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
EXAMPLES OF BIASED KERNELS

2. IX.1 SOLUTION USING 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”)

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
Draw the initial coordinates and speed of the n from the source density
Draw its free flight
if it escapes the reactor, go to 4.
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
Deal with next n in memory (if appropriate) and go to 2.
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*) = 
F(x) 1  x 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)

6. Sampling of the collision kernel
Two steps
Interaction type
Let 
Rem: if i* = f, sampling of the distribution of 
Speed and direction
Depends on the interaction sampled

7. ESTIMATION OF FINITE INTEGRALS1 M
n=0
(xi,yi) uniformly drawn from [a,b], [m,M] resp., i=1…N
yi  f(xi)  n=n+1
(x3,y3)
(x1,y1)
f(x)
(x2,y2)
g(x)
(x4,y4) m a b
Geometric interpretation of the integral:

8. 2 (x) 0  x  [a,b] If xi, i=1…N, drawn independently from (x)
(cf. MC  estimation of an expected value)
If xi, i=1…N, drawn independently from (x)
Then : unbiased estimator of I
Proof:

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:
= 1 for an infinite reactor
Captures not accounted for
  1

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

11. Formal solution using Neumann series
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

12. ESTIMATION OF A REACTION RATE Preliminary problem
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:

13. Solution of the transport equation
Estimation of
with ?
Development in Neumann series 
Algorithm (run i, i = 1…N)
j=0 ; sample Pio from I(P) / wio with
Sample Pi,j+1 from
with
j = j + 1 ;  2 until the n is captured or exits the reactor

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 times
Rather 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
2 : variance of the MC game
E = 1/(2)
 : average time / history

16. IX.2 ESTIMATORS OF A REACTION RATE
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:

17. (Physical interpretation ?)
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
(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’

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: +
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

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 

23. Example: escape rate out of an homogeneous slab of thickness L
We have 
But
Then
 Track-length estimator
(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:
 Capture rate

25. SECOND MOMENT OF THE SCORE
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
MC game with partially non-biased estimators (no fission)
Comparison not really obvious…

26. IX.3 VARIANCE REDUCTION 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.
(x) 0  x  [a,b]
with

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?

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

29. ESTIMATION OF A REACTION RATE Preliminary problem
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

30. Solution of the transport equation
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 

31. Algorithm (run i, i = 1…N)
j=0 ; sample Pio from I(P) / wio with
Sample Pi,j+1 from
Compute the statistical weight
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

32. EQUATION OF THE FIRST MOMENT
MC game based on the definition of the reaction rate
Reminder: 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”

33. MC game with partially non-biased estimators
Reminder: analog case
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

34. Biased case
Estimation with no bias?

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 sampled
Transport kernel biased to prevent this non-informative situation to occur and extend the interesting runs

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 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 !

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

38. References in Monte Carlo simulation
A primer for the Monte Carlo method, Ilya M. Sobol‘, CRC Press, Boca Raton, 1994
Monte Carlo methods, Malvin H. Kalos, Paula A. Whitlock, J. Wiley & Sons, New York, 1986
Monte Carlo particle transport methods: Neutron and photon calculations, Iván Lux, László Koblinger, CRC Press, Boca Raton, 1991