Estimation of the Effective Multiplication Factor by Monte-Carlo Method Using the Importance Function

Transcription

1 Esimaion of he Effecive Muliplicaion Facor by Mone-Carlo Mehod Using he Imporance Funcion MI Gurevich, MA Kalugin, DS Oleynik, DA Shkarovsky Naional Research Cener Kurchaov Insiue,, Akademika Kurchaova pl, Moscow, Russia, 2382, Absrac The work describes he MCU code based realizaion of a mehod ha allows leveling of ecive muliplicaion facor bias in criicaliy calculaions wihou any changes in simulaion and receiving reliable evaluaions of is saisical uncerainy The mehod is based on he calculaion of a discree adjoin funcion of a marix ranspor equaion and is use for he evaluaion The eciviy of he mehod is demonsraed in calculaion of he «Whieside problem» weakly coupled sysem I INTRODUCTION There are hree problems in Mone-Carlo calculaions of k of large sysems in which geomeric dimensions are much bigger han a neuron pah lengh They are relaed o he usage of he generaion mehod wih a fixed number of neurons N TOT in a generaion The problems are: he choice of iniial neuron disribuion or he choice of number of he firs skipped generaions N SKIP ; 2 he esimaion of bias К sysemaic miscalculaion of k ; 3 he esimaion of a saisical error corr aking ino accoun correlaions beween generaions ( 0 denoes he esimaion of he sandard deviaion excluding he correlaions) These problems can be solved wih he usage of he auocorrelaion funcion of he conribuion of generaions o k esimaion [ 4] e s denoe C k as he covariance funcion of conribuion of he curren and he curren plus k-h generaions o k when seady-sae condiion is reached For a saionary random process { k, k2,, kn,, k N}, where k n is he evaluaion of k a generaion number n, he C k evaluaion may be defined using finie sample of N elemens Nk Ck ( kn kn )( knk kn ), N k n where < > denoe he evaluaion Correlaion funcion с k =C k /C 0 normalized o dispersion is called he auocorrelaion funcion of a random process e s denoe as heir lag Then σ σ 2 S, where S corr 0 ck k Usually when calculaing by MCU code [7] he following shall be saisfied N SKIP ; 2 N TOT is such a large number ha [] Δ K Ck k () k is inessenial (as i is shown in [2], 2 3/2 ΔK Ck O( NTOT ) ) ); k k 3 saisical error is calculaed for he so-called series, each of which includes a user-defined number of generaions N BAT N BAT is such a large number ha S is minimal I s imporan o noe [ 6] ha К /N TOT and D k ~ N, S k Δ D k cons; ha s n TOT К n why he bias doesn depend on N and P (number of processors), and depends on N TOT value I is known ha he use of imporance funcion (adjoin funcion) essenially minimizes he correlaion beween generaions and ge he k esimaion wih K =0 and K = 0 Work [2] suggess a mehod o decrease he bias of k evaluaion and correlaion beween he k n evaluaions using spaial ransformaion of ranspor equaion This work suggess a differen approach wihou an inegral operaor modificaion Insead, i is associaed wih he adjusmen of conribuions ino he evaluaion a he sage of allying using he adjoin funcion The approach is realized in he MCU code [7] and numerical resuls of k evaluaion for he «Whieside problem» weakly coupled sysem [8] are obained using normalizaion on he imporance funcion a which K 0 and K 0 II DESCRIPTION OF METHOD The descripion of he mehod of unbiased k evaluaion using he imporance funcion is he developmen of he work [] In his sudy, more aenion is paid o a precise descripion of he changes in he algorihms used in he ranspor equaion modeling Basic conceps e s assume ha he sae S of Mone Carlo simulaion of a neuron ranspor in a criicaliy ask is he se of M neurons each of which is characerized by is weigh w i and a poin in phase space р i (i =,2,,M) Here M is he maximum number of paricles in he sae (M N TOT ) Thus he sae may be represened as a marix S=[w,р], where w is a non-negaive vecor-column of M componens and р is a column wih poins of he phase space

2 The generaion rae refers o a non-negaive densiy in he phase space Three deerminisic represenaions are correc for he saes: oal weigh W(S)=w + +w M ; muliplicaion by a posiive scalar λ[w,р] = [λw,р]; reducion o densiy (v(s))(p) = w δ(p p )+ + +w M δ(p p M ), where p is an arbirary poin in phase space and δ is Dirac dela disribuion I is obvious ha v(λs)=λv(s) and λw(s)=w(λs) e s define wo narrow ses a he variey of saes Generaion F=[w,p] is such a sae ha w = =w NTOT >0, and w NTOT+ = = w M =0 2 Normal generaion G=[w,p] is such a sae ha w = =w NTOT =,and w NTOT+ = = w M =0 I is obvious ha W(G)=N TOT, G=(N TOT /W(F))F is a normal generaion 2 The formalizaion of he modeling process Mone Carlo mehod o simulae he ranspor equaion in he criical asks can be convenionally represened as wo processes, each of which are produced by random variables based on cerain physical quaniies The firs process is a simulaion sep Using he normal generaion G i builds a random sae S G (S under he condiion G) Wherein he condiion of bias absence a a single sep simulaion is saisfied, ie for any G E( v( S G)) H v( G), (2) where H is he secondary neurons generaion operaor, E is he mahemaical expecaion The second process is normalizaion Using he given condiion of he se S=[w S,р S ] i builds a random normal generaion G S=[w G,р G ] Wherein he following condiions are saisfied: any poin in р G componens may be found in р S componens of non-zero weighs, ie any paricle of normal generaion is obained from a paricle of a sae of S; equivalence of weighs, ie le a p be a sum of weighs of paricles of a sae of he S se corresponding o some poin р, and b p is he mahemaical expecaion of a similar amoun for normal generaion, han for any poins р and q a b p >0 i is rue ha a q /a p = b q /b p ; normalizaion is homogeneous wih weigh 0, ie for any λ>0 i is rue ha G S = G (λs) I is followed from he second condiion ha E ( v( G S)) v( S) N TOT / W( S) (3) In he beginning of he ieraion process of simulaion he iniial normal generaion G 0 is chosen e s assume ha normal generaion G is formed Then he sae of secondary neurons is S + =S G according o he said above The nex normal generaion is G + =G S The conribuions o he assessmen of he ecive muliplicaion facor are k + = W(S + )/W(G ) = W(S + )/N TOT Insead, i is suggesed o use he following conribuions o he evaluaion of k k h,+ = (h,v(s + )) /(h,v(g )), (4) where h is a non-negaive funcion in he phase space, which is posiive in all poins wih fuel; round brackes (, ) denoe inegraion of funcion and densiy produc I is obvious, ha k k,, where е is he funcion е idenically equal o one Sequence {S,G } uniquely defines he sequence of generaions {F } Namely, F 0 = G 0, furher ransiion o he nex poin is carried ou according o he following equaions G = F / c, where c = W ( F ) / N TOT, S S G, G G S, F G cw S / N G c k TOT In oher words, c + = k k 2 k + [] A he same ime due o (3) vf S c v G S k cv S E E I follows ha E vf F E( EvF S F c E v S F c E v S G c Hv G due o (2) Thus vf F vf E H Ieraing he las equaion we ge E( v( F ) F0) H v( F0) This formula in differen noaion may be found in oher works, eg [,2] Insead of he k evaluaion one may use k h,, where h is he main adjoin funcion of he operaor H I allows one o avoid bias in he k evaluaion, and zero ou corresponding covariance a saisical uncerainy calculaion The firs follows from E(k h,+ G ) = E((h,ν(S + )) / (h,ν(g )) G ) = = (h,hν(g )) / (h,ν(g )) = (H + h, ν(g )) / (h,ν(g )) = k In paricular, his equaion implies ha for any q 0 E( k G ) E( E ( k G ) G ) k h, q h, q q Zero covariance beween k h,+q+ and k h,+ follows from E( k k G ) E( E ( k k G ) G ) h, q h, h, q h, Based on he fac ha k h,+ is consan a he condiion of G + E( E( k k G ) G ) E( k E( k G ) G ) = h, q h, h, h, q E( k k G ) = k h, 2 2 Thus, covariance is k k k 0

3 3 Implemenaion Thus, if we know he adjoin funcion we may obain non-biased esimaion of k and reliable esimae of saisical uncerainy Implemenaion of he mehod of calculaion of he adjoin funcion is based on he use of fission marix [9,0] The se of poins in phase space conaining fuel is divided ino a number of zones Then he equaion of criicaliy can be expressed as k ψ=tψ, where ψ is a vecor, i-h componen of which corresponds o he neuron generaion rae in zone i, and T is he marix, where T ij is he average number of neurons born in zone i a he condiion ha he fission was caused by he neuron born in zone j In his case he adjoin equaion is k ψ + =T T ψ +, here ransposed marix T has he meaning of he adjoin operaor Marix elemens are calculaed by means of he Mone Carlo mehod during neuron ranspor simulaion and he main adjoin funcion is calculaed by means of any deerminisic ieraion mehod The accuracy of he calculaion of he adjoin funcion using his mehod depends on he deail of he pariion, which can be infiniely increased, bu i also means he increase in he load on he compuing resources Afer he imporance funcion is calculaed he second sage of calculaion begins Generaions are produced by means sandard for he MCU code To evaluae he conribuion ino k using he formula (4) ψ + is used as he funcion h In he process of he simulaion of he generaion number he wo sums are calculaed The firs one includes values of ψ + for neurons of he generaion in heir iniial poins The second sum includes values of ψ + in he poins of secondary neurons generaion muliplied by he corresponding weighs If here is no non-analog simulaion and a generaion in he poins of absorpion his gives he average amoun of fission neurons The conribuion ino k is he raio of he second sum o he firs one Thus, he deviaion from he sandard calculaion akes place no in he simulaion, bu in he allying, which is provided in he MCU code III RESUTS The mehod is approved by calculaion of weakly coupled sysem «Whieside problem» Model descripion The sysem consiss of 9 9 9=729 spheres wih pluonium-239 (see Fig ), which are locaed in mesh poins of he regular cubic laice wih he 60 cm pich The radius of cenral sphere is equal o 5009 cm, he radius of he res ones is equal o 3976 cm The laice of spheres is locaed in he air and is encircled by he waer reflecor of 00 cm hickness The emperaure of all maerials is equal o 300 K In his sysem k = 0035 Fig Scheme «Whieside problem» in OXY plane 2 Calculaion of imporance funcion All 729 spheres are grouped in 5 cubic layers, and he marix dimension is 5 5 Enumeraion of ally zones in OXY plane is shown on Fig (from inner layer (-s zone) o ouer one (5-h zone)) The scheme of enumeraion in OXZ plane is he same Esimaions of he fission marix, he eigenfuncion and he imporance funcion are given in Tables I,II The values of eigenfuncion are divided by uni volume of a sphere and normalized (ψ norm ) The eigenvalue of he marix k is equal o 0035 I should be noed ha in his ask values of ψ norm and ψ + are pracically he same (see Table II) To invesigae he ec of fission marix deailing on calculaion resuls a fission marix of (each sphere is a differen ally) was evaluaed Calculaions by he MCU code have been performed wih he various number of neurons in he generaion N TOT =2000, 5000, 0000, 20000, and Calculaion resuls of ecive muliplicaion facor wih he normalizaion using he imporance funcion k adj (5 zones) and k adj729 (729 zones) are compared wih he sandard calculaion of k sd (see Table III and Fig 2) For convenience of comparing all calculaions use he same

4 number of hisories ( ) Saisical uncerainy (a sandard deviaion) is equal o 0,004 % Table I Fission marix T i\j Table II The eigenfuncion and he imporance funcion No ψ norm ψ Sum Table III Effecive muliplicaion facor esimaions in dependence on N TOT N TOT K sd k adj k adj The resuls wih differen division show ha evaluaions wih higher deailing (729 allies) have smaller bias However, he difference is small and does no exceed 0,0 % These daa indicae ha he imporance funcion can be calculaed wih coarse approximaion Tha is why i is possible o use he imporance funcion, which has been calculaed for similar models Fig 2 Calculaion k of wih normalizaion using he imporance funcion (k adj, k adj729 ) and wihou ha (k sd ) I should be noed essenial irregulariy beween neuron sources in differen spheres I is shown on Fig 3 where calculaion of eigenfuncion and imporance funcion was obained for 729 zones The neuron source in cenral sphere is essenially more han ohers I characerizes weakly coupled sysem Also i should be noed ha eigenfuncion divided by uni volume is nearly equal o imporance funcion for deail marix oo Fig 3 Values of eigenfuncion are divided by uni volume of a sphere and imporance funcion for cenral row by heigh which are obained as a resul of fission marix soluion wih 729 zones Boh ψ norm and ψ + are normalized so ha sum of elemens of each is equal o Calculaion of he k evaluaion bias and is saisical uncerainy The calculaion of bias of k is performed using he formula () The sandard calculaion wih N TOT =2000 gives he bias K =05 % (see Table IV and Fig 4), whereas he calculaion wih he normalizaion on he esimaion of he imporance funcion gives he bias of only K =0,02% Hereafer he resuls are from calculaions wih imporance funcion obained using five ally division Table IV Sandard deviaion and bias calculaion s resuls Calculaion aking ino Sandard calculaion accoun imporance funcion N TOT 0, % corr, K, % % 0, % corr, K, % %

5 Thus, normalizaion using he imporance funcion does no decrease he saisical uncerainy, bu i allows obaining i using he sandard formula for he variance of he random variable offseing he impac of correlaions beween generaions Fig 4 Bias calculaed depending on N TOT Figure 5 illusraes he behavior of k in dependence on he number of simulaed generaions a N TOT =2000 One may see ha he sandard calculaion gives bias while he calculaion wih he normalizaion using he imporance funcion gives almos he correc answer Fig 6 Sums of he firs elemens of auocorrelaion funcion IV CONCUSION Fig 5 Dependence of k on he number of simulaed generaions In he sandard calculaion he sum of =00 elemens of auocorrelaion funcion a NTOT=2000 is equal o (see Fig 6) S c k k 6, wha means ha saisical uncerainy aking ino accoun correlaion beween generaions is σcorr / σ0 2S 2 imes higher han saisical uncerainy obained using he sandard formula for he variance of he random variable Saisical uncerainy of k excluding correlaions beween generaions is 0 =0002%, and corr =0004 % including hem In he calculaion wih he normalizaion using he imporance funcion he saisical uncerainy of k calculaion obained by he sandard formula is 0 =0004% The paper describes a mehod ha allows one o level he bias of he ecive muliplicaion facor calculaion wihou making any changes o he simulaion process and receive reliable value of is saisical error The proposed mehod is very simple o implemen and pracically does no increase he neuron hisory simulaion ime Moreover, is use reduces he number of neurons in he generaion and, consequenly, he oal calculaion ime by order of magniude The eciveness of he mehod is demonsraed by he example of he calculaion of a weakly coupled sysem «Whieside problem» As a resul, he work allows concluding he following The usage of he normalizaion using he esimaion of he imporance funcion reduces significanly he value of he maximum lag Calculaions of he auocorrelaion funcion using he sandard normalizaion gives he value of he lag =00, in case of he normalizaion using he imporance funcion, i is =5 2 The usage of he normalizaion using he imporance funcion reduces significanly he value of he bias of k esimaion A raher small N TOT =2000 such normalizaion gives almos non-biased k esimaion To obain he similar resul saisically i is necessary o use N TOT = The normalizaion using he imporance funcion doesn reduce he saisical error of he calculaion bu allows one o obain i by he sandard formula for he dispersion of a random value wihou aking ino accoun correlaions beween generaions because his correlaion is almos equal o zero

6 4 The soluion of he marix equaion almos coincides wih k obained by Mone-Carlo mehod wih he normalizaion using he imporance funcion REFERENCES V MAYOROV Esimaes of he Bias in he Resuls of Mone Carlo Calculaion of recors and sorages sies for nuclear fuel, Aomic Energy, v99(4), 2005, p T UEKI Inergeneraional Correlaion in Mone Carlo K-Eigenvalue Calculaions, Nuclear Science and Engineering, v4, 2002, p0 0 3 МА KAUGIN, DS OEYNIK, EA SUKHINO- KHOMENKO Evaluaion of he Sysemaic Error of Mone Carlo Calculaions of Neuron-physical Properies Using a Muliprocessor Compuer, Aomic Energy, v(2), 20, p МА KAUGIN, DS OEYNIK, EA SUKHINO- KHOMENKO The Esimaion Techniques of he Time Series Correlaions in Nuclear Reacor Calculaions by he Mone Carlo Mehod Using Muliprocessor Compuers, Physics of Aomic Nuclei, v74(4), 20, p T UEKI Mone Carlo criicaliy calculaion under exreme condiion, Journal of Nuclear Science and Technology, v49, 202, p RJ BRISSENDEN, AR GARICK Biases in he esimaion of k and is error by Mone Carlo Mehods, Annals of Nuclear Energy, v3, 986, p MA KAUGIN, DS OEYNIK, DA SHKAROVSKY Overview of he MCU Mone Carlo Sofware Package, Annals of Nuclear Energy, v82, 205, p EG WHITESIDES Difficuly in Compuing he k of he World, Trans Am Nucl Soc, 4, No2, 97 9 DS OEYNIK Mone Carlo Calculaion of Weakly Coupled Sysems, Aomic Energy, v99(4), 2005, p FB BROWN, SE CARNEY, BC KIEDROWSKI, WR MARTIN, "Fission Marix Capabiliy for MCNP, Par I - Theory", Mahemaics & Compuaion 203, Sun Valley, repor A-UR , alk A-UR (203)