1

CH.IX: MONTE CARLO METHODS FOR TRANSPORT PROBLEMS

SOLUTION USING MONTE CARLO SIMULATION • MONTE CARLO SIMULATION • SIMULATION OF NEUTRON TRANSPORT• SAMPLING• ESTIMATION OF FINITE INTEGRALS• ESTIMATION OF A REACTION RATE• ADVANTAGES AND DRAWBACKS• IMPROVING THE SIMULATION EFFICIENCY

ESTIMATORS OF A REACTION RATE• FIRST MOMENT OF THE SCORE• PARTIALLY NON-BIASED ESTIMATORS• SECOND MOMENT OF THE SCORE

VARIANCE REDUCTION • ESTIMATION OF FINITE INTEGRALS• ESTIMATION OF A REACTION RATE• EXAMPLES OF BIASED KERNELS

2

MONTE CARLO SIMULATION

Introduction

Boltzmann eq: PDE with 7 variablesSolution only for some simplified cases

Reactor: highly heterogeneous medium Classical numerical methods “not fitted” for an exact solution

Monte Carloresorting to random numbers to estimate a quantity as an expected value in a stochastic process associated to the problem at hand ( “survey”)

IX.1 SOLUTION USING MONTE CARLO SIMULATION

3

SIMULATION OF NEUTRON TRANSPORT

Transport process = stochastic process!Estimation of transport-related quantities (e.g. reaction rate) as

their expected value on a large number of evolutions (“runs/histories”) of the neutron population

Algorithm

1. Draw the initial coordinates and speed of the n from the source density

2. Draw its free flightif it escapes the reactor, go to 4.

3. Draw the type of collision* if absorption, go to 4.* if scattering, draw the outgoing speed of the n* if fission, draw the number of n produced and their outgoing

speed + memorize the coordinates of the additional n

4. Deal with next n in memory (if appropriate) and go to 2.5. Go to 1. if there are still runs to play

4

SAMPLING

Principle

Cumulative distribution function (c.d.f.) F of a random variable x = monotonously non-decreasing function on [0,1]

Draw a random number uniformly distributed on [0,1] Inversion of F

x* s.t. F(x*) =

x

F(x)

1

0

x*

5

Sampling of the transition kernel

Negative exponential distribution

For an homogeneous reactor : F(s) = 1 - exp(-ts)

s* s.t. 1 - exp(-ts*) = ’ = 1 -

s* = - (ln )/t

General case

with (infinite reactor)

*

1 srr jj

ln),(.. ** srrqts jjv

6

Sampling of the collision kernel

Two steps

1. Interaction type

Let

Rem: if i* = f, sampling of the distribution of

2. Speed and direction

Depends on the interaction sampled

fscivr

vrp

t

ii ,,,

),(

),(

j

i

jj

i

j

ppqti

**

0

1

0

* ..

7

ESTIMATION OF FINITE INTEGRALS

b

adxxfI )(

Geometric interpretation of the integral:

mabN

nabmMI ).().).((

~

• n=0

• (xi,yi) uniformly drawn from [a,b], [m,M] resp., i=1…N

• yi f(xi) n=n+1

M

m

a b

f(x)

g(x)

1

(x1,y1)

(x2,y2)

(x3,y3)

(x4,y4)

8

N

iixhN

I1

)(1

b

a

b

adxxxhdxxfI )()()(

1)( b

adxx

(cf. MC estimation of an expected value)

If xi, i=1…N, drawn independently from (x)

Then : unbiased estimator of I

Proof:

(x) 0 x [a,b]

•

2

IdxxxhN

xhN

EIE

N

i

b

a iii

N

ii

1

1

)()(1

)(1

)(

9

ESTIMATION OF A REACTION RATE

Transport kernel

Probabilistic transfer function: output of 1 collision entry in the next one

Collision kernel

Probabilistic transfer function: entry in 1 collision output

Compact notation:

)'().'(.'

'

')',(),,',','( 2

),'(

vvrr

rr

rr

evrvrvrT

rr

t

v

)'()','(

)]','(4

)(),',','([

),,',','( rrvr

vrv

vvrvrvrC

t

fs

)',','(';),,( vrPvrP

1)'( dPPPC

)(1)'( vedPPPT

Captures not accounted for

1

= 1 for an infinite reactor

10

Collision densities

Ingoing density: = expected number of n entering /u.t. a collision with coordinates in dP about P

Outgoing density: = expected number of n leaving /u.t. a collision with coordinates in dP about P

Evolution equations

')'()'()( dPPPTPP

dPP)(

)()()( PPP t

dPP)(

')'()'()()( dPPPCPPQP

')'()'()(

"')'()'"()"(')'()'()(

dPPPKPPI

dPdPPPTPPCPdPPPTPQP

11

Rem:

equ. of (P) = (equ. of (P)) x t(P)

Natural interpretation of n transport 1 collision after the other

Formal solution using Neumann series

Let

j(P): ingoing density after j collisions

: solution of the transport equation

Not realistic Basis for solution algorithms

)()( PIPo

...1,')'()'()( 1 jdPPPKPP jj

)()(0

PP jj

12

ESTIMATION OF A REACTION RATE

Preliminary problem

Estimation of

with

and j-1(P) : pdf + K(P’ P) : non-negative function ?

Let and

Algorithm

Sample N values of P’i from j-1(P’)

Sample next the corresponding Pi , i=1..N, from k(P|P’)

= unbiased estimator of Rj

Proof:

dPPPfR jj )()(

dPPPKPw )'()'(

')'()'()( 1 dPPPKPP jj

)'(

)'()'|(

Pw

PPKPPk

)()'(1~

1ii

N

ij PfPwN

R

jj

jj

RPPPKdPPdPf

PPPkPfPdPwdPRE

)'()'(')(

)'()'|()()'(')~

(

1

1

13

Solution of the transport equation

Estimation of

with ?

Development in Neumann series

Algorithm (run i, i = 1…N)

1. j=0 ; sample Pio from I(P) / wio with

2. Sample Pi,j+1 from

with

3. j = j + 1 ; 2 until the n is captured or exits the reactor

with

')'()'()()( dPPPKPPIP dPPPfR )()(

dPPIwio )(

dPPPfRR jj

jj

)()(00

dPPPKPw ijijji )()(1,

)(

)()|(

1, ijji

ijij Pw

PPKPPk

)(1~

1ijij

N

ij PfWN

R

ik

j

kij wW

0

14

ADVANTAGES AND DRAWBACKS

Transport = natural stochastic process No restrictive assumptions on the transport equation Solution of the whole transport problem

Optimisation of a MC game: for the estimation of one reaction rate at a time

Number of runs: large for a given accuracy

Important computer timesRather validation of classical solution schemes than

repeated calculations in industry

15

IMPROVING THE SIMULATION EFFICIENCY

Difficulties related to the estimation of low reaction rates (e.g. transmission probability through a protection wall):

Few histories giving information on the rate to be estimated A large number of histories have to be played for the

statistical accuracy of the estimations

Efficiency E of a simulation algorithm

The shorter the computer time needed by a MC algorithm to reach a given accuracy, the higher its efficiency

Increasing the efficiency

More info collected / history better estimation More interesting events biasing

E = 1/(2) 2 : variance of the MC game

: average time / history

16

FIRST MOMENT OF THE SCORE

Adjoint form of the transport equation

Estimation of

with

Importance H(P) of a n – entering a collision at P – in the estimation of R? (see chap.VI)

Direct contribution due to a collision at point P: f(P) Expected contribution due to the next collisions:

')'()'()(

"')'()'"()"(')'()'()(

dPPPKPPI

dPdPPPTPPCPdPPPTPQP

IX.2 ESTIMATORS OF A REACTION RATE

dPPPfR )()(

')'()'( dPPHPPK

17

Adjoint equation: H(P) *(P)

Expression of the reaction rate based on importance? n emitted by the source, then transported to a 1st collision

Expected contribution to the score due to a n emitted at P?

Thus

1st moment?

with

dPPPI

dPPPfR

)(*)(

)()(

')'(*)'()()(* dPPPPKPfP

')'(*)'()(1 dPPPPTPM dPPMPQR )()( 1

')'()'(')'()'()( 11 dPPMPPLdPPfPPTPM PdPPCPPTPPL~

)'~

()~

()'( (Physical interpretation ?)

18

General set of estimators

Consider the following MC algorithm:

Samplings are performed from kernels T and C Score collected along a history based on estimators

associated to each possible event:

Event Estimator

Free flight from P to P’ f(P,P’)

Capture at P’ fc(P’)

Scattering from P’ to P” fs(P’,P”)

Fission (with k secondary n) at P’ with n emitted at P”

fk(P’,P”)

19

Explicit form of the collision kernel

with ci(P’): proba that the collision at P’ is of type i , i = c,s,f

P: point outside the domain of interest (capture) Cs(P’P”): scattering kernel – distribution of the outgoing

coordinates P”, given a scattering collision takes place at P’ qk(P’): proba that a fission at P’ produces k secondary n

Ck(P’P”): fission kernel – distribution of the coordinates P” of the secondary n, given a fission producing k n takes place at P’

)"'()'()'(

)"'()'()"()'()"'(

1

PPCPkqPc

PPCPcPPPcPPC

kkk

f

ssc

20

Expected score M1’(P) from a starter at P

Contribution due to the 1st collision:

Free flight:

Capture:

Scattering:

Fission:

Contribution due to the next collisions:

')',()'( dPPPfPPT

')'()'()'( dPPfPcPPT cc

'")",'()"'()'()'( dPdPPPfPPCPcPPT sss

'")",'()"'()'()'()'(1

dPdPPPfPPCPkqPcPPT kkkk

f

)"(')"'()'('" 1 PMPPCPPTdPdP +

M1’(P)=

21

PARTIALLY NON-BIASED ESTIMATORS

Definition

For the estimation of a reaction rate

{f(P,P’), fc(P’), fs(P’,P”), fk(P’,P”)} = set of partially non-biased estimators iff

M1(P) M1’(P) P

with

and

dPPPfR )()(

')'()'(')'()'()( 11 dPPMPPLdPPfPPTPM

')'(')'(

")",'()"'()'()'(

")",'()"'()'(

)'()'()',()'(')('

1

1

1

dPPMPPL

dPPPfPPCPkqPc

dPPPfPPCPc

PfPcPPfPPTdPPM

kkkk

f

sss

cc

22

Necessary and Sufficient Condition: independent terms equal

Particular cases

Estimator f(P) in the definition of R Case without fission

Free-flight estimator

At the start of each free flight, score = expected contribution over all possible free flights

")",'()"'()'()'(

")",'()"'()'(

)'()'()',()'('')'()'()(

1

dPPPfPPCPkqPc

dPPPfPPCPc

PfPcPPfPPTdPdPPfPPTPI

kkkk

f

sss

cco

0)",'()",'()'(

)'()',(

PPfPPfPf

PfPPf

ksc

0)",'()",'()'(

)()',(

PPfPPfPf

PIPPf

ksc

o

)'()",'()'(

0)",'()',(

PfPPfPf

PPfPPf

sc

k

23

Example: escape rate out of an homogeneous slab of thickness L

We have

But

Then

Track-length estimator

dPPPfdPHxHLxPR xx )()())()()()()((

))()()()(()(

1)( xx

t

HxHLxP

Pf

)'().'(.)','()',',',,( /'xx

xxtxx vvevxvxvxT xt

')'()'()( dPPfPPTPIo

)(.),(exp)(.),(exp xx

txx

t Hx

vxHxL

vx

(H(x) = 1 if x 0, 0 else)

24

Intuitive, binary estimator of the capture rate in a volume V (analog MC algorithm)

Simulation of the free flights and collision types, and unit contribution to the score when a capture is sampled

Partially non-biased estimator associated to the free flight

Corresponding reaction rate:

0)",'()",'()',(

)'()'(

PPfPPfPPf

rPf

ks

Vc

0)",'()",'()'(

)'().'()'().'(/)'()'()',(

PPfPPfPf

rPcrPPPfPPf

ksc

VcVtc

dPPPdPPrP

P

dPPPfR

c

V

Vt

c )()()()()(

)(

)()(

Capture rate

25

SECOND MOMENT OF THE SCOREComparison of the efficiency of different estimators

Reference MC game with f(P)

2nd moment: expected value of (f(P’)+s(P”))2, where s(P”) = score obtained starting from P”, leaving the 1st collision

MC game with partially non-biased estimators (no fission)

Comparison not really obvious…

)"()"'()'('

)"()"'(")'()'('2')'()'()(

2

12

2

PMPPCPPTdP

PMPPCdPPfPPTdPdPPfPPTPM

)"(')"'()'('

)"())",'()',()("'(")'('

))',()'()('()'(')('

2

221

0

22

PMPPCPPTdP

PMPPfPPfPPCdPPPTdPC

PPfPfPcPPTdPPM

rr

srr

cc

26

ESTIMATION OF FINITE INTEGRALS

Analog estimation

Let

If xi, i=1…N, are sampled independently from (x)

Then : unbiased estimator of I, i.e.

Variance

Estimator?

s.t.

1)( b

adxx

IX.3 VARIANCE REDUCTION

b

a

b

adxxxhdxxfI )()()( (x) 0 x

[a,b]

•

N

iixhN

I1

)(1

dxxIxhsb

a)())(( 22

2

1

2 ))((1

1Ixh

Ns k

N

k

22

1

))((1

1Ixh

NN

Nk

N

k22 )( ssE

with

IIE )(

27

Importance sampling

Let

with : probability density function

If xi, i=1…N, sampled independently from

Then : unbiased estimator of I

Variance:

: better or worse?

)(~

)()(

1

1 k

kN

kki

x

xxh

NI

dxxIx

xxhs

b

ai )(~

))(

~)()(

( 22

dxxx

xxhdxxxhI

b

a

b

a)(

~

)(~

)()()()(

)(~x

)(~x

dxxx

xxhss

b

ai )())(

~)(

1)((222

28

Particular case

zero variance: !

But applicable only if the solution I is already known…

Practical use: choice of based on an approximation of I

Better variance

Statistical weight

w(xk) = corrective factor of the estimator h(x) due to changing the pdf used for the sampling

02 isI

xxhx

)()()(

~

)(~x

)(~

)()(

1

1 k

kN

kki

x

xxh

NI

)()(1

1k

N

kk xwxh

N

29

ESTIMATION OF A REACTION RATE

Preliminary problem

Estimation of

with

and j-1(P): pdf + K(P’ P): non-negative function?

Sampling?

Based on a kernel s.t.

Objective: artificially increase the number of samplings favorable to the estimation of Rj, in order to increase the statistical quality of its estimation

dPPPfR jj )()(

')'()'()( 1 dPPPKPP jj

)'(~

PPK 1)'(~ dPPPK

30

Algorithm

Sample N values of P’i from j-1(P’)

Sample the corresponding Pi , i = 1..N, from

Let the corresponding statistical weight

= unbiased estimator of Rj

Proof:

Solution of the transport equation

Estimation of

with ?

Sampling from a modified kernel

Solution in Neumann series

jj

jj

RPPPKdPPdPf

PPPKPfPPdPwdPRE

)'()'(')(

)'()''(~

)(),'(')~

(

1

1

)'(~

)'(),'(

PPK

PPKPPw

)(),'(1~

1iii

N

ij PfPPwN

R

)'(~

PPK i

')'()'()()( dPPPKPPIP dPPPfR )()(

)'(~

PPK

dPPPfRR jj

jj

)()(00

31

Algorithm (run i, i = 1…N)

1. j=0 ; sample Pio from I(P) / wio with

2. Sample Pi,j+1 from

Compute the statistical weight

3. j = j + 1 ; 2 until n is captured or exits the reactor

with

Remark

Impact of biasing the kernel on the accuracy of the results?

Cases favored by resorting to the modified kernel w < 1 Cases unfavored by resorting to the modified kernel w

> 1

A couple of unfavored samplings might ruin the statistical accuracy

Biasing: dangerous if not cautiously used

)(~

PPK ij

dPPIwio )(

)(~

)(),(

1,

1,1,1,

jiij

jiijjiijji

PPK

PPKPPw

)(1~

1ijij

N

ij PfWN

R

ik

j

kij wW

0

32

EQUATION OF THE FIRST MOMENT

MC game based on the definition of the reaction rateReminder: analog case

Biased case

Let be the first moment of the score obtained from a starter n emitted at P with a unit statistical weight

Let W: statistical weight of the n at P

W’(P,P’): weight after a free flight from P to P’

W”(P’,P”): weight after a collision at P’ exited at P”

)"()"'(")'('

')'()'()(

1

1

PMPPCdPPPTdP

dPPfPPTPM

)"(~

")"'(~

")'(~

'

')'(')'(~

)(~

1

1

PMWPPCdPPPTdP

dPPfWPPTPMW

)(~

1 PM

33

MC game with partially non-biased estimatorsReminder: analog case

Biased caseW: statistical weight of the n at PW’(P,P’): weight after a free flight from P to P’W”(P’,P”): weight after a collision from P’ to P”

Wc(P,P’): weight due to the capture at P’ of a n emitted at P

Ws(P’,P”): weight due to a scattering from P’ to P” of 1 n emitted at P

Wk(P’,P”): weight due to a fission from P’ to P” of 1 n emitted at P

)"()"'(")'('

")",'()"'()'()'(

")",'()"'()'(

)'()'()',()'(')(

1

1

1

PMPPCdPPPTdP

dPPPfPPCPkqPc

dPPPfPPCPc

PfPcPPfPPTdPPM

kkkk

f

sss

aa

34

Biased case

Estimation with no bias?

)"(~

")"'(~

")'(~

'

")",'()"'(")'()'(~

")",'()"'(")'(~

)'()'(~)',(')'(')(~

1

1

1

PMWPPCdPPPTdP

dPPPfPPCWPkqPc

dPPPfPPCWPc

PfPcWPPfWPPTdPPMW

kkkkk

f

ssss

ccc

dPPMPQdPPMPQ )()()(~

)(~

11

35

EXAMPLES OF BIASED KERNELS

Estimation of the escape probability (see above)

Slab of thickness L, 1D-model

analog case:

Track-length estimator

Expected value of the escape probability accounted for from the start of any free flight

No additional info if this event of leak is actually sampledTransport kernel biased to prevent this non-informative

situation to occur and extend the interesting runs

')()(exp)'()'(

x

xx

Ltt

duuuxxxT

)(.

)(.)()(.)(exp

)(exp)'(

)'(~

'

x

Hdu

uHdu

u

duux

xxT L

x

x

ox

tx

L

xx

t

x

xx

tt

36

Estimation of the capture rate in a volume V (see above)Analog case: use of the collision kernel

Estimator associated to the free flight and scoring cc(P’)Expected value of the capture probability at the end of each

free flightNo additional info if capture is actually sampledCollision kernel biased to prevent this non-informative

situation to occur and extend the interesting runs

Remark

In both cases, “risk-free” biasing: Augmentation of the number of favorable cases No loss of information Statistical accuracy ok (all weights < 1)

BUT no stopping criterion of a history !

37

Russian roulette

If the weight W of a history goes below a threshold Wo:

Sampling of a random number , uniformly distributed on [0,1]

If < Wo, then the history goes on with a weight W / Wo

Else, the history is killed

Bias?

Expected value of the weight after a roulette:

E(W) = (W / Wo).P(history kept) + 0.P(history killed)

= W