![Page 1: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/1.jpg)
Probabilistic Approaches to Neutron
Transport Problems
Minmin Wang, University of Sussex
Growth & Division in Mathematics & Medicine, 4-6 November, UCL
![Page 2: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/2.jpg)
What is this talk about
• Some probability tools for spatial branching processes.
• Applications to neutron transport problems.
Based on joint work with Alexander M.G. Cox, Simon C. Harris, Emma Horton
and Andreas E. Kyprianou.
1
![Page 3: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/3.jpg)
Neutron flux
c©Wood.
ψt(r, v): neutron flux inside the nuclear
reactor core at time t emitted
from the configuration (r, v).
time t
time 0
(r,v)
2
![Page 4: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/4.jpg)
Neutron flux
c©Wood.
ψt(r, v): neutron flux inside the nuclear
reactor core at time t emitted
from the configuration (r, v).
time t
time 0
(r,v)
2
![Page 5: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/5.jpg)
Criticality
• Growth rate of ψt(r, v):
Supercritical Critical Subcritical
Put another way, ψt(r, v) � eλ0t , where λ0 = lead eigenvalue.
Criticality of the reactor is determined by sgn(λ0).
• How to evaluate λ0 for a given reactor design?
Monte-Carlo methods
3
![Page 6: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/6.jpg)
Criticality
• Growth rate of ψt(r, v):
Supercritical Critical Subcritical
Put another way, ψt(r, v) � eλ0t , where λ0 = lead eigenvalue.
Criticality of the reactor is determined by sgn(λ0).
• How to evaluate λ0 for a given reactor design?
Monte-Carlo methods
3
![Page 7: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/7.jpg)
Criticality
• Growth rate of ψt(r, v):
Supercritical Critical Subcritical
Put another way, ψt(r, v) � eλ0t , where λ0 = lead eigenvalue.
Criticality of the reactor is determined by sgn(λ0).
• How to evaluate λ0 for a given reactor design?
Monte-Carlo methods
3
![Page 8: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/8.jpg)
Criticality
• Growth rate of ψt(r, v):
Supercritical Critical Subcritical
Put another way, ψt(r, v) � eλ0t , where λ0 = lead eigenvalue.
Criticality of the reactor is determined by sgn(λ0).
• How to evaluate λ0 for a given reactor design?
Monte-Carlo methods
3
![Page 9: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/9.jpg)
Monte-Carlo methods
• A variety of Monte-Carlo methods:
• Neutron branching process (basic)
• Neutron random walk (many-to-one)
• h-neutron random walk (h-transform)
• Modelling of the neutron process as a
spatial branching process.
4
![Page 10: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/10.jpg)
Neutron Process
Motion of a neutron is governed by
• Fission
• Scattering
• Absorption
5
![Page 11: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/11.jpg)
Neutron Process
Motion of a neutron is governed by
• Fission
• Scattering
• Absorption
6
![Page 12: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/12.jpg)
Neutron Process
Motion of a neutron is governed by
• Fission
• Scattering
• Absorption
7
![Page 13: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/13.jpg)
Neutron Process
Motion of a neutron is governed by
• Fission
• Scattering
• Absorption
8
![Page 14: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/14.jpg)
Neutron Process
Motion of a neutron is governed by
• Fission
• Scattering
• Absorption
9
![Page 15: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/15.jpg)
This is a spatial branching process!
• branching fission & absorption
• spatial motion transport, scattering & absorption
Remark. No neutron-neutron interactions.
Inhomogeneous branching and scattering rates.
10
![Page 16: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/16.jpg)
This is a spatial branching process!
• branching fission & absorption
• spatial motion transport, scattering & absorption
Remark. No neutron-neutron interactions.
Inhomogeneous branching and scattering rates.
11
![Page 17: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/17.jpg)
This is a spatial branching process!
• branching fission & absorption
• spatial motion transport, scattering & absorption
Remark. No neutron-neutron interactions.
Inhomogeneous branching and scattering rates.
12
![Page 18: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/18.jpg)
This is a spatial branching process!
• branching fission & absorption
• spatial motion transport, scattering & absorption
Remark. No neutron-neutron interactions.
Remark. Inhomogeneous branching and scattering rates.
13
![Page 19: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/19.jpg)
1st MC method: Neutron branching process
• Run a simulation of the neutron
branching process starting from the
configuration (r, v).
• Repeat for k times.
N it = number of the surviving neutrons at
time t from the i-th simulation.
• ψt(r, v) = E[N it ] ≈ 1
k
∑ki=1 N
it . Then
λ0 ≈1
tlogψt(r, v) ≈ 1
tlog( 1
k
k∑i=1
N it
).
14
![Page 20: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/20.jpg)
Neutron Branching Process: Analysis
• Relatively easy to implement and
geometry insensitive.
• Slow convergence and costly to run.
c©Wood.
15
![Page 21: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/21.jpg)
Towards other MC methods
• We only employ the first moment:
ψt(r, v) = E[N it ].
• Can we find another random variable Nt also satisfying
ψt(r, v) = E[Nt ]
but at the same time
• either easier to simulate
• or having a smaller variance
16
![Page 22: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/22.jpg)
Towards other MC methods
• We only employ the first moment:
ψt(r, v) = E[N it ].
• Can we find another random variable Nt also satisfying
ψt(r, v) = E[Nt ]
but at the same time
• either easier to simulate
• or having a smaller variance
16
![Page 23: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/23.jpg)
Many-to-one & neutron random walk
We can suppress branching by
• simulating a single neutron path (rs , vs)
• then applying the many-to-one formula:
ψt(r, v) = E[e∫ t0β(rs ,vs )ds 1{survival at t}
]where β depends only on the branching parameters.
× weight
17
![Page 24: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/24.jpg)
Neutron random walk: Analysis
• Quick/cheap to run.
• The exponential weight e∫ t0β(rs ,vs )ds could increase variance.
18
![Page 25: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/25.jpg)
h-transform & h-neutron random walk
• Applying an h-transform to the law of the neutron path yields
ψt(r, v) = E[
exp
(∫ t
0
Lh(rs , vs)
h(rs , vs)+β(rs , vs)ds
)h(r, v)
h(rt , vt)1{survival at t}
]where (rs , vs) is a neutron path which scatters at rate ∝ 1
h and
where L is some differential-integral operator.
1h
19
![Page 26: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/26.jpg)
h-transform & h-neutron random walk (Cont’)
• Optimal choice of h:
(L + β)h = λ0h
In general, such a solution is not known explicitly.
• Instead, we substitute with a guess.
• Since the h-transform formula is valid for a general h, there is no loss
in accuracy.
• As soon as h = 0 at boundary, the scattering will force the neutron
to stay inside.
20
![Page 27: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/27.jpg)
h-transform & h-neutron random walk (Cont’)
• Optimal choice of h:
(L + β)h = λ0h
In general, such a solution is not known explicitly.
• Instead, we substitute with a guess.
• Since the h-transform formula is valid for a general h, there is no loss
in accuracy.
• As soon as h = 0 at boundary, the scattering will force the neutron
to stay inside.
20
![Page 28: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/28.jpg)
Some questions
• What would be a practically good choice of h?
• It seems more efficient to have h = 0 at boundary.
On the other hand, this causes a divergence in the scattering rate.
Will this be a hurdle?
• How to update our knowledge on the eigenfunction and use it to
improve the convergence?
21
![Page 29: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/29.jpg)
Some questions (Cont’)
• Other Monte-Carlo methods?
Consider the following method:
• Start with k particles.
• Let the system evolve for some time T0. Denote by K0 the number
of neutrons in the system. If K0 > k, discard K0 − k particles. If
K0 < k, sample new particles from the distribution given by the
current states.
• Iterate.
Long-time behaviour? Genealogy of the particles?
22
![Page 30: › ~ucakwat › gd › wang.pdf · Probabilistic Approaches to Neutron Transport ProblemsProbabilistic Approaches to Neutron Transport Problems ... Neutron Branching Process: Analysis](https://reader033.vdocuments.net/reader033/viewer/2022041819/5e5ca912e31ec7628a73c9b2/html5/thumbnails/30.jpg)
Some questions (Cont’)
• Ergodicity of the neutron branching process?
Denote by (ris , vis), 1 ≤ i ≤ Ns , the configurations of the neutrons at time
s. Let Xs = 1Ns
∑i δ(ris ,vis ).
• In the supercritical regime, Harris, Horton & Kyprianou recently
showed that Xs converges to some deterministic measure. What is
the speed of this convergence?
• What about critical/subcritical regimes (conditioned upon survival)?
• Other applications for the many-to-one or h-transform methods?
Note that there is an analogous theory for discrete-time/generation-based
branching processes (ask Emma Horton!)
23