

5.1. Configuration2

Let us consider another square cavity which dimensions are setalong the x andy axes andare infinite along thez axis. The medium insidethe square,in thedomain



,is semi-transparent. Thefourwallsofthesquarecavitiesareblack-bodiesandonlythe upperwallisemitting(_Rup):

⎧

⎪

⎨

⎪

⎩

I

(

x_F,u,

π

)

=0 orx_F=

(

x,0,z

)

I

(

x_Rl,u,

π

)

=0 orxRl=

(

0,y,z

)

I

(

xRr,u,

π

)

=0 orxRr=

(

L,y,z

)

I

(

xRup,u,

π

)

=Ib

(

T

)

orxRup=

(

x,L,z

)

(19)

Fig. 8. Radiative flux on the bottom-wall J ( π) and its sensitivity ∂πJ ( π) to the height of the up-wall. The phase function in the semi-transparent medium is isotropic p U(  ui) = 41π and the probability density function of scattering extinction

is p (li) = k s exp (−k s l i) . Cavity dimensions are set by a variation δL of L influencing the optical thickness k e L . Single scattering albedo is uniform on the cavity kkse = 0 . 5 .

Estimations of the radiative flux and its sensitivity are obtained for 1.10 6 realiza-

tions N of the corresponding Monte-Carlo weight function.

Fig. 9. Schema of the concentrating solar tower and of the heliostat field with an illustration of the heliostat size deformation. The sun surface is named S, the tower

τ, the heliostats surface H and the receiver surface R .

Thegeometricalparameter

π

inducesanextensionoftheheightof thesquare cavityby a displacementoftheupper-wall ina direc-tioncolineartotheupwardnormalvectorn_Rup (Fig.7).The objec-tivefunctionistheradiativefluxcrossingthebottom-wall(F)and wewanttoestimateitssensitivityto

π

.

The sensitivity source is only at the modified boundary, i.e. alongtheupperwall.Ithasfourcontributionsassociatedto:

• extinction of radiation by the medium in the vicinity of the modifiedboundary,

• scatteringsourcesinthesamevicinity,

• temperature discontinuity at the left side of the modified boundary,

• temperature discontinuity at the right side of the modified boundary.

The lasttwo contributionsareDiracsources (involvinga Dirac distributionat0andL): s_{∂(π )}

(

x_{R(π )},u,

π

)

=−ka+ks  u.nRup Ib

(

T

)

+ ks  u.n_Rup  4π p

(

u

|

u

)

duI

(

x,u,

π

)

+Ib

(

x,u,

π

,T

)

 u.ex  u.nRup

δ

(

x− 0

)

H

(

L− x

)

H

(

u.ex

)

−Ib

(

x,u,

π

,T

)

 u.ex  u.n_Rup

δ

(

x− L

)

H

(

x− 0

)

H

(

−u.ex

)

(20)

The second contribution was already discussed in Configura-tion 1. The first contribution didnot appear because the square surface was neither emitting nor reflecting (no incoming radia-tion). Hereit appearsbecause thegeometrically modified bound-ary is a black-body at temperature T. Both the first and second contributions are distributedall along the upper wall. The other twocontributions areatthesingularinfinitecornerlines(dotsin

Fig.7b).BecauseoftheHeavisidetermsH

(

u.ex

)

andH

(

−u.ex

)

,for

agivendirectionu,onlyoneofthetwoisactive:that correspond-ingtothecornerfromwhichuisavalidincomingdirection.They correspond to the fact that from a given location inside the do-main,whenconsideringradiationcomingfromthecornerexactly, increasing

π

(i.e. raising the top surface) implies that the non-emitting lateral face is viewed,whereas lowering it implies that theemittingtopsurfaceisviewed.TheDiracsourcesofsensitivity translatethisdiscontinuityindistributionterms.

We implemented a very standard forward Monte Carlo algo-rithminwhichthesesourcesare propagatedfromtheupper-wall intothedomainalong multiple-scatteringpaths,attenuated expo-nentially with respect to the absorption optical thickness of the path. Only the paths reaching the bottom-wall contribute to the sensitivityestimate.Fig.8displaystheresultsforboththe bottom-wall radiative flux as function of the top surface height. In the same graph the sensitivityto the height is plottedtwice: as es-timated using the present sensitivity Monte Carlo algorithm and evaluatedfromfinitedifferences.Usingthesensitivitymodel con-vergence is lessthan 1% for106 _{sampledpaths, in contrastwith}

the finitedifference estimatesthat are muchlessaccurate as ex-plainedinSection4.

5.2. Configuration3

Inthisconfigurationshape sensitivities areimplemented fora concentrating solar tower application[9].The configuration is il-lustratedonFig.9.Themirrorsarespecularreflectingflatsquares andthesurroundingmedium istransparent.Theintensity bound-aryconditionsforallincomingdirectionsarestatedoneachofthe followinggeometricalobjects:thesunS,thetowerT,thethermal receiverRandtheheliostatsH:

⎧

⎪

⎨

⎪

⎩

I

(

xS,u,

π

)

=Is I

(

x_T,u,

π

)

=0 I

(

x_R,u,

π

)

=0 I

(

xH,u,

π

)

=

ρ

I

(

xH,uspec,

π

)

(21)

The incomingintensities are nullat all theother limits(skyand ground).The simulationobjectiveistoevaluate theradiative flux atthereceiverandunderstandtheimpactoftheheliostatfield de-signintermsofthefollowingthreeopticalphenomena:anoptical path froma pointin thesolar diskto apoint on one ofthe he-liostatsmay

• beshadedbyasurfacelocatedbetweenthesunandthe helio-stat(theshadingphenomenon);

• beblockedbyasurfacelocatedbetweenthereflectionlocation andthereceiver(theblockingphenomenon);

• missthereceiverduetoopticalerrorsduringthereflection(the spillagephenomenon).

We here study the influence of the variation of the heliostat size(whichismodifiedaccordingtotheparameter

π

,Fig.10).The enlargementof amirror symbolized inFig.D.17 by thearrow D,

Fig. 10. Deformation of an heliostat mirror with respect to the parameter π. The parameter influences the size of the mirror area surface.

Fig. 11. Thermal power sensitivity to the size of the mirrors of an heliostat field. The medium is transparent and the deformation the width of the mirrors increases by π. The size is represented by the width of the square heliostat mirrors.

• anincreaseofthethermalpowercollectedduetotheincrease inreflectiveareaFig.D.17a

• a decreaseofthe thermalpower collecteddueto the appear-anceofashadingphenomenonFig.D.17b

• a decreaseofthe thermalpower collecteddueto the appear-anceofablockingphenomenonFig.D.17c

Asthey aretheonlydeformedsurfaces,intermsofsensitivity modelingtheheliostatsmirrorsaretheonlysensitivitysources.On eachheliostattheincomingsensitivityhasfivecontributions:

• one associated to the reflection of sensitivity sources coming fromotherheliostats;

• fourassociatedtotheextensionofeachoftheheliostatedges. Using a Cartesian coordinate system(O, x,y) centered on the heliostatandalignedwithedges,

s_{∂(π )}

(

x_{H(π )},u,

π

)

=

ρ

s

(

x,uspec,

π

)

H

(

π

2 − x

)

H

(

π

2 − y

)

H

π

2 +x



H

π

2 +y



+

(

ρ

I

(

x,uspec,

π

)

− I

(

x,u,

π

)

)

2

δ

π

2 − x



H

π

2 − y



×H

π

2 +x



H

π

2 +y



+

(

ρ

I

(

x,uspec,

π

₂

)

− I

(

x,u,

π

)

)

δ

(

π

₂ − y

)

H

π

₂ − x



×H

π

2 +x



H

π

2 +y



+

(

ρ

I

(

x,uspec,

π

)

− I

(

x,u,

π

)

)

2

δ

π

2 +x



×H

π

2 − y



H

π

2 − x



H

π

2 +y



+

(

ρ

I

(

x,uspec,

π

)

− I

(

x,u,

π

)

)

2

δ

π

2 +y



×H

π

2 − x



H

π

2 +x



H

π

2 − y



(22)

Thefirstcontributionwasnotencounteredinconfiguration1and configuration2becausenoneofthesurfaceswerereflecting.Here forthesensitivitymodel,thereflectingheliostatssimplyreflectthe impacting sensitivitiesexactly asthey reflectintensityin the ini-tialradiative transfer model.It isnot always the case. When the geometricparametercreatesatranslationofthereflective bound-ary,othertermsappear(notshown).See,forinstance,the general-izationofreflectivesensitivityboundaryconditionsthathavebeen done in [21] for deterministic method. The four other terms are very similar tothe Dirac sources ofconfiguration 2: they trans-lateinsensitivitytermsthe intensitydiscontinuityatthelimit of modifiedboundary.

As far as the sensitivity Monte Carlo implementation is con-cerned, weused aforward approach,whichwasnot thecasefor theinitialradiativetransferalgorithm.Indeed,whenevaluatingthe flux collected by the receiver, standard concentrated solar algo-rithm donot start atthe sun:the firststep is thesamplingof a locationontheheliostatsurfacesandonlythenalocation onthe solardiskissampledtorepresentthe solarradiativesources ina backwardMonteCarloapproach.Hereforthesensitivity,westick toadirectsamplingofthesensitivitysourcesthemselves,without anyintermediatestatisticalstep.Thismeansthatwestartby sam-plinga locationalong allthe edgesofall theheliostats.The sen-sitivitysourcesareindeedonlyattheedges:thefirstcontribution in(22),spreadoveralltheheliostatsurface,isonlyareflectionof sensitivities emitted elsewhere (at the edge of an heliostat) and propagated to the mirror surface. Therefore in a forward Monte Carloalgorithmforsensitivities,pathswillbestartedattheedges andrefectionwill simplyoccurwhena pathencountersamirror. Theonlysubtlepointisthatthesources ateachedgelocation in-volvetheknowledge oftwointensities: onein thedirectionuspec

beforereflection,one intheconsidereddirectionucorresponding to radiation propagating at the edge location without impacting it,thereforecoming fromanotherheliostat.These twointensities arenotknownanddoublerandomizationisused(asalready illus-tratedinSections4and5.1),meaningthattworadiativepathsare initiatedatthesamplededgelocation.Apartfromthisdouble ran-domizationtechnique,thatwillberequiredforallsensitivity algo-rithmsinvolvingeitherreflectionorscattering,thesensitivity algo-rithmisfullystandard,thesourcesbeingpropagatedalongstraight lines,reflected by the heliostats,until they reach the receiver or arestoppedbyanabsorbingsurface.

Fig.11displayssimulationresultsforanheliostatfieldincluding 250heliostatsetup witharadial staggeredlayout[23].The ther-mal power sensitivities are estimatedfor an heliostat size range between0.8to1.2mwithastepof0.02m.Theresultspresented intheFig.11startatthevalueof0.8m.Indeedwhenthereisno overlapoftheheliostats(blocking,shading)thecollectedpoweris proportionaltothesizeoftheheliostats.Thisresultsinaconstant sensitivityofthe collectedpowerto the sizeof theheliostats. In thisfigurewethereforeonlypresentthesizesofheliostatmirrors whichinduceoverlap.

Theresultshavebeencrossvalidatedwithfinitedifferencesand show an exact match. The standard erroris significantly smaller (0.03%for106 _{samples)thanthatofthefinitedifferenceestimate,}

asexplainedinSection4.Oncethedoublerandomizationscheme was clarified, there were no particular difficulties associated to theimplementationitselfandnospecificconvergenceissueswere identified.

Fig. 12. Complex geometry academic configuration representing an emitting sphere inside a supershape ( Fig. 12 a) and the receptive part of the supershape surface ( 12 b).

Fig. 13. Radiative flux received by a part of the surpershape surface ( Fig. 12 b) J ( π) and its sensitivity ∂πJ ( π) to the sphere radius. The phase function in the semi- transparent medium is isotropic p U(  ui) = 41π and the probability density function of

scattering extinction is p (li) = k s exp (−k s l i) . Cavity dimensions are set by a vari- ation δr of r . Single scattering albedo is uniform on the surpershape ks

ke = 0 . 5 . Es-

timations of the received radiative flux and its sensitivity are obtained for 1.10 6

realizations N of the corresponding Monte-Carlo weight function.

5.3.Configuration4

In thisconfiguration we considera sphere, describedin para-metricterms, inside a surpershape geometry,descried asa large ensembleoftriangles.Thesphereisstatedasanisothermal emit-tingblack-bodyattemperatureTandthe boundaryofthe super-shapeisalsoablack-bodybutnotemitting.Themediumbetween thesupershapeandthesphereissemi-transparent andnon emit-ting.

Theobjectivefunctionistheradiativepowerreceivedbyapart of the supershape boundary: that is all the supershape surface above theslice inthe Fig. 12b.The goal is to estimate its sensi-tivitytothesphereradius.Theintensitymodelisstatedfromthe RTEwithLambertianboundaryconditionatthespheresurface:



I

(

x_Ssphere,u,

π

)

=Ib

(

T

)

with u.nsphere>0

I

(

x_Sshape,u,

π

)

=0 with u.nshape>0

(23)

with all the normal vectors oriented toward the medium. The radiative power emitted by the sphere surface is 4

π

r2

_σ

_T4_{. This}

quantityisattenuatedduringitstransportinthesemi-transparent medium. Since only thesphere radius r is influenced by the ge-ometrical parameter

π

only the sphere surface will be emitting

sensitivity. The sensitivity source hasthree distinct contributions associatedto:

• extinction of radiation by the medium in the vicinity of the sphere,

• scatteringsourcesinthesamevicinity,

• shadowingoftangentradiation.

Usingstandardnotationsofpolarandazimutalangles,

θ

and

φ

forincomingdirections,

θ

and

φ

 foroutgoingdirections,

s_{∂(π )}

(

x_{R(π )},

θ

,

φ

,

π

)

=−ka+ks cos

(

θ

)

Ib

(

T

)

+ ks cos

(

θ

)

 2π d

φ

  π 2 0 d

θ

p

(

θ

,

φ|

θ

,

φ



)

I

(

x,

θ

,

φ

,

π

)

+1_r



σ

T4

π

− I

(

x,

θ

,

φ

,

π

)



δ

(

θ

−π2

)

cos

(

θ

)

sin

(

θ

)

(24)

The three contributions are distributed all over the surface and thedistinction betweenthemis essentiallyangular. Thefirst two terms of Eq. (24)corresponds to the effectof medium diffusion and absorption in the additional infinitesimal volume. The third oneonlyexistinthetangentdirections(seethetermin

δ

(

θ

−π2

)

in Eq. (24)) and come from the intensity discontinuity between theboundaryemissionandthevolumeout-comingintensityatthe samepoint,inthatdirection.

We implemented a very standard forward Monte Carlo algo-rithminwhichthesecontributionsarepropagatedthroughoutthe mediumaccordingtoamultiplescatteringalgorithm,with contin-uousextinctionbyabsorptionalongthepath,untilreachingonof theabsorbing boundaries.The pathonlycontributestothe sensi-tivity if it ends at the surfaceof interest (the surface above the slicein Fig. 12b). Fig. 13 displays theresults of the Monte Carlo estimatesvalidatedagainst thefinitedifference estimates.We en-counterednoalgorithmicdifficulty andtherewasnoconvergence issue.A perspective forcurved parametric surfaces is theformal identificationoftheboundaryconditionbythe methodology pre-sentedinAppendixA,whichwe could notyetachieved(Eq.(24)

wasonly constructedin an ad-hocmanner. This wouldopen the waytoasimilarhandlingofanysmoothsurface.

6. Conclusion

AsMonteCarlostartsfromexpressingtheobjectivefunctionas anintegral(anexpectation),thefirstapproachestotheevaluation ofsensitivitiesinside aMonteCarloalgorithm consistedintaking thederivativeofthisintegral.Thisleadtoasolutionthatalthough