https://data-science.llnl.gov/sites/data_science/files/anna_matsekh_machine_learning_for_memory_reduction_in_the_implicit_monte_carlo_simulations_of_the_thermal_radiative_transfer_0.pdf
Machine
Learning for IMC
Anna Matsekh
Machine Learning for Memory Reduction in
the Implicit Monte Carlo Simulations of the
Thermal Radiative Transfer 1
Anna Matsekh
Luis Chacon, HyeongKae Park, Guangye Chen
Theoretical Division
Los Alamos National Laboratory
1 LA-UR-18-26613
Machine
Learning for IMC
Anna Matsekh
Thermal Radiative Transfer
• Thermal Radiative Transfer (TRT) equations
• describe propagation and interaction of photons with the
surrounding material
• are challenging to solve due to the stiff non-linearity and
high-dimensionality of the problem
• TRT applications at LANL include simulations of
• Inertial Confinement Fusion experiments
• astrophysical events
(a) ICF
(b) supernova
Figure: TRT applications
Machine
Learning for IMC
Anna Matsekh
Implicit Monte Carlo Simulations
• Advantages, compared to the deterministic case
• easier to extend to complex geometries and higher dimensions
• easier to parallelize
• Disadvantages
• Monte Carlo solutions to IMC equations exhibit statistical
variance and IMC convergence rate is estimated to be
p
O(1/ Np )
where Np is the number of simulation particles
• Even when advanced variance reduction techniques employed,
Monte Carlo simulations
• exhibit slow convergence
• prone to statistical errors
• require a very large number of simulation particles
• Implicit Monte Carlo codes are typically very large, long running
codes with large memory requirements at checkpointing &
restarting
Machine
Learning for IMC
Anna Matsekh
Machine Learning for IMC
Project Goal: use parametric Machine Learning methods in order to
reduce memory requirements at checkpointing & restarting in the
IMC simulations of Thermal Radiative Transfer using
• Expectation Maximization and Weighted Gaussian Mixture
Model-based approach for ‘particle-data compression’,
introduced in Plasma Physics to model Maxwellian particle
distributions by Luis Chacon and Guangye Chen
• Expectation Maximization with Weighted Hyper-Erlang Model in
order to compress isotropic IMC particle data in the frequency
domain
• Expectation Maximization and von Mises Mixture Models for
compression of anisotropic directional IMC data on a circle
(work-in-progress)
Machine
Learning for IMC
TRT Equations
Anna Matsekh
Consider 1-d Transport Equation without scattering and without
external sources in the Local Thermodynamic Equilibrium (LTE):
∂Iν
1
1 ∂Iν
+µ
+ σν Iν = σν Bν
c ∂t
∂x
2
coupled to the Material Energy Equation
ZZ
Z
∂T
cv
=
σν Iν dν dµ − σν Bν dν
∂t
(1)
(2)
where the emission term
Bν (T ) =
2 h ν3
1
c 2 e kh Tν − 1
(3)
is the Planckian (Blackbody) distribution and
• Iν = I (x, µ, t, ν) - radiation intensity
• ν - frequency, T - temperature
• σν - opacity, cv - material heat capacity
• k - Boltzmann constant, h - Planck’s constant, c - speed of light
Machine
Learning for IMC
Anna Matsekh
Implicit Monte Carlo Method of
Fleck and Cummings
Transport Equation
∂Iν
1
1 ∂Iν
+µ
+ (σνa + σνs ) Iν = σνa c urn +
c ∂t
∂x
ZZ 2
1
σνs (bν /σp )
σν 0 Iν 0 dν 0 dµ
2
(4)
Material Temperature Equation (T n = T (tn ) ≈ T (t), tn ≤ t ≤ tn+1 )
Z tn+1 Z Z
cν T n+1 = cν T n − f σp c ∆t urn + f
dt
σν 0 Iν 0 dν 0 dµ (5)
tn
• f = 1/(1 + α β c ∆t σp ) - Fleck factor
• σνa = f σν - effective absorption opacity
• σνs = (1 − f ) σν - effective scattering opacity
• ur - radiation energy density, bν (T ) - normalized Planckian
R
• σp = σν bν dν - Planck opacity
• α ∈ [0, 1] s ...