anssummer2015

MotivationObjectives

Background of Supporting Algorithms and TheoryNumerical tests and results

ConclusionsAcknowledgements

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Multi-level Reduced Order Modeling with Robust ErrorBounds

Mohammad G. Abdoand

Hany S. Abdel-Khalik

School of Nuclear EngineeringPurdue University

[email protected] and [email protected]

June 10, 2015

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Motivation

ROM is indispensiblefor analysis withrepetitive executions.

ROM premised on theassumption: intrinsicdimensionality�nominaldimensionality.

ROM discardscomponants withnegligible impact onreactor attributes ofinterest and hencemust be equippedwith error metrics.

Can extract "active subspaces" from a reduced complexity model thatundergoes similar physics. 2 / 51




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Objectives

Apply the reduction and identify the active subspaces in a much moreefficient methodology.

Equip the reduced model with a robust error bound that can test therepresentitaveness of the active subspaces and hence define avalidation domain that includes different conditions and differentscenarios.

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ROMAlgorithmsError Estimation

Definition

A nonlinear function f is said to be reducable if there exist matricesUrx ∈ Rn×rx and/or Ury ∈ Rm×ry such that:∥∥∥∥∥ f (x)− Ury UT

ry f (Urx UTrx x)

f (x)

∥∥∥∥∥ ≤ ε4 / 51




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Reduction Algorithms

In our context, reduction algorithms refer to two different algorithms [4],each is used at a different interface:

Gradient-free Snapshot Reduction Algorithm (Reduces response interface).Gradient-based Reduction Algorithm(Reduces parameter interface).

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Gradient-free Snapshot Reduction Algorithm

Consider the reducible model under inspection to be described by:

φ = f (x) , (1)

1 Generate k random parameters realizations:{xi}k

i=1.

2 Execute the forward model k times and record the corresponding kresponses:

{φi = f

(xi)}k

i=1 , and aggregate them into:

8 =[φ1 φ2 · · · φk

]∈ Rm×k .

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Gradient-free Snapshot Reduction Algorithm

Consider the reducible model under inspection to be described by:

φ = f (x) , (1)

1 Generate k random parameters realizations:{xi}k

i=1.2 Execute the forward model k times and record the corresponding k

responses:{φi = f

(xi)}k

i=1 , and aggregate them into:

8 =[φ1 φ2 · · · φk

]∈ Rm×k .

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Snapshot Reduction (cont.)

3 Calculate the singular value decomposition (SVD):

8 = Uy Sy VTy ;where Uy ∈ Rm×k .

4 Collect the first ry columns of Uy in Ury to span the active responsesubspace, where ry ≤ min (m, k).

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Gradient-based Reduction

This algorithm may be described by the following steps:1 Execute the adjoint model k times to get:

G =

[dRpseudo

1dx

∣∣∣∣x1

· · ·dRpseudo

kdx

∣∣∣∣xk

].

2 From SVD of: G = Ux Sx VTx , one can pick the first rx columns of Ux

(denoted by Urx ) to span the active parameter subspace.

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Error Estimation

To estimate the error resulting from the reduction, the ij th entry of theoperator E can be written as:

[E]ij =fi(xj)− Ury (i, :)UT

ry (i, :) fi(Urx UT

rx xj

)fi(xj) ,

where Urx ∈ Rn×rx and Ury ∈ Rm×ry are matrices whose orthonormalcolumns span the parameter and response spaces respectively.

We need to estimate an upper bound for the error in each response.

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Error Estimation (cont.)

The 2-norm of E (or each row in E) can be estimated using:

P{‖E‖ ≤ η max

i=1,2,...s‖Ew (i)

‖

}≥ 1−

(∫ 1η2

0pdfw2

1(t)dt

)s

(2)

where E ∈ Rm×N with N being the number of sampled responses and wis an N-dimensional random vector sampled from a known distribution DThis probabilistic statement has its roots in Dixon’s theory [1983], wherehe sampled w (i) from a standard normal distribution and found ananalytic value for the probability in terms of the multiplier η

It is intuitive that if the user presets a probability of sucess then the valueof the multiplier η depends solely on D from which w is sampled.

careful inspection showed that the estimated error can be multiple orderof magnitudes larger than the actual error (unneccessarily conservative).

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Error Estimation (cont.)

This motivated the numerical inspection of many distributions and theselection of the most practical one (i.e. which gave the least multiplier η).

The inspection showed that the distribution which gave the least η is thebinomial distribution. [2, 3]

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Distribution Selection

Figure : Uniform Distribution. Figure : Gaussian Distribution.

Estimated norm is orders of magnitude off.

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Distribution Selection [cont.]

The binomial shows a linearstructure arround the 45-degreesolid line.

This means that even if the caseis a failure case (i.e. The actualnorm is greater than thebound),the estimated norm willstill be very close to the actualnorm

This appealing behaviourmotivates the use of the binomialdistribution to get rid ofunneccessarily conservativebounds.

Figure : Binomial Distribution

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Case Study 1Case Study 2Case Study 3

Case Study 1

Benchmark lattice for Peach Bottom Atomic Power Station Unit2 (PB-2,1112MWe BWR designed by OECD/NEA and manufactured by GeneralElectric).

Figure : 7x7 BWR Benchmark.14 / 51




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Case Study 1 [cont.]

The idea is that for such an assembly, running the forward model and theassembly model rx times to identify the active parameter subspace isstill very expensive !! .

Can we extract the active subspace from running only subdomain of theproblem? Or a reduced complexity model?

Figure : Calculation Levels.

http://www.nrc.gov/about-nrc/emerg-preparedness/images/fuel-pellet-assembly.jpg

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Why don’t we extract the parameters active subspace for each of thenine pin cells and try to visualize them ?!.

(x ∈ R49784)!!!!

We are not visualizing to see what does the topology of thisn-dimensional surface look like ! We are only trying to see how far iseach subspace from the other ! How different !

Figure : Scatter Visualization for the active subspaces.

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Why don’t we extract the parameters active subspace for each of thenine pin cells and try to visualize them ?!. (x ∈ R49784)!!!!



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Case Study [cont.]

The previous figure defends that active parameter subspaces for the 9pins are pretty close.

This motivates that the active parameter subspace for the wholeassembly might be revealed from sampling a pin or more !!

Tests Description:

Identify the parameter active subspace for one (or more) pin cells

Construct an error bound for each response

test the identified subspace on different pins then on the whole assembly.

If successful, test it in different conditions !!

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Case Study [cont.]

Nominal dimension for the parameter space n = 127nuclides×7reactions ×56 energy groups = 49784

3 pins are depleted to 30 GWd/MTU then used to extract the subspace(2.93% UO2 with 3 % gd, 1.94%UO2,2.93%UO2 ) and rx is taken to be1500

Test1: The subspace is tested on the highest enrichment pin and acompletely different one.

First two figures will show the error in flux within two selective energyranges: 1.85-3.0 MeV and 0.625-1.01 eV for the most dominant Pin Cell.

Then a figure showing the maximum and mean errors over all energiesand hence are taken as the bounds.

Test2: This is repeated for another pin cell (Mixture 4)

Test3: The extracted subspace is then employed on the whole assemblydepleted to the same point (30GWd/MTU). The results for this test isshown in 9 figures showing the maximum and mean error for the 9unique mixtures. 27 / 51




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Test 1

Figure : Fast Flux Error (Mix 500, LF). Figure : Thermal Flux Error(Mix 500, LF).

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Test 2

Figure : Error Boubds (Mix 500, LF). Figure : Actual Errors(Mix 4, LF).

The left figure shows the typical bounds doesn’t exceed 3% formaximum error and is less than 0.7% for mean error.

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Test 3

Figure : Actual Errors (Mix 1, HF). Figure : Actual Errors(Mix 2, HF).

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Test 3 [cont.]


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Test 3 [cont.]


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Test 3 [cont.]

Figure : Actual Errors (Mix203, HF). Figure : Actual Errors(Mix 212, HF).

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Test 3 [cont.]

Figure : Actual Errors (Mix 213, HF).

Figures show that thesubspace extracted fromthe low-fidelity model fullyrepresent the fullassembly at 30GWd/MTU and hotconditions.

the maximum error didnot exceed 3% and themean error is always lessthan 0.7% which isexactly consistent withwhat we had from thelow-fidelity model.

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Case Study 2

The assembly is depleted to (60 GWd/MTU)

The next 9 figures show the maximum and mean errors for the 9 differentmixtures.

This test aims to inspect the behaviour of the active subspace extractedfrom the low fidelity model at different composition due to differentburnup.

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Case Study 3

The assembly is depleted to the End of the first cycle (20 GWd/MTU)and at Cold conditions

The next 9 figures show the maximum and mean errors for the 9 differentmixtures.

This test aims to inspect the behaviour of the active subspace extractedfrom the low fidelity model at different composition due to differentburnup and at different temperature.

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Conclusions

ROM errors are reliably quantified using realistic bounds (i.e., actualerror is close to error bound).

Provide-a-first-of-a-kind approach to quantify errors resulting fromdimensionality reduction in nuclear reactor calculations.

Can be used to experiment with different ROM techniques to determineoptimum performance for application of interest.

Quantify errors resulting from homogenization theory (a form ofdimensionality reduction). Multi-physics ROM, where one physicsdetermines active subspace for next physics.

Efficient rendering of active subspaces for expensive model, MLROM

Using MLROM enables the application of ROM on all models that wasexpensive enough to prohebit executing the model to extract thesubspace.

This enables the determination of the validation space and One canmake a statement about how good is the subspace if used withconditions different than those used in the sampling process.

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Conclusions

ROM errors are reliably quantified using realistic bounds (i.e., actualerror is close to error bound).

Provide-a-first-of-a-kind approach to quantify errors resulting fromdimensionality reduction in nuclear reactor calculations.

Can be used to experiment with different ROM techniques to determineoptimum performance for application of interest.

Quantify errors resulting from homogenization theory (a form ofdimensionality reduction). Multi-physics ROM, where one physicsdetermines active subspace for next physics.

Efficient rendering of active subspaces for expensive model, MLROM

Using MLROM enables the application of ROM on all models that wasexpensive enough to prohebit executing the model to extract thesubspace.

This enables the determination of the validation space and One canmake a statement about how good is the subspace if used withconditions different than those used in the sampling process. 47 / 51




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Acknowledgements

I’d like to acknowledge the support of the Department of NuclearEngineering at North Carolina State University to complete this work insupport of my PhD.

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Bibliography I

SCALE:A Comperhensive Modeling and Simulation Suite for NuclearSafety Analysis and Design,ORNL/TM-2005/39, Version 6.1, Oak RidgeNational Laboratory, Oak Ridge, Tennessee,June 2011. Available fromRadiation Safety Information Computational Center at Oak RodgeNational Laboratory as CCC-785.

M. G. ABDO AND H. S. ABDEL-KHALIK, Propagation of error boundsdue to active subspace reduction, ANS, 110 (2014), pp. 196–199.

, Further investigation of error bounds for reduced order modeling,ANS MC2015, (2015).

Y. BANG, J. HITE, AND H. S. ABDEL-KHALIK, Hybrid reduced ordermodeling applied to nonlinear models, IJNME, 91 (2012), pp. 929–949.

J. D. DIXON, Estimating extremal eigenvalues and condition numbers ofmatrices, SIAM, 20 (1983), pp. 812–814.

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Bibliography II

N. HALKO, P. G. MARTINSSON, AND J. A. TROPP, Finding structure withrandomness:probabilistic algorithms for constructing approximate matrixdecompositions, SIAM, 53 (2011), pp. 217–288.

P. G. MARTINSSON, V. ROKHLIN, AND M. TYGERT, A randomizedalgorithm for the approximation of matrices, tech. report, Yale University.

J. A. TROPP, User-friendly tools for random matrices.

S. S. WILKS, Mathematical statistics, John Wiley, New York, 1st ed.,1962.

F. WOOLFE, E. LIBERTY, V. ROKHLIN, AND M. TYGERT, A fastrandomized algorithm for the approximation of matrices, preliminaryreport, Yale University.

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Questions/Suggestions?

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anssummer2015

Documents

different algorithms

theory numerical tests

error metrics

reduced model

reducible model

different interface

rom discards componants

robust error bounds