Propagation of Error Bounds due to Active Subspace Reduction
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This document summarizes the propagation of error bounds due to active subspace reduction in computational models. It presents two algorithms for performing active subspace reduction: one that is gradient-free and reduces the response or state space, and one that is gradient-based and reduces the parameter space. It then develops a theorem for propagating error bounds across multiple reductions, both in the parameter and response spaces. Numerical experiments on an analytic function and a nuclear reactor pin cell model are used to validate the error bound approach.
![196
Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014
Uncertainty Quantification and Sensitivity Analysis Methods
Propagation of Error Bounds due to Active Subspace Reduction
Mohammad G. Abdo and Hany S. Abdel-Khalik
Department of Nuclear Engineering, NC State University, Raleigh, NC, 27695-7909
mgabdo@ncsu.edu; abdelkhalik@ncsu.edu
INTRODUCTION
Reduced order modeling (ROM) has been recognized
to be an essential tool in support of modern predictive
strategies relying on the use of high fidelity and often
computationally expensive models [1]. ROM provides an
efficient manner by which these models can be executed
repeatedly for various engineering analyses, such as
design optimization and uncertainty characterization and
propagation. In our developments, ROM employs
mathematical subspace transformations to reduce the
effective dimensionality of the various model interfaces.
To assure a robust reduction for all envisaged model
variations, this summary develops probabilistic upper-
bounds that propagate all reduction errors across all
model interfaces at which reduction is rendered. This is
particularly important in a multi-physics setting, where
the output of one model is passed as an input to the next
model in the chain. The new error bound is tested using
both an analytic nonlinear function and a representative
PWR pin cell model.
DESCRIPTION OF THE ACTUAL WORK
ROM seeks to reduce the effective dimensionality of
the various model interfaces. For example in neutronics
calculations, the model interfaces include the input
parameters (e.g., cross-sections), the state function (i.e.,
flux), and the responses of interest (e.g., pin power,
reaction rates and detector response). The premise is that
a small number of degrees of freedom r in the parameter
space is sufficient to characterize to a high degree of
accuracy all possible variations in the state function and
responses of interest, whose original dimensionalities are
often much higher than r.
Our error metric development relies on two reduction
algorithms that were introduced in earlier work [1]. The
first algorithm uses a Gradient-free technique to reduce
the dimensionality of the response or state space without
the necessity of computing the derivatives (in contrast to
the second algorithm). By executing the forward model k
times we can generate k realizations and the
corresponding responses ^ `1 1
and{ } ( )
kk
i i i i i
x y f x ,
respectively. These can be aggregated in a matrix whose
left singular vectors yrU span the range of the active
output space. In practice the reduced dimensionality of
this space yr is increased until the error upper-bound](https://image.slidesharecdn.com/propagationoferrorboundsduetoactivesubspacereduction-150709235229-lva1-app6892/85/Propagation-of-Error-Bounds-due-to-Active-Subspace-Reduction-2-320.jpg)
![.x x
T
x r re xI W W
where xe is the error resulting from constraining x to the
active subspace.
Notice that discarding components in the parameter
space will give rise to errors in the response space even if
no reduction in the response space is rendered, (If only
the second algorithm was employed). Estimating the total
response error resulting from reduction at both parameter
and response levels represents another goal of this
summary.
The proposed error bound is based on a theorem
developed in the applied mathematics community [2, 3];
they are introduced below for the sake of a complete
discussion. The interested readers may consult the cited
references for more details.
Theorem 1: Let ^ `: | 1; 2n T
S x x x n t|||n T
|||||||| be a unit
hyper sphere, and consider an n-dimensional random
vector n
x n
whose entries are iid and are sampled from
a uniform distribution, (0,1)(0,1) over S. For a general
positive definite real matrix n nu
A n nu
with eigenvalues:
1 2 0,nO O Ot t t !nO !nOnn one can show that [2]:
^ `1
2
Prob 1 .T T n
x x x xO T
S T
d d t A A (2)
where T is a scalar multiplier greater than 1.0.](https://image.slidesharecdn.com/propagationoferrorboundsduetoactivesubspacereduction-150709235229-lva1-app6892/85/Propagation-of-Error-Bounds-due-to-Active-Subspace-Reduction-4-320.jpg)
![197
Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014
Uncertainty Quantification and Sensitivity Analysis Methods
Now if m nu
B m nu
such that T T
A LL B B and
2 2
= nT D
S
§ ·
¨ ¸
© ¹
then (2) can be re-written as [2, 3]:](https://image.slidesharecdn.com/propagationoferrorboundsduetoactivesubspacereduction-150709235229-lva1-app6892/85/Propagation-of-Error-Bounds-due-to-Active-Subspace-Reduction-5-320.jpg)
Propagation of Error Bounds due to Active Subspace Reduction
- 2. 196 Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 Uncertainty Quantification and Sensitivity Analysis Methods Propagation of Error Bounds due to Active Subspace Reduction Mohammad G. Abdo and Hany S. Abdel-Khalik Department of Nuclear Engineering, NC State University, Raleigh, NC, 27695-7909 mgabdo@ncsu.edu; abdelkhalik@ncsu.edu INTRODUCTION Reduced order modeling (ROM) has been recognized to be an essential tool in support of modern predictive strategies relying on the use of high fidelity and often computationally expensive models [1]. ROM provides an efficient manner by which these models can be executed repeatedly for various engineering analyses, such as design optimization and uncertainty characterization and propagation. In our developments, ROM employs mathematical subspace transformations to reduce the effective dimensionality of the various model interfaces. To assure a robust reduction for all envisaged model variations, this summary develops probabilistic upper- bounds that propagate all reduction errors across all model interfaces at which reduction is rendered. This is particularly important in a multi-physics setting, where the output of one model is passed as an input to the next model in the chain. The new error bound is tested using both an analytic nonlinear function and a representative PWR pin cell model. DESCRIPTION OF THE ACTUAL WORK ROM seeks to reduce the effective dimensionality of the various model interfaces. For example in neutronics calculations, the model interfaces include the input parameters (e.g., cross-sections), the state function (i.e., flux), and the responses of interest (e.g., pin power, reaction rates and detector response). The premise is that a small number of degrees of freedom r in the parameter space is sufficient to characterize to a high degree of accuracy all possible variations in the state function and responses of interest, whose original dimensionalities are often much higher than r. Our error metric development relies on two reduction algorithms that were introduced in earlier work [1]. The first algorithm uses a Gradient-free technique to reduce the dimensionality of the response or state space without the necessity of computing the derivatives (in contrast to the second algorithm). By executing the forward model k times we can generate k realizations and the corresponding responses ^ `1 1 and{ } ( ) kk i i i i i x y f x , respectively. These can be aggregated in a matrix whose left singular vectors yrU span the range of the active output space. In practice the reduced dimensionality of this space yr is increased until the error upper-bound
- 3. y y T y r re I y U U meets a user-defined error tolerance. One goal of this summary is to extend this upper-bound to smooth nonlinear models. The second algorithm uses a Gradient-based reduction technique in which the active input subspace is constructed via k executions of the adjoint model. The derivatives of the pseudo responses with respect to the input parameters are then computed and aggregated in: 1 1 . k pseudops x x eudo n kkdRdR dx dx u ª º « » « » ¬ ¼ G .n k pse dRkdRp dRk dxdx Similarly to the previous algorithm, we can use the SVD of G to select the first xr left singular vectors xrW to span the active subspace for the parameters space such that:
- 4. .x x T x r re xI W W where xe is the error resulting from constraining x to the active subspace. Notice that discarding components in the parameter space will give rise to errors in the response space even if no reduction in the response space is rendered, (If only the second algorithm was employed). Estimating the total response error resulting from reduction at both parameter and response levels represents another goal of this summary. The proposed error bound is based on a theorem developed in the applied mathematics community [2, 3]; they are introduced below for the sake of a complete discussion. The interested readers may consult the cited references for more details. Theorem 1: Let ^ `: | 1; 2n T S x x x n t|||n T |||||||| be a unit hyper sphere, and consider an n-dimensional random vector n x n whose entries are iid and are sampled from a uniform distribution, (0,1)(0,1) over S. For a general positive definite real matrix n nu A n nu with eigenvalues: 1 2 0,nO O Ot t t !nO !nOnn one can show that [2]: ^ `1 2 Prob 1 .T T n x x x xO T S T d d t A A (2) where T is a scalar multiplier greater than 1.0.
- 5. 197 Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 Uncertainty Quantification and Sensitivity Analysis Methods Now if m nu B m nu such that T T A LL B B and 2 2 = nT D S § · ¨ ¸ © ¹ then (2) can be re-written as [2, 3]:
- 6. 1,..., 2 Prob max 1 . k i i k n xD D S ½° ° d t ® ¾ ° °¯ ¿ B B (3) This theorem represents the basis for randomized linear algebra techniques which attempt to estimate matrix properties such as eigenvalues and condition number by relying solely on randomized matrix-vector products. From a high level, this theorem asserts that one can bound the maximum singular values for any matrix to a high probability using a small number of matrix-vector products. The only requirement is that these vectors are selected randomly. For example, with D=10 and k=3 , one can determine with probability 99.9% that the maximum singular value of matrix B will not be larger than the limit determined by Eq. (3), using only three random matrix- vector products. Next, to prove that the m output responses can be constrained to a subspace, we need to re-write the individual components of the response vector from Eq. (1) as follows: @
- 7. qq y f x , 1,..., .q m where @q y refers to the qth component of a vector y. Assume that the functions @q y are integrable everywhere in the parameter space, which implies that m functions, zq, exists such that: @
- 8. q qq l z y f x x w w , 1,...,q m for any l. (4) Note that this equation is applied for any l since the functions @q y are integrable everywhere. Without loss of generality, we pick l = q. Further we assume that the integrand functions zq are smooth in the sense that they can be Taylor-series expanded. In earlier work [1], we showed that any Taylor-series expandable function can be expressed in terms of another expansion, referred to as the tensor-free expansion, of the form:
- 12. 1 1 ( ) ( ) ( ) 1, , , , , , 1 ,. .., 1 .. .. .l k l k n k T k T k T q q j q l q j q k q j q k j j j z x x x x E E E f ¦ ¦ whose derivatives with respect to x are given by a vector:
- 20. 1 1 1, , , , , , , 1 ,.. ., 1 .. .. .l k l l k n k T k T k T k q q j q l q j q k q j q j q k j j j z x x x x E E E E f c’ ¦ ¦ This expansion was developed in support of the gradient- based reduction algorithm described above. For full details on its construction, the reader may consult a previous journal article [1]. The derivative expression implies that the derivative at any point in the parameter space may be viewed as a linear combination of a number of E vectors. By randomly sampling the derivatives, one could identify a basis of reduced dimension that approximates the derivatives anywhere in the parameter space, and via Theorem 1, upper-bound the error resulting from the reduction. Next form a super vector that aggregates all the derivatives for all the zq functions as follows:
- 22. 1 1 .... . T TT mn mg z x z x uª º’ ’ « »¬ ¼ 1 .mnu Since one can find a basis in the parameter space that can upper-bound the construction error for each of the
- 23. qz x’ vectors, one can find another super basis that upper bounds the construction error for the vector g. This was proved in earlier work using the idea of pseudo- response [1]. Now, design a matrix K such that: @1 ... .m mn m u K K Km mnu Km mnm mn where m n q u K m nu is the null matrix except the qth diagonal element is given by: 1.0.q qq ª º¬ ¼K This matrix is designed to pick the derivative of the function zq with respect to the qth vector in order to take advantage of Eq. (4): 1 1 ... T m m zz g y x x ª ºww « » w w¬ ¼ K . This equation implies that given a subspace that approximates the vector g with the reduction error-upper- bounded, one can find a subspace for the vector y, which also bounds its reduction error. Now, we can propagate the error resulting from parameter space reduction to the response space, we introduce the following theorem. Theorem 2: Consider a physical model: ( ); where : .n m y f x f .n m Let @ @1 2 1 2and .N Nx x x y y yX Y@ @a d . @N N@ 1 2@ 1 21 N1 21@ 1 21@ andx @@ andx @ and@x @ and where the vectors n ix n are randomly generated realizations for the parameters, whose corresponding responses are given by: ( ).i iy f x Let x x x T r rX W W X , where x x n r r u W xn rxu defines a basis for the parameters active subspace. Next, define: 1 2 .x x x x Ny y yª º¬ ¼Y .ºx ¼Ny ººx Ny with ( )x x x T i r r iy f xW W . Note that the vector x iy corresponds to the response value resulting from a single reduction at the parameter space. Next, define: .y y y T r rY U U Y
- 24. 198 Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 Uncertainty Quantification and Sensitivity Analysis Methods where yrU is the matrix associated with the response active subspace calculated based on the snapshots matrix Y (obtained by algorithm #1). Let .y y xy T x r rY U U Y This equation implies two reductions, one embedded in Yx due to reduction in the parameter space, followed by another reduction in the response space via multiplication by the matrix yrU . To calculate the total errors resulting from both reductions, define the following error bounds:
- 25. 11 1,2, , 2 maxx x y i k iN wD S Y Y1k 1, k k 2x 2 y 1D S D , the response error due to parameter reduction, and
- 26. 22 1,2, , 2 maxy y y i k iN wD S Y Y2k 2,k k 2y 2 y 2 S D , the response error due to response reduction. Then the theorem states that: ^ ` 1 2 21prob (1 )(1 )k kxy y x y y D D d t Y Y ` (1`y y ` (1`y x (1(1` (5) Proof. From theorem 2 one can write: ^ ` 1 1Pr .ob 1 kx x y D d t Y Y ` 1`x y D1` (6) ^ ` 2 2ob .Pr 1 ky y y D d t Y Y ` 1`y y D1` (7) But ( ) ( )y y xy y xT r r U UY Y Y Y Y Y , and hence: . .y y T r r xy y x d Y Y Y U Y - YUY (8) Provided that yrU has orthonormal basis, we know that: 1.y y T r rU U Plugging this into (8), then taking the probability and substituting from Eqs. (6), (7), we get (5). RESULTS Two numerical tests are employed to test the proposed error bound. The first is an analytical nonlinear function f where: 15 10 such that:n m y f ( x ); f : ; n , m;n m 1 3 8 2 2 2 2 3 1 2 3 4 4 5 5 26 6 7 7 8 7 2 28 9 10 9 10 9 10 10 9 1 4 1 5 1 1 0 8 1 6 = 1 0 1 0 2 T T T T T T a x T T T T T T a x T T T T T T a x ( a x ) ( . * a x . * a x )y y y e y cos( . a x . a x) y y (a x a )* [(a x) sin(a x)] y y ( . e )* [(a x) (a x) ] y y a x . a x a x a ª º « » « » « » « » « » « » « » u « » « » « » « » « » « » « »¬ ¼ B 108 T ; x a x ª º « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » « » ¬ ¼ where 1 2 and is a random matrixn ia ;i , , m m m . uB; , ,1 2 and is1 21 2 d in 1 2 d i1 2 and is1 2 The second case study simulates neutron transport in a PWR pin cell model. The objective is to check the probability of the proposed error bounds due to reductions at both the parameter and response spaces. The computer code employed is TSUNAMI-2D, a control module in SCALE 6.1 [4], wherein the derivatives are provided by SAMS, the sensitivity analysis module for SCALE 6.1. Case Study 1: The dimension of the parameter space is 15n , and the response space 10m , and a user defined tolerance of 6 10 is selected. The parameter active subspace is found to have a size of 9;xr whereas the response active subspace is 9.yr Fig. 1 shows the function behavior plotted along a randomly selected direction in the parameter space. Fig. 1. Function behavior along a random input direction.
- 27. 199 Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 Uncertainty Quantification and Sensitivity Analysis Methods The following table shows the minimum theoretical probabilities predicted by the theorem Probmin and the actual probability resulting from the numerical test Prob number of successes total number of tests . Table I. Analytic Function Results. Prob Probmin x x yH dY Y 1.0 0.9 y y yH dY Y 0.998 0.9 xy x x y yH H d Y Y 1.0 0.81 Next we show the relative error xy Y Y Y due to both reductions vs. the theoretical upper bound predicted by the theory x y y yH H Y . Fig. 2. Theoretical and actual errors for case study 1. It is important to notice that theoretical probability predicted by theorem 1 is designed such that even if the matrix was of rank-one the minimum probability is still satisfied, and the higher the higher the rest of the singular values get the higher the actual probability is. Which explains why the actual probability is higher than the theoretical one. Case Study 2: For the pin cell model the full input subspace (cross sections) had a size of 1936n , whereas the output (material flux) was of size 176m . The cross-sections of the fuel, clad, moderator and gap were perturbed by 30% (relative perturbations). Based on a user defined tolerance of 5 10 , the sizes of the input and output active subspaces are 900xr and 165yr , respectively. The following table shows the minimum theoretical probabilities predicted by the theorem and the probability resulted from the numerical test. Table II. Transport Model Results. Prob Probmin x x yH dY Y 1.0 0.9 y y yH dY Y 1.0 0.9 xy y x y yH H d Y Y 1.0 0.81 Next we show the relative error xy Y Y Y due to both reductions vs. the theoretical upper bound predicted by the theory x y y yH H Y . Fig. 3. Test vs. theoretical error bound for case study 2. CONCLUSIONS This summary has equipped our previously developed ROM techniques with probabilistic error metrics that bound the maximum errors resulting from the reduction. Given that reduction algorithms can be applied at any of the various model interfaces, e.g., parameters, state, and responses, the developed metric effectively aggregates the associated errors to estimate an error bound on the response of interest. This functionality will prove essential in our ongoing work focusing on extension of ROM techniques to multi-physics models. REFERENCES 1. Y. BANG, H. ABDEL-KHALIK, J. HITE “Hybrid Reduced Order Modeling Applied to Nonlinear Models”, IJNME, 91, 929-949, (2012). 2. J. DIXON, “Estimating Extremal Eigenvalues and Condition Numbers of Matrices”, SIAM, 20, 2, 812- 814, (1983). 3. N. HALKO, P. MARTINSSON, J. TROPP, “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions”, SIAM, 53, 4, 217-288, (2011). 4. SCALE: A Comprehensive Modeling and Simulation Suit for Safety Analysis and Design, ORNL/TM- 2005/39, Version 6.1, Oak Ridge National Laboratory, Oak Ridge, Tennessee, (2011).