PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
IX.1 SOLUTION USING MONTE
CARLO SIMULATION
MONTE CARLO SIMULATION
Introduction
Boltzmann eq: PDE with 7 variables
 Solution only for some simplified cases
Reactor: highly heterogeneous medium
 Classical numerical methods “not fitted” for an exact solution
Monte Carlo
resorting to random numbers to estimate a quantity as an
expected value in a stochastic process associated to the
problem at hand ( “survey”)
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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 flight
if 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
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
SAMPLING
Principle
Cumulative distribution function (c.d.f.) F of a random variable x
= monotonously non-decreasing function on [0,1]
F(x)
1

0
x*
x
 Draw a random number  uniformly distributed on [0,1]
 Inversion of F
 x* s.t. F(x*) = 
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
 rj 1  rj  s*
with
s* t.q.  v (rj , rj  s*)   ln 
(infinite reactor)
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Sampling of the collision kernel
Two steps
1. Interaction type
 i (r , v)
p

, i  c, s , f
Let
i
 t (r , v)
i * 1
i*
j 0
j 0
*
 i t.q.  p j     p j
Rem: if i* = f, sampling of the distribution of 
2. Speed and direction
Depends on the interaction sampled
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
ESTIMATION OF FINITE INTEGRALS
b
I   f ( x)dx
1
a
M
(x1,y1)
• n=0
(x3,y3)
f(x)
(x4,y4)
• (xi,yi) uniformly drawn from
[a,b], [m,M] resp., i=1…N
(x2,y2)
g(x)
• yi  f(xi)  n=n+1
m
a
b
Geometric interpretation of the integral:
n
~
I  ( M  m).(b  a).  (b  a).m
N
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
2
b
b
a
a
I   f ( x)dx   h( x) ( x)dx
• (x) 0
•
 x  [a,b]
b
  ( x)dx  1
a
(cf. MC  estimation of an expected value)
If xi, i=1…N, drawn independently from (x)
Then
Proof:
1
I
N
N
 h( x )
i 1
i
: unbiased estimator of I
 1 N

E ( I )  E
h
(
x
)

i 
 N i 1

1 N b

 h( xi ) ( xi )dxi  I
N i 1 a
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
ESTIMATION OF A REACTION RATE
Transport kernel
 Probabilistic transfer function: output of 1 collision  entry
in the next one
 ( r ', r )


r

r
'
. (v  v' ). (    ' )
T ( r ' , v ' ,  '  r , v,  )   t ( r , v ' )
   
2
r  r ' 
r  r' 
e
v
Collision kernel
 Probabilistic transfer function: entry in 1 collision  output
C ( r ' , v ' ,  '  r , v,  ) 
[  s ( r ' , v ' ,  '  v,  ) 
 (v )
 f (r ' , v' )]
4
 (r  r ' )
 t (r ' , v' )
Compact notation: P  (r , v, ) ; P'  (r ' , v' , ' )
 v (  )
T
(
P
'

P
)
dP

1

e
 = 1 for an infinite reactor

 C ( P'  P)dP  1
 Captures not accounted for
1
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Collision densities
 ( P)dP = expected number of n entering
Ingoing density:
/u.t. a collision with coordinates in dP about P
 ( P)  t ( P) ( P)
Outgoing density:  ( P)dP = expected number of n leaving /u.t.
a collision with coordinates in dP about P
Evolution equations
 ( P)  Q( P)    ( P' )C ( P'  P)dP'
 ( P)    ( P' )T ( P'  P)dP'
 ( P)   Q( P' )T ( P'  P)dP'    ( P" )C ( P"  P' )T ( P'  P)dP' dP"
 I ( P)    ( P' ) K ( P'  P)dP'
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
 o ( P)  I ( P)
Let
 j ( P)    j 1 ( P' ) K ( P'  P)dP' , j  1...
 j(P): ingoing density after j collisions

  ( P)    j ( P) : solution of the transport equation
j 0
 Not realistic
 Basis for solution algorithms
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
ESTIMATION OF A REACTION RATE
Preliminary problem
Estimation of R j   f ( P) j ( P)dP
with  j ( P)    j 1 ( P' ) K ( P'  P)dP'
and j-1(P) : pdf + K(P’  P) : non-negative function ?
w( P' )   K ( P'  P)dP
Let
and k ( P | P' ) 
K ( P'  P)
w( P' )
Algorithm
 Sample N values of P’i from j-1(P’)
 Sample next the corresponding Pi , i=1..N, from k(P|P’)
1
~
Rj 
N
N
 w( P' ) f ( P )
i 1
i
= unbiased estimator of Rj
~
E ( R )    dP' dPw( P' ) f ( P)k ( P | P' ) ( P' )
Proof:
  dPf ( P)  dP' K ( P'  P) ( P' )  R
i
j 1
j
j 1
j
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Solution of the transport equation
Estimation of R   f ( P) ( P)dP
with
 ( P)  I ( P)    ( P' ) K ( P'  P)dP' ?


j 0
j 0
Development in Neumann series  R   R j    f ( P) j ( P)dP
Algorithm (run i, i = 1…N)
1. j=0 ; sample Pio from I(P) / wio with
k ( P | Pij ) 
2. Sample Pi,j+1 from
with
wio   I ( P)dP
K ( Pij  P )
wi , j 1 ( Pij )
wi , j 1 ( Pij )   K ( Pij  P)dP
3. j = j + 1 ;  2 until the n is captured or exits the reactor
1
~
Rj 
N
j
N
W
i 1
ij
f ( Pij )
with
Wij   wik
k 0
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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 times
 Rather validation of classical solution schemes than
repeated calculations in industry
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
E=
1/(2)
2 : variance of the MC game
 : average time / history
Increasing the efficiency
 More info collected / history  better estimation
 More interesting events  biasing
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
IX.2 ESTIMATORS OF A REACTION
RATE
FIRST MOMENT OF THE SCORE
Adjoint form of the transport equation
Estimation of R   f ( P) ( P)dP
with  ( P)   Q( P' )T ( P'  P)dP'    ( P" )C ( P"  P' )T ( P'  P)dP' dP"
 I ( P)    ( P' ) K ( P'  P)dP'
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:
 K ( P  P' ) H ( P' )dP'
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
 Adjoint equation: H(P)  *(P)
 * ( P)  f ( P)   K ( P  P' ) * ( P' )dP'
Expression of the reaction rate based on importance? n
emitted by the source, then transported to a 1st collision
R   f ( P) ( P)dP
  I ( P) * ( P)dP
Expected contribution to the score due to a n emitted at P?
M 1 ( P)   T ( P  P' ) * ( P' )dP'
Thus R   Q( P) M 1 ( P)dP
1st moment? M 1 ( P)   T ( P  P' ) f ( P' )dP'  L( P  P' ) M 1 ( P' )dP'
~
~
~ (Physical interpretation ?)
with L( P  P' )   T ( P  P )C ( P  P' )dP
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
Free flight from P to P’
Capture at P’
Estimator
f(P,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”)
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Explicit form of the collision kernel
C ( P'  P" )  cc ( P' ) ( P" P )  cs ( P' )Cs ( P'  P" )

 c f ( P' ) kqk ( P' )Ck ( P'  P" )
k 1
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’
19
Expected score M1’(P) from a starter at P
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Contribution due to the 1st collision:
Free flight:
 T ( P  P' ) f ( P, P' )dP'
Capture:
 T ( P  P' )c ( P' ) f ( P' )dP'
c
 T ( P  P' )c ( P' )  C ( P'  P" ) f ( P' , P" )dP" dP'
Scattering:
Fission:
c
s
 T ( P  P ' )c
s
s

f
( P' )  kqk ( P' )  Ck ( P'  P" ) f k ( P' , P" )dP" dP'
k 1
Contribution due to the next collisions:
 dP"  dP'T ( P  P' )C ( P'  P" )M ' ( P" )
1
+
=
M1’(P)
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
PARTIALLY NON-BIASED ESTIMATORS
Definition
For the estimation of a reaction rate R   f ( P) ( P)dP
{f(P,P’), fc(P’), fs(P’,P”), fk(P’,P”)} = set of partially non-biased
estimators iff
M1(P)  M1’(P)  P
with
M 1 ( P)   T ( P  P' ) f ( P' )dP'  L( P  P' ) M 1 ( P' )dP'
and M 1 ' ( P)   dP' T ( P  P' ) f ( P, P' )  cc ( P' ) f c ( P' )
 cs ( P' )  Cs ( P'  P" ) f s ( P' , P" )dP"

 c f ( P' )  kqk ( P' )  Ck ( P'  P" ) f k ( P' , P" )dP"  
k 1
  L( P  P' ) M 1 ' ( P' )dP'
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Necessary and Sufficient Condition: independent terms
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
equal
I o ( P)   T ( P  P' ) f ( P' )dP'   dP' T ( P  P' ) f ( P, P' )  cc ( P' ) f c ( P' )
 cs ( P' )  Cs ( P'  P" ) f s ( P' , P" )dP"

 c f ( P' )  kqk ( P' )  Ck ( P'  P" ) f k ( P' , P" )dP"  
k 1
Particular cases
Estimator f(P) in the definition of R
Case without fission
 f ( P, P' )  f k ( P' , P" )  0
 f ( P, P' )  f ( P' )
 

f
(
P
'
)

f
(
P
'
,
P
"
)

f
(
P
'
,
P
"
)

0
s
k
 c
 f c ( P' )  f s ( P' , P" )  f ( P' )
Free-flight estimator
 f ( P, P' )  I o ( P)

 f c ( P' )  f s ( P' , P" )  f k ( P' , P" )  0
 At the start of each free flight, score = expected contribution
over all possible free flights
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Example: escape rate out of an homogeneous slab of thickness
L
We have R    ( P)( ( x  L) H ( x )   ( x) H ( x )) dP   f ( P) ( P)dP

f ( P) 
1
( ( x  L) H ( x )   ( x) H ( x ))
 t ( P)
But T ( x, v,  x  x' , v' ,  x ' )  t ( x' , v' )e
Then
t
x x' / x
. (v  v' ). ( x   x ' )
I o ( P)   T ( P  P' ) f ( P' )dP'
 

 
L  x 
x
.H ( x )
.H ( x )  exp   t ( x, v)
 exp   t ( x, v)

 x 
 x 
 
 
 Track-length estimator
(H(x) = 1 if x  0, 0 else)
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
 f c ( P' )  V (r ' )
 
 f ( P, P' )  f s ( P' , P" )  f k ( P' , P" )  0
Partially non-biased estimator associated to the free flight
 f ( P, P' )  f ( P' )  c ( P' ) / t ( P' ).V (r ' )  cc ( P' ).V (r ' )
 
 f c ( P' )  f s ( P' , P" )  f k ( P' , P" )  0
Corresponding reaction rate:
R   f ( P) ( P)dP

 c ( P)
V (r ) ( P)dP    c ( P) ( P)dP  Capture rate
 t ( P)
V
24
SECOND MOMENT OF THE SCORE
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
 Comparison 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
M 2 ( P)   T ( P  P' ) f 2 ( P' )dP'2  dP'T ( P  P' ) f ( P' )  dP" C ( P'  P" ) M 1 ( P" )
  dP' T ( P  P' )  C ( P'  P" ) M 2 ( P" )
MC game with partially non-biased estimators (no fission)
M 2 ' ( P)   dP' T ( P  P' )cc ( P' )( f c ( P' )  f ( P, P' )) 2
1
  Cr2  dP'T ( P  P' )  dP" C ( P'  P" )( f ( P, P' )  f s ( P' , P" )) 2 r M r ( P" )
r 0
  dP' T ( P  P' )  C ( P'  P" ) M 2 ' ( P" )
Comparison not really obvious…
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
IX.3 VARIANCE REDUCTION
ESTIMATION OF FINITE INTEGRALS
Analog estimation
b
b
a
a
Let I   f ( x)dx   h( x) ( x)dx
with
• (x) 0
 x  [a,b]
b
•
  ( x)dx  1
a
If xi, i=1…N, are sampled independently from (x)
1 N
Then I   h( xi ) : unbiased estimator of I, i.e. E ( I )  I
N i 1
Variance s 2  b (h( x)  I ) 2  ( x)dx

a
Estimator?
s.t. E (s 2 )  s 2
N
N
N
1

1
2
2
2

(
h
(
x
))

I


s2 
(
h
(
x
)

I
)

k

k
N  1  N k 1
N  1 k 1

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PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Importance sampling
Let
with
b
b
a
a
I   h( x) ( x)dx  
~
 ( x) ~
h( x) ~  ( x)dx
 ( x)
 ( x ) : probability density function
~
If xi, i=1…N, sampled independently from  ( x )
1
Ii 
N
Then
Variance:

 ( xk )
h( xk ) ~

 ( xk )
k 1
N
s 
2
i
a
s s  
2
2
i
b
b
a
: unbiased estimator of I
h( x) ( x)
2 ~
( ~
 I )  ( x)dx
 ( x)
 ( x)
h ( x)(1  ~ ) ( x)dx
 ( x)
2
: better or worse?
27
Particular case
h( x) ( x)
 ( x) 
I
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
~
 zero variance:
si2  0
!
But applicable only if the solution I is already known…
~
Practical use: choice of  ( x ) based on an approximation of I
 Better variance
Statistical weight
1
Ii 
N
 ( xk )
1 N
h( xk ) ~
  h( xk )w( xk )

N k 1
 ( xk )
k 1
N
 w(xk) = corrective factor of the estimator h(x) due to changing the
pdf used for the sampling
28
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
ESTIMATION OF A REACTION RATE
Preliminary problem
Estimation of R j   f ( P) j ( P)dP
with  j ( P)    j 1 ( P' ) K ( P'  P)dP'
and j-1(P): pdf + K(P’  P): non-negative function?
Sampling?
~
Based on a kernel K ( P '  P ) s.t.

~
K ( P'  P)dP  1
Objective: artificially increase the number of samplings
favorable to the estimation of Rj, in order to increase the
statistical quality of its estimation
29
Algorithm
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
 Sample N values of P’i from j-1(P’)
~
 Sample the corresponding Pi , i = 1..N, from K ( Pi '  P)
Let
K ( P'  P)
w( P' , P)  ~
K ( P'  P)
1
~
Rj 
N
the corresponding statistical weight
N
 w( P' , P ) f ( P )
i 1
i
i
i
Proof:
= unbiased estimator of Rj
~
E(R j )  
~
dP
'
dPw
(
P
'
,
P
)
f
(
P
)
K
( P'  P' ) j 1 ( P' )

  dPf ( P)  dP' K ( P'  P) j 1 ( P' )  R j
Solution of the transport equation
Estimation of R   f ( P) ( P)dP
with
 ( P)  I ( P)    ( P' ) K ( P'  P)dP' ?
Sampling from a modified kernel
Solution in Neumann series 

~
K ( P'  P)

R   Rj  
j 0
j 0
 f ( P)
j
( P)dP
30
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Algorithm (run i, i = 1…N)
1. j=0 ; sample Pio from I(P) / wio with wio   I ( P)dP
~
2. Sample Pi,j+1 from K ( Pij  P)
K ( Pij  Pi , j 1 )
Compute the statistical weight wi , j 1 ( Pij , Pi , j 1 )  ~
K ( Pij  Pi , j 1 )
3. j = j + 1 ;  2 until n is captured or exits the reactor
1
~
Rj 
N
Remark
j
N
W
i 1
ij
f ( Pij )
with
Wij   wik
k 0
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
31
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
EQUATION OF THE FIRST MOMENT
MC game based on the definition of the reaction rate
Reminder: analog case
M 1 ( P)   T ( P  P' ) f ( P' )dP'
  dP' T ( P  P' )  dP"C ( P'  P" ) M 1 ( P" )
Biased case
~
Let M 1 ( P ) 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”
~
~
WM 1 ( P)   T ( P  P' )W ' f ( P' )dP'
~
~
~
  dP' T ( P  P' )  dP"C ( P'  P" )W " M 1 ( P" )
32
MC game with partially non-biased estimators
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Reminder: analog case
M 1 ( P)   dP' T ( P  P' ) f ( P, P' )  ca ( P' ) f a ( P' )
 cs ( P' )  Cs ( P'  P" ) f s ( P' , P" )dP"

 c f ( P' )  kqk ( P' )  Ck ( P'  P" ) f k ( P' , P" )dP"  
k 1
  dP' T ( P  P' )  dP" C ( P'  P" )M 1 ( P" )
Biased case
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 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
33
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Biased case
~
WM 1 ( P)   dP' T ( P  P' ) W ' f ( P, P' )  Wc c~c ( P' ) f c ( P' )
 c~s ( P' )  Ws " Cs ( P'  P" ) f s ( P' , P" )dP"

 c~f ( P' )  kqk ( P' )  Wk " Ck ( P'  P" ) f k ( P' , P" )dP"  
k 1
~
~
~
  dP' T ( P  P' )  dP"C ( P'  P" )W " M 1 ( P" )
Estimation with no bias?
~
~
Q
(
P
)
M
1 ( P ) dP   Q ( P ) M 1 ( P ) dP

34
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
EXAMPLES OF BIASED KERNELS
Estimation of the escape probability (see above)
Slab of thickness L, 1D-model
 analog case:
 x'
du 

T ( x  x' )   t ( x' ) exp     t (u ) L (u )

x

x


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 sampled
 Transport kernel biased to prevent this non-informative
situation to occur and extend the interesting runs
 x'
du 

 t ( x' ) exp     t (u )
x
 x 
~

T ( x  x' ) 
. L ( x)
 L

x
du
du

exp     t (u ) .H ( x )    t (u )
.H ( x ) 
x
o
x
x


35
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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 flight
 No additional info if capture is actually sampled
 Collision 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 !
36
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
37
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
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
39