Mc2015 posterlandscape

Further Investigation of Error Bounds

for Reduced Order ModelingMohammad Abdo2, Congjian Wang1,2 and Hany Abdel-Khalik1,2

1School of Nuclear Engineering, Purdue University, IN, USA2Nuclear 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 xE

f x

Q Q Q Q

• Probabilistic Error bounds:

1, N

iw wwhere

,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.