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FurtherInvestigationofErrorBoundsforReducedOrderModeling

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ANS MC2015 - Joint International Conference on Mathematics and Computation (M&C), Supercomputing in Nuclear Applications (SNA) and the Monte Carlo (MC) Method • Nashville, TN • April 19-23, 2015, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2015)

Further Investigation of Error Bounds for Reduced Order Modeling

Mohammad Abdo and Hany S. Abdel-Khalik.

School of Nuclear Engineering, Purdue University

Corresponding Address

abdo@purdue.edu; abdelkhalik@purdue.edu

ABSTRACT

This manuscript investigates the level of conservatism of the bounds developed in earlier work to capture the errors resulting from reduced order modeling. Reduced order modeling is premised on the fact that large areas of the input and/or output spaces can be safely discarded from the analysis without affecting the quality of predictions for the quantities of interest. For this premise to be credible, ROM models must be equipped with theoretical bounds that can guarantee the quality of the ROM model predictions. Earlier work has devised an approach in which a small number of oversamples are used to predict such bound. Results indicated that the bound may sometimes be too conservative, which would negatively impact the size and hence the efficiency of the ROM model.

Key Words: Error bounds, reduced order modeling.

INTRODUCTION

Reduced order modeling (ROM) denotes any process by which the complexity of the model can be reduced to render its repeated execution computationally practical. Almost all engineering practitioners rely on ROM in some form. To warrant the use of the ROM model in lieu of the original model, the quality of its predictions must be assessed against the original model predictions. This is in general a difficult problem and it largely depends on how the ROM model is constructed. In our earlier work, we have relied on the use of randomized techniques to identify the so-called active subspace that can be used to reduce the effective dimensionality of the models [Abdel-Khalik, et al., 2012]. In doing so, the same physics model is employed, however its input parameters and output responses are constrained to their respective active subspace. This approach for building the ROM models has many advantages, among which is the ability to construct an upper-bound on the error resulting from the reduction. This bound provides a warranty to the user that future predictions of the ROM model will be within the bound when compared to the original model predictions.

Earlier work has shown that the construction of the bound depends on the probability distribution function (PDF) from which the oversamples are drawn. This was inspired by the analytic work by Dixon [1983], where he uses a normal distribution to estimate the 2-norm of a matrix. We have noticed that the use of this PDF results in an overly conservative bound, implying that the actual reduction errors can be one order of magnitude less than the nominal bound. This manuscript investigates the effect of the PDF choice on the level of conservatism of the bound, and whether there exists other distributions that render a more realistic bound estimate.

Abdo, M., and Abdel-Khalik, H.

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DESCRIPTION OF PROPOSED APPROACH

Consider a model of reactor physics simulation of the form:

y f x (1)

where nx are n-reactor physics parameters, e.g., cross-sections, my are m-reactor responses of interest e.g., eigenvalue, peak clad temperature, etc. ROM approach is aiming to approximate the original model by f to replace it in any computationally expensive analysis, such as uncertainty characterization. To certify the robustness of the ROM, the process must be equipped with an error criterion:

b userf x f x x S (2)

where b is an error upper-bound to be determined based on the reduction spaces. Whereas useris user-defined tolerance and S is the parameter active space extracted by any ROM technique.

Using our ROM approximation f will be:

f x f x Ν Κ

where both N and K are rank-deficient transformation matrices such that:

T m my y

N Q Q , dim R yrN , and min( , )yr m n ,

T n nx x

K Q Q , dim R xrK , and min( , )xr m n .

Now let’s look at the error operator as an unknown black box which can be sampled and aggregated in a matrix E where thij element of E represents the error in the thi response of the thj sample, which can be written as:

,: ,: ( )[ ]

T Ti j y y i x x j

ij

i j

f x i i f xE

f x

Q Q Q Q

(3)

The matrix E calculates the component of the function f that is truncated, i.e., discarded, by the ROM application. If the norm of the matrix E can be estimated, an upper-bound on the resulting ROM errors in the function evaluation can be computed. Note that each row of the matrix E represents a response, implying that if one treates each row as a matrix, it is possible to calculate a different error bound for each response. This is important since each response is expected to have its own reduction error.

Earlier work has shown that a matrix norm can be estimated using randomized inner products, which is due to [Dixon, 1983]. This approach may be described as follows. Consider a matrix

m NE and a random vector nx such that ix , where is a known distribution. Then

x can be used to estimate the 2-norm of E via:

,x p E E (4)

Further Investigation of Error Bounds for Reduced Order Modeling

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where 1

x . It is intuitive that p depends on both and the probability distribution

function of x. In this work we chose to perform that numerically and not to constrain the distribution to the normal or even the uniform distributions. In fact the main goal of this work is to inspect the sensitivity of the error statement to the distribution used. To do this we fixed p to be 0.9 and then find that makes 90% of the test cases satisfy the bound. Repeat that for many distributions and pick the distribution with the smallest multiplier .

The analytic version of this statement was first introduced by Dixon (1983), when he used a normal distribution to estimate the 2-norm of a matrix, Dixon has shown that in case of normal

distribution the multiplier 2

corresponds to a probability of 11 , which means that

with an value of 10, the multiplier will be 8 with a probability of success of 0.9. As we estimate the rank of an ROM model, the magnitudes of the singular values typically become closer to each other implying that xE is expected to be close to E , thereby rendering a

multiplier of 8 impractical. To overcome this situation, we have inspected other distributions and numerically tested the multiplier in each case to determine the distribution with the smallest multiplier.

CALCULATIONAL PROCEDURE AND RESULTS

To find the multiplier corresponding to each distribution the following procedure is deployed: 1. Generate a random matrix E and a random vector x sampled from the distribution under

inspection. 2. Compute E and xE .

3. Repeat steps 1, 2 million times.

4. Compute the multiplier such that 90% of the cases satisfies that: xE E

(i.e. 0.9x E E ).

The previous procedure is repeated for different distributions, the multiplier is computed; the computed multiplier is then tested with a different sample set. Eq. (4) is verified using a scatter plot and the number of failures is reported and compared to the probability predicted by the theory. The multipliers are shown in the following table followed by the scatter plots of the distributions that gave the largest and the smallest multipliers.

Table I. Multipliers of Different Distributions

Gaussian(0,1) 7.98 Chi‐square 1.31

Uniform(‐1,1) 13.2 Log‐normal 1.50

Binomial(N,0.9) 1.02 Beta(0.5,(N‐1)/2) 1.65

Poisson 1.67 Beta(1,10) 1.44

Exponential 1.49

Abdo, M., and Abdel-Khalik, H.

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From the previous table we conclude that the binomial distribution (1 p)n x n xxC p has the

smallest multiplier which provides a realistic bound.

Figure 1. Gaussian distribution

Figure 2. Uniform distribution

The x-axis in the top right two figures show the calculated bound using the Gaussian PDF and the actual error, with the red points indicating failure, where the actual error exceeds the estimated upper-bound. The trend shows that the bound could be much bigger than the actual error, therefore justifying our current investigation for better distributions. Fig. 3 presents similar results to Fig. 2 but now calculated using the binomial distribution which calculates more realistic bounds.

Further Investigation of Error Bounds for Reduced Order Modeling

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Figure 3. Binomial distribution

CONCLUSIONS

The ROM error bound estimation requires sampling of the actual error using a user-defined PDF. Several oversamples are drawn from such PDF to calculate the corresponding exact errors of the ROM model with the maximum of these errors multiplied by a scalar to establish an upper-bound on the ROM error. Earlier work has noticed that the scalar multiplier could result in an unnecessarily conservative bound, and identified the reason for that being the use of Gaussian PDF to select the oversamples. This work employed numerical experiments to show that one may use a binomial distribution to calculate a more realistic bound, i.e., one that is closer to the actual reduction errors.

ACKNOWLEDGEMENTS

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.

REFERENCES

1. Mohammad G. Abdo and Hany S. Abdel-Khalik, Propagation of Error Bounds due to Active Subspace Reduction, Transactions of American Nuclear Society, Summer 2014.

2. S. S. Wilks, Mathematical statistics, John Wiley, New York, 1st ed. 1962. 3. John D. Dixon, Estimating extremal eigenvalues and condition numbers of matrices, SIAM

1983; 20(2): 812–814. 4. Abdel-Khalik, H., et al., “Overview of Hybrid Subspace Methods for Uncertainty

Quantification and Sensitivity Analysis,” Annals of Nuclear Energy, 52, pp.28-46 (2013).A Tutorial on Applications of dimensionality reduction and function approximation.