effect of random variable distributions on the probabilistic Error Bound

1. Further Investigation of Error Bounds
for Reduced Order Modeling
Mohammad Abdo2, Congjian Wang1,2 and Hany Abdel-Khalik1,2
1School of Nuclear Engineering, Purdue University, IN, USA
2Nuclear Engineering Department, North Carolina State University, NC, USA
Motivations
• To practically perform intensive
reactor physics tasks:
‒ Employ Reduced Order
Modeling
‒ Build high accuracy surrogates
‒ Identify active
subspaces/manifolds.
‒ Equip with error analysis
modules.
• To reduce complexity
‒ Use simplified/decoupled physics
‒ Use homogenization, MLROM
Quarter-core virtual geometry model
of a PWR
http://www.casl.gov/image_gallery.sht
ml
Fuel Rod,
Control Rod,
Burnable
Poison Rod
Assembl
ies
Structural
Materials
Overview of Reduction Algorithms
• Reduction can be performed on various interfaces (parameter, state space
and/or response of interest).
‒ Gradient-based (Parameter reduction):
o Requires Sensitivity information.
o Requires Adjoint runs.
‒ Gradient-free Snapshots (Response reduction):
o Requires forward runs.
Gradient-Based Reduction Algorithm
• Randomly perturb the input parameters.
• Sample the sensitivities of the pseudo responses w.r.t the input parameters
(G).
• Find the Range of the sensitivity profile R(G) Using any linear algebra
approach.
Snapshots (Gradient-Free) Reduction Algorithm
• State variable interface reduction.
• Can be represented as a reduction in model A that is passed directly to Model
B (loosely-coupled physics).
• Response of Model A is randomly sampled.
• Active response subspace is identified.
Error Analysis
• Consider the physics model :  y f x
• Reduction Error :
       
 
,: ,:
[ ]
T T
i j y y i x x j
ij
i j
f x i i f x
E
f x


Q Q Q Q
• Probabilistic Error bounds:
1
,N
iw w
where
  ,w p E E
• Using Normal distribution can make the bound 1-2 orders of magnitude larger
than actual error.
Current Contribution
• Many distributions are inspected and the multiplier  is computed such
that the actual norm is less than the estimated norm in 90 % of the cases.
  0.9w B B
• A random matrix B is used instead of the error matrix E to reduce the
analysis cost.
• The distribution with the least multiplier is picked as the corresponding
estimated norm will be the least conservative and hence the most practical
(estimated norm is the closest to the actual norm).
Numerical Results
Uniform (-1,1) 13.2 Poisson 1.67 Log-normal 1.50
Normal (0,1) 7.98 Exponential 1.49 Beta(0.5,(N-1)/2) 1.65
Binomial (N,0.9) 1.02 Chi-square 1.31 Beta(1,10) 1.44
• The analysis shows that the numerically computed multiplier for the normal
distribution agreed with the analytic value proposed by [Dixon 1983].
• Binomial distribution gives the least multiplier and hence is chosen for
future error bound estimation.
Numerical Results
• Both Gaussian and Normal
distributions give non practically
conservative bounds.
• Binomial distribution:
‒ Least multiplier
‒ Linear pattern
‒ Slope is close to 1.0 (Even for
cases that the prediction fails
the estimated norm is very close
to the actual norm).
Conclusions
• ROM error estimation requires sampling of the actual error and matrix-vector
multiplications by randomly sampled vectors.
• Using the binomial distribution can remove the unnecessary conservativeness
of the bounds coming from the Gaussian or Uniform distributions.
• These results were employed on realistic neutronic problems.
• Getting rid of the impractical bounds eased the use of this approach in
propagating error bounds across different levels of reduction and hence the use
on loosely-coupled multi-physics problems.
• This is also in MLROM where the subspace was extracted from a pin cell then
deployed for a full assembly with no violations for the error bound.