Multi-level ROM uses a low-fidelity model to extract the active feature subspace.

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Development of Multi-Level Reduced Order
Modeling
CONFERENCE PAPER · JUNE 2015
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Mohammad Abdo
North Carolina State University
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2. Development of Multi-Level Reduced Order Modeling Methodology
Mohammad G. Abdo and Hany S. Abdel-Khalik
School of Nuclear Engineering, Purdue University
400 Central Drive, Purdue Campus, NUCL Bldg., West Lafayette, IN 47906
abdo@purdue.edu; abdelkhalik@purdue.edu
INTRODUCTION
Over the past decade, the nuclear engineering
community in the United States and abroad has been heavily
investing in the development and implementation of
advanced science-based (i.e., employing fine-mesh, first-
principles, tightly coupled multi-physics) modeling and
simulation of nuclear reactor systems. Although these
models are expected to improve the understanding of the
reactors behavior, their performance is expected to be
handicapped by the enormous computational cost required
for their execution, especially when computationally
intensive analyses are of interest such as uncertainty
quantification and design optimization. To address this
challenge, reduced order modeling (ROM) techniques [1, 2]
development has been underway to help realize the goals of
advanced science-based modeling and simulation.
Past work has demonstrated a number of algorithms
that can render reduction for reactor physics calculations
[3]. The general outcome of this research is that one may
render reduction for a high fidelity model by executing such
model a number of times r (referred to as the effective
dimensionality of the model and is typically in the order of
few hundreds for reactor models), which is several orders of
magnitude smaller than the original model dimensionality m
which measures either the size of the state space (i.e., flux),
the parameters (e.g., cross-sections), or responses (e.g., pin
power distribution), all expected to be very large in
dimensionality. Notwithstanding this significant reduction,
the execution of high fidelity codes few hundred times is
still impractical even on leadership supercomputers. We
therefore present a new development in ROM, referred to as
multi-level ROM, which allows one to extract the effective
dimensionality by executing the high fidelity model in a
small sub-domain of the overall problem domain, i.e., pin
cell or 2D lattice vs. whole core. We show that this
approach is as capable as the previously developed ROM
approach in capturing the errors resulting from the reduction
and establishing realistic upper-bounds on their magnitude
over the expected range of variations for the high fidelity
model.
DESCRIPTION OF THE ACTUAL WORK
Unlike conventional surrogate construction techniques,
or low order physics approximations, ROM offers a
rigorous approach by which the effective dimensionality of
the physics model is determined without altering the
formulation of the high fidelity physics model. The idea is
to assume that the state variables, the input parameters, and
the output responses, are varying over lower dimensional
manifolds (referred to as active subspaces). See below for
the mathematical description of ROM techniques.
Any type of reduction introduces errors due to the
discarding of non-influential components. Capturing these
errors, and more importantly upper-bounding them, is very
crucial to lend credibility to the reduction process. Past
work has arrived at two important results. The first result
deals with the straightforward application of ROM to the
high fidelity model, wherein the active subspace is
constructed based on few hundred executions of the high
fidelity model. In this case, upper-bounds on the reduction
errors can be established with sufficiently high probability,
i.e., 0.99999. We refer to this active subspace as the ideal
active subspace, since it is demonstrated to have the
smallest size as compared to the subspaces constructed in
this work. The second result of past work has proved that
one could develop an upper-bound on the ROM error for
any user-defined active subspace. Of course, if the subspace
employed is a bad approximation to the ideal active
subspace, the error bounds would be too high. We employ
this second result in this summary by employing a physics-
informed application of the high fidelity model to sub-
regions of the problem domain to extract the active
subspace. This approach is referred to as multi-level ROM,
or MLROM. The physics of the problem will be employed
to inform our choice of the sub-domain(s) used to represent
the full order model. In particular, we consider a 2D lattice
model depleted to 60 GWd/MTU to represent the full high
fidelity model. The goal is to employ few static (i.e.,
without depletion) pin cell models to identify the active
subspace. Error bounds will be constructed using the pin-
cell-determined active subspace, which will be verified
numerically by comparing against the non-reduced (i.e., full
order) simulation of the lattice model. Static evaluation of
the flux for a pin cell model few hundred times is
computationally inexpensive. In future work, this idea will
be applied to core-wide calculations, wherein some
representative lattices may also be needed to capture the
core-wide active subspace. This will be investigated in
future work.

3. MATHEMATICAL DESCRIPTION OF MLROM
Consider a reactor physics model, abstractly defined as:
 y f x (1)
where xm
x are reactor physics parameters, e.g., cross-
sections, ym
y   are reactor responses of interest e.g.,
eigenvalue, peak clad temperature, reaction rates, etc., and
mx and my are the numbers of parameters and responses,
respectively.
The ROM replaces the original simulation with a
reduced order representation f such that:
   y xf x f x T T (2)
where both Tx and Ty are rank-deficient linear mappings for
parameters and response spaces, respectively, described by:
z zm mT
z z z

 T Q Q  for ,z x y
The effective dimensionalities of these operators, i.e.
  dim R zz mrT - with R being the range of the operator
Tz - are typically much smaller than the original dimensions
of the parameters and responses spaces, i.e.,
max( , ) min( , )x ym m x yr r m m . The columns of the matrix
x mx
m r
x

Q  define the range of all possible input parameter
perturbations that have dominant impact on the response of
interest, whereas the matrix
y my
m r
y

Q  define the range of
all possible response variations resulting from all possible
parameter variations. Depending on the number of
parameters and responses, one may elect to render a single
reduction in the parameter or response spaces, or
simultaneous reduction in both spaces. A number of
algorithms have been developed to achieve that; the reader
may consult earlier work for more details on the
construction of the Q matrices [3, 4].
To ensure the ROM model is credible, one must bound
the errors resulting from discarding components in the input
parameter space and/or the response space. To achieve that,
we define the following error matrix E which calculates the
exact reduction errors using a number of samples. The ijth
element of E defines the error in the ith
response recorded in
the jth
random sample:
     
 
,: ,: ( )
[ ]
T T
i j y y i x x j
ij
i j
f x i i f x
f x


Q Q Q Q
E (3)
The elements of this error matrix can be used to
calculate an upper-bound on the error via the following
theoretical result due to Dixon [5]:
 
   
2
2
1
1
01,2,
max 1 ( ) ,
i s
xi s
x pdf t dt



   E E

 (4)
where the multiplier 1 x  is arbitrary. Recent work has
determined a numerical value of 1.0164 for this multiplier
which ensures that the bound is not too conservative. The
straightforward application of Dixon theory employs a
multiplier of approximately 8. Details on this aspect of the
error bound determination may be found in the following
references [6, 7].
In this work we assume that no reduction is done in the
response space, i.e., y T I is the identity matrix, while xT
is determined using gradient-based ROM approach applied
to a pin cell model, for more details about this approach
reader can consult the work in [3]
NUMERICAL TESTS AND RESULTS
For this demonstration, we employ a benchmark lattice
model for the Peach Bottom Atomic Power Station Unit 2
(PB-2), which is a 1112 MWe Boiling Water Reactor
(BWR) manufactured by General Electric with the fuel rods
arranged over a 7x7 grid. This benchmark is designed by
OECD/NEA and documented in [8]. Given the requirement
to have flat pin power distribution, several pin cell design
(different enrichments and gad content) are typically
employed in BWR lattices.
The 2.93% enriched UO2 with 3% gadolinium pin cell
is first depleted to 30 GWd/MTU. The resulting
composition is used as the reference model for constructing
the subspace. This pin is assumed to be the most
representative of all pins in the lattice. Note here that our
goal is to identify the dominant cross-sections for all the
pins, hence this step requires familiarity with the model.
After the subspace is constructed, we employ it to model
other fuel pins to establish whether our initial assumption of
representativity is adequate. If the assumption is adequate,
we move to the next level, and test its adequacy to represent
the entire lattice; if not, other pins are added to construct a
more representative subspace.
The nominal dimension mx of the parameter subspace
for the reference pin cell is (93 fuel nuclides + 15 clad
nuclides + 2 moderator nuclides + 1 gap nuclide) * 3
reactions * 44 energy groups =14,652 parameters. The
effective dimensionality is selected to be 600. Note here that
the error bounds are expected to vary from one response to
the next because of two reasons. First, depending on the
quality of the active subspace, different responses are
expected to be captured to different degrees of accuracy.
Second, because ROM attempts to identify the most
influential components, non-influential responses such as
the very fast and very thermal groups are expected to have
higher relative errors than the rest of the groups.
Figures 1 through 8 illustrate how the subspace
extracted from the low-fidelity model behaves when used
with different pin cells. Even-numbered figures display the
errors in the flux range of (1.85 – 2.35 MeV), whereas the
odd-numbered figures are for the range (0.625 – 1.0 eV).
Figures 1 through 4 show how the subspace behaves if used
to identify the parameters of a 1.33% enriched UO2 pin cell,
whereas figures 5 through 8 are for a 2.93% enriched UO2

4. pin cell. Figures 1 and 2 show the actual error vs. the error
bound due to parameter reduction in a 1.33% enriched UO2
pin cell after depleting to 15 GWd/MTU. In both figures the
red dots represent failed cases where the blue ones represent
the success, i.e., when the error bound is exceeded by the
actual error. All the cases showed a failure probability of
less than 0.1 as the theory predicts. The red solid 45-degree
line separates the failure and success regions. In this
demonstration, we picked a low probability of success, i.e.,
0.9, in order to realize some failure. In practice however, the
success probability is set to at least 0.99999.
Fig. 1. Fast Flux Error (15 GWd/MTU ).
The previous figure shows that the maximum actual error in
the fast flux range is 0.12%.
Fig. 2. Thermal Flux Error (15 GWd/MTU).
Fig. 3. Fast Flux Error (45 GWd/MTU).
Fig. 4. Thermal Flux Error (45 GWd/MTU).
Figure 2 shows similar results for the thermal range. Figures
3 and 4 examine the active subspace applied on the same pin
cell if the initial composition resulted from a 45 GWd/MTU
depletion.
The previous figures show that the maximum error in the
specified fast and thermal ranges are 0.28% and 0.42%,
respectively. Figures 5 and 6 show the fast and thermal flux
errors for the 2.93% enriched UO2 pincell after depleting to
15 GWd/MTU.
Fig. 5. Fast Flux Error (15GWd/MTU).
Fig. 6. Thermal Flux Errors (15 GWd/MTU).
Maximum errors in fast and thermal flux are found to be
0.3% and 0.33% respectively. Finally, figures 7 and 8 plot
the errors if the pin cell is initially depleted to 45
GWd/MTU.

5. Fig. 7. Fast Flux Errors (45 GWd/MTU).
Again the previous figure shows that constraining the cross
sections to the active space extracted from the reference pin
cell model resulted in an error that does not exceed 0.3% for
the specified fast range of (1.85 – 2.35 MeV).
Fig. 8. Thermal Flux Errors (45 GWd/MTU).
The thermal range errors follow the same pattern and
recorded a maximum value of 0.3%.
CONCLUSIONS
This exploratory studies investigate a new idea to construct
the active subspace for a model that is computationally
expensive to execute. The basic idea is to employ the
physics in a sub-domain, akin to lattice physics calculations,
and let the reduction theory determine the error bounds
resulting from this application to other pin cell models.
Initial results indicate that the idea is sound for the lattice
model employed representing an LWR reactor. The next
step will be to apply this idea to the lattice level. We would
like to point out that while the idea here is similar to
standard homogenization theory (at least in spirit, since no
actual homogenization is applied), we are able to calculate
error bounds on the active subspace employed, which is not
possible with existing homogenization theory techniques. If
successful, this approach could help realize the potential of
employing high fidelity simulation tools, currently under
development in many institutions around the country, in a
practical manner that can benefit the end-users, i.e., nuclear
practitioners.
ACKNOWLEDGMENT
The first author would like to acknowledge the support
received from the department of nuclear engineering at
North Carolina State University to complete this work in
support of his PhD.
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