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Multiscale Uncertainty Quantification with Arbitrary Polynomial Chaos

Nick Peppera, Francesco Montomolia, Sanjiv Sharmab

a: UQlab, Dept of Aeronautics, Imperial College of London

b: Airbus, United Kingdom

Abstract

This work presents a framework for upscaling uncertainty in multiscale models. The problem is

relevant to aerospace applications where it is necessary to estimate the reliability of a complete part

such as an aeroplane wing from experimental data on coupons. A particular aspect relevant to

aerospace is the scarcity of data available.

The framework needs two main aspects: an upscaling equivalence in a probabilistic sense and an

efficient (sparse) Non-Intrusive Polynomial Chaos formulation able to deal with scarce data.

The upscaling equivalence is defined by a Probability Density Function (PDF) matching approach. By

representing the inputs of a coarse-scale model with a generalized Polynomial Chaos Expansion (gPCE)

the stochastic upscaling problem can be recast as an optimisation problem.

In order to define a data driven framework able to deal with scarce data a Sparse Approximation for

Moment Based Arbitrary Polynomial Chaos is used. Sparsity allows the solution of this optimisation

problem to be made less computationally intensive than upscaling methods relying on Monte Carlo

sampling. Moreover this makes the PDF matching method more viable for industrial applications

where individual simulation runs may be computationally expensive. Arbitrary Polynomial Chaos is

used to allow the framework to use directly experimental data.

Finally, the difference between the distributions is quantified using the Kolmogorov-Smirnov (KS)

distance and the method of moments in the case of a multi-objective optimisation. It is shown that

filtering of dynamical information contained in the fine-scale by the coarse model may be avoided

through the construction of a low-fidelity, high-order model.

Keywords

Uncertainty quantification; multiscale modelling; stochastic upscaling; polynomial chaos expansions;

SAMBA; PDF matching

Contact

Nick Pepper

Imperial College London

Department of Aeronautics

United Kingdom

nicholas.pepper16@imperial.ac.uk

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1. Introduction

Computational modelling plays an extensive part in aircraft design: Computational Fluid Dynamics (CFD) models are used

from the preliminary design phase onwards to predict the aerodynamic performance of aircraft designs; Finite Element

Method (FEM) models are used in the aero-elastic analysis of the aircraft structure; heat transfer models are used in the

design of propulsion systems and even the economic impact of a new aircraft design can be modelled with market

penetration models [1]. Such models may be multiscale in nature. For instance, in the case of multiscale models of composite

materials the scales may range from the microscale through to the component level [2]. The multiscale modelling of

composite materials is a popular area of research [3]–[7]. Computational fluid dynamics (CFD) models and heat transfer

models may also be multiscale [8], [9]. Aircraft designs must be validated with rigorous physical tests before being certified.

Such tests are expensive and time consuming [10]. There is motivation then in ensuring that computer generated preliminary

designs are reliable before physical testing occurs.

In recent years the potential of Uncertainty Quantification (UQ) to accelerate the process of aircraft design and component

certification has come to be recognised [11]. UQ is used to understand how parametric uncertainties affect output quantities

of interest by propagating uncertainties through computational models [12]. To conduct UQ on multiscale models it is

therefore necessary to develop computationally efficient means of propagating uncertainties through scales. Figure 1 shows

a schematic of the stochastic upscaling problem: from fine-scale PDF data on coupons we need to forecast the properties of

a complete part. The problem can also be seen as the inverse one, in which properties of the materials (as distributions) are

needed to achieve a specific airplane design. There is no difference in the problem formulation, although in the case of the

inverse formulation the solution attained may not be unique. In this work we will follow the schematic in Figure 1.

The aim of the framework presented here is to use stochastic model reduction through probabilistic equivalence where possible in order to reduce the number of stochastic parameters at each scale as the multi-scale model moves from more fundamental scales to the level of a complete part or even to the level of the entire aircraft. Reducing the number of stochastic dimensions makes uncertainty quantification of the entire structure more computationally efficient. Whether the method discussed here is appropriate at every stage of this upscaling process will naturally depend on the nature of the model at each scale. However, there is some precedent in the literature for applying stochastic model upscaling to models of large composite structures. For instance, Sasikumar et al. (2015) argue that when the properties of each individual laminae in a composite plate is modelled as an independent random field the number of stochastic parameters becomes prohibitive. In this case, stochastic model reduction through probabilistic equivalence is achieved in two methods: on the nodal response and on the probabilistic characteristics of the matrix [13]. Gorguluarslan and Choi (2014) demonstrate that stochastic model order reduction can be implemented at the level of a complete part: in this particular case by up-scaling from a fine-scale, mesostructured model of a hydrogen tank to one where it is treated as homogeneous [14]. While not an aeronautics case per se, such an upscaling may be thought of as analogous to wing structural design in aeronautics, where both finite element and analytical, lumped mass models may be used to model a wing. Thus far, works on stochastic model upscaling have focussed on a two-scale problem: a quantity of interest common to both the fine and coarse scale models is identified and probabilistic equivalence is achieved for this quantity of interest. In order to achieve the full component roll-up illustrated in Figure 1 it is necessary to propagate uncertainty through multiple scales and upscaling through probabilistic equivalence may be used more than once in the course of the roll-up. However, the quantity for which the probabilistic equivalence is found will not be the quantity that is passed up to coarser scales and certainly will not be the quantity of interest of the model at the coarsest scale. A thorough error analysis of a realistic problem involving multiple scales is necessary in order to determine that employing probabilistic equivalence at several stages of the roll-up of scales does not introduce significant model errors at the coarsest scale. An error analysis such as this will be the subject of future publications on this topic. In other works in the literature, the problem of stochastic upscaling in multiscale models has commonly been studied from

the perspective of modelling groundwater flow through heterogeneous porous media [15]–[19]. Achieving accurate

predictions from these models is important in contaminant spread, nuclear waste disposal and oil recovery analyses.

However, uncertainties exist due to the variation in permeability with length scale and also due to epistemic uncertainties

arising from lack of information about the system [15]. Both intrusive and non-intrusive methods have been proposed for

propagating uncertainty in stochastic multiscale models such as the multiscale porous heterogeneous flow problem.

Intrusive methods involve changing the underlying model in order to propagate uncertainty through scales and will be

discussed first. Non-intrusive methods, which treat the model at each scale as a black box, will then be discussed.

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Figure 1: The stochastic upscaling problem- how to propagate uncertainties from the finest to the coarsest scales?

Intrusive approaches to multiscale stochastic upscaling problems typically recast the underlying model equations in the form

of stochastic partial differential equations (SPDEs). Once in this form these equations may be solved through one of a number

of established methods such as Monte Carlo simulation [20], spectral stochastic finite element method (SSFEM) [21] or the

stochastic collocation method. Stochastic collocation is a popular choice because of its fast convergence rates and has been

applied frequently to flow problems in heterogeneous media [18]. By representing the random coefficients in the SPDEs

through polynomial chaos expansions and the Karhunen-Loeve (KL) expansion it is possible to generate a set of sample points

through Smolyak sparse grid collocation [15], [17] or through the probabilistic collocation technique [18]. The resulting

deterministic equations may be solved through an appropriate strategy such as the multiscale finite element method [19],

[22] or the multiscale discontinuous Galerkin method [23]. In multiscale finite elements, basis functions are generated at the

coarse grid level which are consistent with the small scale subgrid structures [19]. The use of stochastic collocation coupled

with multiscale finite elements (MFE) is a common approach in the literature [17], [18], [24]. As an alternative to stochastic

collocation, Dostert et al. [16] uses Markov Chain Monte Carlo (MCMC) methods in the situation where the prior distribution

of the stochastic equation coefficients is known. If the measurement error is assumed to follow a Gaussian distribution then

a sampling target distribution may be constructed from which a Markov Chain may be generated.

Another intrusive technique, Coarse graining, is used to find a low-fidelity model which may be matched to a complex, high-

fidelity model. In future evaluations the low-fidelity model is used in the place of the high-fidelity model; Bilionis and Zabaras

[25] introduces a stochastic optimization framework to find an effective coarse grained potential 𝑈𝐶𝐺 which matches the

output of a high-fidelity model based on a number of Monte Carlo samples. Such a technique is reminiscent of the PDF

matching technique described here, although our technique is non-intrusive and requires fewer high-fidelity model

realisations than would be the case when using Monte Carlo sampling, as used in Bilionis and Zabaras [25]. Lastly, a fuzzy

stochastic global-local algorithm is proposed in Babuska and Motamed [26] that upscales uncertainty through a non-

stationary fuzzy-stochastic field, which is evaluated to find a quantity of interest at the coarse scale. This method is currently

limited to the case of upscaling in a one-dimensional fibre composite and is not readily generalizable. In this paper non-

intrusive methods, which do not change the underlying model, are preferred as they are compatible with models already

used in industry.

Non-intrusive techniques for uncertainty propagation attempt to quantify the information loss between scales in a hierarchy

of models. Early works sought to integrate tools for uncertainty quantification that were already available in the literature

with multiscale problems. For instance, a non-intrusive stochastic solver for multiscale composites is outlined in Fish and Wu

[27]. The KL expansion is used to reduce the dimensionality of probability space and a stochastic collocation method in

combination with a massively parallel computer architecture are employed to reduce the computational cost of the

uncertainty analysis. More recent works attempt to quantify the information loss between scales and to model the cross

scale dependencies within the model. For example a multiscale PCE method was proposed in Mehrez et al. [28] to model the

dependencies of the outputs at a particular level in a hierarchical structure of models on the inputs at finer scales. A

generalized hidden Markov model (GHMM) is employed in Wang [29] for the same purpose. Multi response Gaussian

processes (MRGPs) are used in Bostanabad et al. [30] to model the aleatory and epistemic uncertainties arising in multiscale

models of woven composites. A common non-intrusive stochastic technique is that of probabilistic equivalence, where at

each interface between scales the coarse-scale inputs are searched for that will produce a statistically equivalent output to

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that of the fine-scale model [14], [31], [32]. The resulting optimisation problem is usually solved with a genetic algorithm.

Choi et al. [32] employs a hybrid optimisation procedure with the genetic algorithm and sequential quadratic programming.

Recent works have sought to apply Machine Learning to the problem of upscaling uncertainty in multiscale problems in both

intrusive and non-intrusive methodologies. For instance in Chan and Elsheikh [33] a Neural Network is used to generate the

coarse-scale basis functions in MFE in order to reduce the computational cost of repeated model evaluations. Trehan and

Durlofsky [34] introduced a Machine Learning framework to model the upscaling error between a high-fidelity model of a

reservoir and a low-fidelity model. Simulations of the high-fidelity model are used for a high-dimensional regression to model

the error based on user defined features. A random forest is the chosen statistical regression. Scheidt and Caers [35] similarly

addressed the problem of modelling the upscaling error in reservoir models, this time through the use of kernel clustering.

Every method for uncertainty propagation in multiscale models must strike a balance between repeating high-fidelity

simulations at the lowest scales in order to understand how uncertainties at the lowest scale impact the coarse-scale

quantities of interest with the computational resources available. The method presented here builds upon the upscaling

algorithm presented in Arnst and Ghanem [31] through the use of the Sparse Approximation of Moment-Based Arbitrary

Polynomial Chaos (SAMBA) method. The application of SAMBA to this algorithm, as opposed to using Monte Carlo sampling,

greatly increases the computational efficiency by reducing the number of model evaluations needed for the upscaling at

both the coarse and fine scale.

1.1. Fine and coarse-scale probabilistic models

The uncertainty affecting a computational model may be thought of as belonging to one of two categories: model uncertainty

and parametric uncertainty. Model uncertainties refer to uncertainties arising from the truncation of the infinite-dimensional

real world system that the model has been created to represent [36]. These uncertainties are difficult to quantify but will

not be considered here; this work will focus on the propagation of parametric uncertainties between scales in multiscale

models. Parametric uncertainties refer to incomplete knowledge of constants, boundary conditions or initial conditions in

the computational model. It is assumed that such parameters can be identified and the uncertainty represented through a

PDF [37].

The upscaling procedure detailed here propagates uncertainty from a fine-scale model to a coarse-scale model at the next

highest scale in a hierarchy of models that can be used to represent the whole aircraft structure. In so doing the impact of

uncertainties at the most fundamental scales on the complete structure may be quantified. Note that it is also possible to

reverse the order of the hierarchy of models for use in inverse problems: the coarse-scale model can be used to explore the

design space rapidly and identify interesting features which may be modelled in detail using the fine-scale model.

The fine-scale model occupies the probability space denoted by the triplet (Ωh, ℱh,𝒫h) where Ωℎ denotes the sample space

of the fine model, ℱℎ the set of events and 𝒫ℎ the probabilities assigned to these events. A realisation �⃗⃗� ℎ in the fine model

sample space produces a fine random input vector 𝜉 = 𝜉(�⃗⃗� ℎ). Evaluating the fine model, when the uncertain parameters

take the values of the components of 𝜉 , yields a fine output vector �⃗� ℎ:

�⃗� ℎ = �⃗⃗� (𝜉 ), (1)

where 𝑤 is the model quantity of interest. Through repeated model realisations at the fine-scale it is possible to generate a

PDF for the output statistics of the fine model quantity of interest, donated by 𝑓ℎ(�⃗⃗� ). Similarly, the coarse model occupies

the probability space occupied by the triplet (Ωc , ℱc ,𝒫c) and a realisation �⃗⃗� 𝑐 produces a coarse random input vector 𝜂 =

𝜂(�⃗⃗� 𝑐). Evaluating the coarse model with this input vector yields a coarse-scale output vector �⃗� 𝑐, where:

�⃗� 𝑐 = �⃗⃗� (𝜂 ). (2)

The PDF of the coarse-scale output is then donated by 𝑓𝑐(�⃗⃗� ) [32]. The goal of the upscaling procedure is to find the input

distributions at the coarse-scale such that the coarse model has a statistically equal output to the fine model.

1.2. Problem Formulation

The problem of propagating uncertainty from a model to another model higher up in the hierarchy of scales is depicted in a

diagrammatic form in Figure 2. A PDF matching algorithm is used to achieve this. To do so outputs common to both models

are identified so that �⃗� ℎ and �⃗� 𝑐 may be defined. In order to achieve a roll up from coupon level to component level as

depicted in Figure 1 it is necessary that there will always be some overlap in the predictions between the models.

Given that the input distributions at the fine-scale are known, a moment based method, SAMBA, is used to generate a set of

𝑁 collocation points; propagating the corresponding random input vectors, �⃗⃗� = {𝜉1, …, 𝜉𝑁}, through the fine-scale model

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and reconstructing an output distribution 𝑓ℎ for the variable common to both the fine and coarse models. The PDF matching

algorithm then searches for the random input vector, 𝜂 , which when simulated 𝑠 times and propagated with SAMBA through

the coarse-scale model produces an output distribution 𝑓𝑐 that is statistically equivalent to that of the fine-scale model i.e.:

∀�⃗⃗� ∈ 𝑊: 𝑓𝑐(�⃗⃗� (𝜂 )) = 𝑓ℎ(�⃗⃗� (𝜉 )). (3)

Achieving exact equivalence between the two distributions is a computationally intractable problem so in practice an input

vector 𝜂 is sought which minimises the statistical distance between the two distributions i.e.

𝜂 = argminη⃗⃗

𝑑 (𝑓ℎ (�⃗⃗� (𝜉 )) , 𝑓𝑐(�⃗⃗� (𝜂 ))). (4)

𝜂 may be approximated as an 𝑛-dimensional truncated generalized polynomial chaos expansion (gPCE) of order 𝑝, i.e.

𝜂 (𝑝) = ∑ �⃗� 𝛼 Ψ𝛼(𝑧 )𝑃𝑡

𝛼,|𝛼|=0 , (5)

where the number of terms, Pt =(n+p)!

n!p! and 𝛼 is a multi-index. 𝑧 is an independently distributed (iid) random vector; the

distribution from which 𝑧 is sampled from will depend on the orthogonal polynomials used in the PCE. The multivariate

orthogonal polynomials Ψ𝛼 are calculated as the product of univariate orthogonal polynomials, 𝜓𝛼 (the choice of which is

discussed further in section 3.4):

Ψ𝛼(𝑧1 , 𝑧2 ,… , 𝑧𝑛) = 𝜓𝛼1(𝑧1) × …𝜓𝛼𝑛

(𝑧𝑛). (6)

For Hermite polynomials, the 𝑛 components of 𝑧 are normal Gaussian variables. The estimation of the parameter set �⃗� 𝛼 is

thus the goal of the optimisation i.e.

�⃗� 𝛼̂

= argmin�⃗� 𝛼

𝑑 (𝑓ℎ (�⃗⃗� (𝜉 )) , 𝑓𝑐(�⃗⃗� (𝜂 ))), (7)

where �⃗� 𝛼̂

is the numerical approximation to the parameter set. Increasing the order of the gPCE improves the quality of the

PDF matching, however, a trade-off exists as the optimiser must then search a higher-dimensional PCE parameter space,

which will be more computationally intensive. As one would expect, the choice of metric, 𝑑, used to characterise the

statistical distance between 𝑓ℎ and 𝑓𝑐affects the quality of the matching, this is discussed further in section 3. Figure 3

illustrates the steps of the algorithm used. The PDF matching process is discussed in more detail below.

Figure 2: Illustration of the PDF matching algorithm. The coarse inputs, η, are searched for which, when propagated through the coarse model, give a statistically equal output to that of the fine model when known fine inputs, ξ, are propagated

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Figure 3: Flowchart of the PDF matching algorithm

2. Propagating uncertainty using SAMBA

A sparse arbitrary polynomial chaos formulation [38] is used as an alternative to Monte Carlo sampling to propagate

parametric uncertainty through the fine and coarse-scale models. There are two main reasons to use this approach against

others suggested in the literature: the ability to deal with scarce data, as in aerospace problems, and computational

efficiency. The approach consists of three steps: firstly a sparse grid for sampling is constructed using anisotropic Smolyak

quadrature rules [39]; the model is then evaluated at each of these sampling points and finally a polynomial chaos expansion

(PCE) is constructed for the model output. Through Monte Carlo simulation of this PCE an output PDF is derived. The main

relations of the SAMBA method are summarised below, a fuller description of the method may be found in [38].

2.1. Construction of a sparse grid using anisotropic Smolyak quadrature rules

Having been given data from 𝑁𝑑 input distributions, the first stage in propagating uncertainty with SAMBA is to find the

collocation points in the 𝑁𝑑-dimensional input parameter space. The model is then sampled at these points such that an

approximation to the model 𝑓(𝜉 ) for a random input vector, 𝜉 , may be derived. To find the collocation points, the optimal

Gaussian quadrature rules are calculated for each input distribution. Having obtained these quadrature rules, Smolyak’s

algorithm is applied to find a sparse grid for sampling through the tensor products of the Gaussian quadrature rules. Finding

the optimal Gaussian quadrature rules is thus the first stage of the process; through the formation of the Hankel matrix of

moments it is possible to directly compute the quadrature rules. Given a set of 𝑁 random samples of an input distribution

𝜁1 …𝜁𝑁 the 𝑘th statistical moment, 𝜇𝑘, may be calculated using:

𝜇𝑘 =1

𝑁∑ 𝜁𝑖

𝑘 𝑁𝑖=1 . (8)

The Hankel matrix of moments can be formed from the statistical moments:

𝑀 =

[ 𝜇0 𝜇1 ⋯ 𝜇𝑝

𝜇1 𝜇2 𝜇𝑝+1

⋮ ⋱ ⋮𝜇𝑝 𝜇𝑝+1 ⋯ 𝜇2𝑝 ]

. (9)

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Being positive definite, the Cholesky decomposition of the Hankel matrix, 𝑀 = 𝑅𝑇𝑅, may be calculated. As discussed above,

PCEs are based on the weighted sum of a set of orthogonal polynomials, 𝜓𝑗. Through application of the Mysovskih theorem

it is found that the coefficients of the optimal orthogonal polynomials 𝜓𝑗 can be found as the entries of 𝑅−1. To avoid

computing the inverse of 𝑅 the explicit analytic formulas derived by Rutishauser may be used to obtain these polynomial

coefficients from the Cholesky matrix entries 𝑟𝑖𝑗 [40]. In so doing three term recurrence coefficients are derived:

𝑎𝑗 =𝑟𝑗,𝑗+1

𝑟𝑗,𝑗−

𝑟𝑗−1,𝑗

𝑟𝑗−1,𝑗−1, 𝑏𝑗 =

𝑟𝑗+1,𝑗+1

𝑟𝑗,𝑗. (10)

The recurrence coefficients form the entries of the symmetric, tridiagonal Jacobi matrix J, the eigenvalues of which are the

optimal Gaussian collocation points:

J=

[ 𝑎1 𝑏1

𝑏1 𝑎2 𝑏2

𝑏2 𝑎3 𝑏3

⋱ ⋱ ⋱𝑏𝑝−2 𝑎𝑝−1 𝑏𝑝−1

𝑏𝑝−1 𝑎𝑝 ]

. (11)

Optimal Gaussian weights are then found through

𝜔𝑖 = 𝑣1,𝑖 2 , (12)

where 𝑣1,𝑖 is the first component of the eigenvector corresponding to the 𝑖𝑡ℎ eigenvalue of 𝐽. Having found the optimal

Gaussian quadrature points and weights, Smolyak’s algorithm is applied in order to generate a sparse grid to sample on. The

motivation for doing so is to remedy the curse of dimensionality, in which the number of sample points increases dramatically

as the dimension of the sample space is increased. For a sequence of 𝑁𝑑 one-dimensional quadrature rules, {𝑈𝑖𝑗}𝑗=1…𝑁𝑑

with collocation points 𝜉𝑘

𝑖𝑗 denoted by:

𝑈𝑖𝑗 = ∑ 𝑤 (𝜉𝑘

𝑖𝑗)𝜔𝑘

𝑖𝑗 ,𝑚𝑖𝑗

𝑘=1 (13)

where 𝑚𝑖𝑗𝑗 ∈ {1,…𝑁𝑢} is the maximum adaptive order for each univariate quadrature rule and 𝑤 (𝜉

𝑘

𝑖𝑗) refers to a model

evaluation at the collocation point 𝜉𝑘

𝑖𝑗 . A Smolyak quadrature of level 𝑙 is found through:

𝐴𝑆 = ∑ (𝑙 − 1)𝑙+𝑁𝑢−|𝑖| (𝑁𝑢 − 1

𝑙 + 𝑁𝑢 − |𝑖|) ⊗𝑘=1

𝑁𝑈 𝑈𝑖 .𝑙+1≤|𝑖|≤𝑙+𝑁𝑢 (14)

Increasing the level of the quadrature increases the accuracy of the result by adding more points to the Smolyak grid, but at

increased computational expense as there are more points to sample at. |𝑖| is the norm of the vector 𝑖 = {𝑖1 ,… 𝑖𝑁𝑢}

representing the sum of the 𝑗𝑡ℎ of the index matrix 𝐼𝑗𝑘. An example of a level 3 Smolyak grid created for a Gaussian input

distribution and a lognormal histogram input is shown in Figure 4.

1.1. Construction of an output PDF

Having obtained a sparse grid of 𝑁𝑠𝑝 sample points through Smolyak’s algorithm the model is then evaluated at these points.

As has been previously mentioned, the PCE representation of a model output, 𝑤(𝜉 ), for a random input vector 𝜉 is of the

form:

𝑤(𝜉 ) ≈ ∑ 𝛼𝑘 . 𝜓𝑘(𝜉 ),𝑁𝑠𝑝

𝑘=1 (15)

where 𝜓𝑘 refers to the 𝑘𝑡ℎ order polynomial in a family of orthogonal polynomials. In SAMBA the Fourier coefficients, 𝛼𝑘,

of the PCE representation of the model output are approximated numerically through:

𝛼𝑘 =∑ 𝑤(𝜂𝑖)𝜓𝑘(𝜂𝑖)𝜃𝑖

𝑁𝑠𝑝𝑖=1

∑ 𝜓𝑘(𝜂𝑖)𝜃𝑖𝑁𝑠𝑝𝑖=1

, (16)

where the sparse collocation points and weights are 𝜂𝑖 and 𝜃𝑖. Recall that the orthogonal polynomials 𝜓𝑘, evaluated at the

collocation points, were found through the application of the Mysovskih theorem as part of the calculation of the optimal

Gaussian quadrature rules. By sampling at each of the collocation points the model evaluations 𝑤(𝜂𝑖) were obtained. The

PCE may be simulated repeatedly through Monte Carlo sampling to create an output distribution for the model [38], [41].

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Figure 4: one-dimensional collocation points for a Gaussian PDF and a lognormal histogram (left). On the right, a level 3 Smolyak grid produced using SAMBA

2. Quantifying the statistical distance

For the upscaling procedure to work it is necessary to have a measure of the quality of the PDF matching. As can be seen in

Figure 3 the evaluation of the statistical distance between the two distributions 𝑓ℎ(𝑤) and 𝑓𝑐(𝑤) is necessary in order to

determine the convergence of the PDF matching algorithm. The way in which the statistical distance is defined has a

significant impact on the speed of convergence and the quality of the matching. In [31] both the generalised method of

moments and the Kullback-Leibler divergence are suggested as measures of statistical distance. In this work the Kolmogorov-

Smirnov (KS) distance and the method of moments are used.

2.1. Kolmogorov-Smirnov (KS) distance

The Kolmogorov-Smirnov (KS) distance is defined as:

𝑑𝐾𝑆 = sup�⃗⃗�

|𝐹𝑐(�⃗⃗� |𝑝 ) − 𝐹ℎ(�⃗⃗� )|, (17)

where 𝐹𝑐(�⃗⃗� |𝑝 ) and 𝐹ℎ(�⃗⃗� ) are the Cumulative Distribution Functions (CDFs) of the common variable for the coarse-scale

and fine-scale respectively [42]. Note that the CDF is defined as the integral of its PDF i.e. for an arbitrary PDF, 𝑓𝑐(𝑥), the

CDF, 𝐹𝑐(𝑥), is given by:

𝐹𝑐(𝑥) = ∫ 𝑓𝑐(𝑡)𝑑𝑡.𝑥

−∞ (18)

Thus, the CDF may be thought of as an alternative representation of the PDF. A KS distance of less than 0.03, meaning the

two PDFs are from the same distribution to the 3% significance level, was considered to be a ‘good’ matching for our

purposes. The KS distance between two CDFs is illustrated in Figure 5.

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Figure 5: CDFs of two arbitrary distributions- the KS distance (arrow in red) is defined as the supremum distance between the two CDFs

The ability to attach a normalised significance level to the PDF matching makes this definition of the statistical distance

appealing. The KS distance also allows matching of higher order statistical moments, although this may come at the expense

of fitting the first two statistical moments. However, the KS distance is only a meaningful distance metric if there is some

overlap between the PDFs. This point is demonstrated in Figure 6, in which four data sets have been created from sampling

different Gaussian distributions: while the CDF of B clearly lies further away from the target distribution than that of C,

because neither distribution overlaps with the target CDF the KS distance for both is 1. It can therefore be inappropriate to

use the KS distance as a measure of statistical distance if there is uncertainty on the upper and lower bounds of the parameter

set �⃗� 𝛼. By tabulating the statistical distances between the distributions A, B, and C it is clear that the method of moments is

a more appropriate distance measure for when the target and trial distributions are not overlapped. However, the KS

distance gives a more meaningful statistic once an overlap between the distributions is established as the statistic can be

related to a confidence level.

Distance metric Statistical distance between distributions

Target-A Target-B Target-C

Kolmogorov-Smirnov (KS)

distance

0.2 1 1

Method of moments (𝛾 = 1)

0.2165 14.34 26.24

Table 1: A comparison of the statistical distances of the distributions A, B, and C from Figure 6 from the target distribution for different

measures of statistical distance

2.2. Method of moments

The statistical distance, quantified by the Method of Moments, is defined in Arnst and Ghanem (2008) as:

𝑑𝐺𝑀𝑀 (𝑓ℎ(�⃗⃗� ), 𝑓𝑐(�⃗⃗� |𝑝 )) = ||�⃗⃗� − �⃗⃗̃� (𝑝)||22 + 𝛾||𝐶 − �̃�(𝑝 )||2

2, (19)

where 𝛾 is a user defined weighting parameter to weight the importance of the first statistical moment against the second

[31]. The vectors �⃗⃗� and �⃗⃗̃� contain the mean values of the output quantity of interest, whilst the matrices C and �̃�

represent the covariance matrices of the fine-scale and coarse-scale quantities of interest respectively. Mathematically

these quantities are defined as:

[10]

�⃗⃗� = 𝐸{�⃗⃗� ℎ(𝜉 )},

�⃗⃗̃� (𝑝) = 𝐸{�⃗⃗� 𝑐(𝜂 (𝑝))},

𝐶 = 𝐸{(�⃗⃗� ℎ(𝜉 ) − 𝑚) ⊗ �⃗⃗� ℎ(𝜂 (𝑝) − �⃗⃗� (𝑝))},

�̃�(𝑝) = 𝐸 {(�⃗⃗� 𝑐(𝜂 (𝑝)) − �⃗⃗̃� (𝑝)) ⊗ (�⃗⃗� 𝑐(𝜂 (𝑝)) − �⃗⃗̃� (𝑝))}. (20)

Defining the statistical distance in this way means that the fitting of higher order statistical moments is sacrificed in favour

of fitting the first two moments. Whether this yields satisfactory results will depend on the form of the PDF being matched:

for highly skewed distributions it may be more appropriate to use a weighted sum of KS distances, as opposed to the method

of moments.

Figure 6: PDFs (left) and CDFs (right) of four data sets, sampled from Gaussian distributions. The CDFs of B and C do not overlap with that

of the target distribution, hence both have a KS distance of 1.

2.3. Selection of optimisation strategy

The choice of optimisation strategy will determine the manner in which the PCE coefficients, 𝑃𝛼, are updated in the PDF

matching algorithm. If there is only a single common variable between the models i.e. �⃗⃗� (𝜉 ) ∈ ℝ1 and �⃗⃗� (𝜂 ) ∈ ℝ1 then there

is great freedom in the choice of optimisation strategy. In this work a solver based on the Nelder-Mead algorithm was used

to find the local optimum about a chosen start point in the PCE parameter space. If there are multiple variables in common

between the models i.e. �⃗⃗� (𝜉 ) ∈ ℝ𝑁𝑐 and �⃗⃗� (𝜂 ) ∈ ℝ𝑁𝑐 then a multi-objective optimisation strategy is used to find the Pareto

solution for 𝑁𝑐 objective functions. A genetic algorithm is the chosen multi objective optimisation method in [31] and [43].

The PDF matching algorithm was tested for the following test cases. In the first case the algorithm is used to upscale

uncertainty from a wing box designed in Abaqus to a simplified Bernoulli beam model. The algorithm is shown to achieve

satisfactory probabilistic equivalence for both Askey and non-Askey scheme input PDFs. Secondly, probabilistic equivalence

is achieved through the construction of a high-order, low-fidelity model with Fourier interpolation for a case when the coarse

model lacks sufficient degrees of freedom to achieve probabilistic equivalence.

2.4. Choice of Polynomial Chaos Expansion (PCE) basis function

The polynomial chaos expansion for 𝜂 in Equation 5 is formulated as a generalized polynomial chaos expansion (gPCE), as

introduced by Xiu et al. (2002) [44]. Optimal orthonormal polynomial bases exist for a selection of parametric distributions,

known as the Askey scheme. Table 2 displays several common distributions in the Askey scheme, the interval on which the

distributions are supported, and the corresponding univariate polynomial bases.

[11]

The choice of which univariate basis to use for the PCE will likely depend on the form of the fine-scale input PDFs, for instance

in the case of Gaussian input distributions as in section 4.1, the logical choice of univariate basis is the Hermite polynomials.

Equation 3 may thus be rewritten as a Wiener polynomial chaos expansion for 𝜂 :

𝜂 (𝑝) = ∑ �⃗� 𝛼 𝐻𝛼(𝑧 )𝑃𝑡

𝛼,|𝛼|=0 , (21)

where 𝐻𝛼(𝑧 ) = ℎ𝛼1(𝑧1) × …ℎ𝛼𝑛

(𝑧𝑛). ℎ𝛼𝑗 is a normalised Hermite polynomial of order 𝛼𝑗. In the case of Wiener polynomial

chaos 𝑧 is a normal random vector: the components are sampled from the standard normal distribution 𝑁(0,1).

Alternatively, in Legendre-chaos the components of 𝑧 are sampled from the Uniform distribution 𝑈[−1,1] and in Laguerre-

chaos the gamma distribution Γ[𝛼, 1] is used for sampling. 𝛼 is a shape parameter selected by the user (𝛼 > 0). As will be

seen in Section 4.3, the choice of which univariate orthogonal basis to use can lead to differing convergence rates and

qualities of PDF matching for cases where the input distributions do not belong to the Askey scheme.

Askey scheme distribution Interval Orthonormal polynomial

Ψ𝛼(𝑧 )

Distribution for 𝑧

Uniform [𝑎, 𝑏] Legendre 𝑈(−1,1)

Gaussian [−∞,∞] Hermite 𝑁(0,1)

Gamma [0,∞] Laguerre Γ(𝛼, 1)

Table 2: Common types of parametric distributions and their corresponding univariate orthogonal polynomials in the Askey scheme

3. PDF matching with a single common variable

The capacity of the PDF matching algorithm to propagate uncertainty from a fine-scale model to a coarse-scale model of

reduced stochastic dimension is demonstrated for the case of a wing box. The wing box is a useful test case in aeronautics

and has been used as a benchmark test in a number of publications such as in Arnst and Ghanem (2008) and Riccio et al.

(2013) [31], [45]. The particular example of propagating uncertainty from a wing box model to a Bernoulli beam to

demonstrate multiscale uncertainty propagation was inspired by [31]. Only a single variable, the first natural frequency, is

common to both models. At the fine, more fundamental, scale a wing box model was created in Abaqus and a vibration

analysis performed to find the first natural frequency of the beam. At the coarse scale, the wing box was considered to be

part of a larger structure where it was instead modelled as a simple cantilever beam using the Bernoulli beam equation.

Figure 8 shows a cross section of the wing box: the material properties of each of the 6 strips joining the plates to the ‘I’

beams were characterised as random input variables. Considering the Young’s modulus and Poisson ratio of each strip to be

random variables resulted in the fine model being of stochastic dimension 12. In the case of the Bernoulli beam model, the

density and Young’s modulus of the entire beam were considered to be random material properties, hence the coarse-scale

model was of stochastic dimension 2. Having propagated known fine-scale PDFs using SAMBA through the wing box model

to create a PDF of values for the first natural frequency of the wing box, the PDF matching algorithm was used to find PDFs

for the material properties of the Bernoulli beam such that the PDF of its first natural frequency matched that of the wing

box model.

3.1. PDF matching for normal input distributions (Askey scheme)

Figure 9 illustrates the results of the PDF matching algorithm when the random material properties in the fine model were

sampled from two normal distributions: 𝑁(2.1 × 108, 5 × 107) for the Young’s modulus and 𝑁(0.3, 0.01) for the Poisson’s

ratio of each strip. Given that the fine-scale material properties are sampled from normal distributions, Hermite polynomials

were chosen as the orthonormal basis in the gPCE representations of the coarse-scale material properties. It was found that

it was possible to achieve good probabilistic equivalence between the output PDFs using a first-order Wiener PCE: the KS

distance between the two outputs was 0.0163, significantly lower than the target of 0.03. This demonstrates that it is possible

to achieve PDF matching using a coarse model of reduced stochastic dimension for input PDFs belonging to the Askey

scheme.

[12]

Figure 7: Wing box, with mesh, created in Abaqus

Figure 8: Cross sectional view of the meshed wing box. The material properties of the thin strips between the plates and I beams (coloured in red) are considered to be random, hence the model is of stochastic dimension 12

Figure 9: PDF matching for the case of Gaussian input distributions using a first-order PCE (𝑑𝐾𝑆 = 0.0163)

[13]

3.2. PDF matching for non-Askey scheme input distributions

The PDF matching algorithm was also demonstrated for non-Askey scheme input distributions. Non-Askey, heavy tailed

distributions such as the Cauchy distribution are useful in rare events simulation and so important for designing reliable

components [41]. PDF matching of the first natural frequency was repeated for the wing box case study but with non-Askey

scheme input distributions.

Figure 10 shows the result of the PDF matching when the Young’s moduli of the 6 strips were selected from a Cauchy

distribution with parameters 𝐶(0.1 × 108 , 2.1 × 108) and the Poisson’s ratios from a stable distribution with

parameters 𝑆(1, 1, 0.008,0.3). In this case, a second-order Wiener PCE was used in order to achieve satisfactory PDF

matching. The KS distance, 0.0268, is below the target of 0.03 for PDF matching.

The example case of a wing box modelled in Abaqus serves as a proof of concept for the PDF matching algorithm using

SAMBA. The ability to achieve PDF matching for both Askey and non-Askey scheme input PDFs implies that probabilistic

equivalence should be achievable for any reasonable input PDF. Future work will aim to increase the complexity of the case

study to account for higher modes and, for non-linear cases, the interactions amongst these higher modes.

Figure 10: PDF matching with non-Askey scheme input distributions using a second-order PCE (𝑑𝐾𝑆 = 0.0268)

3.3. Choice of univariate orthogonal polynomials

As has been discussed in section 3.4, the choice of which orthonormal polynomials to use as the basis for the gPCE can have

an impact on the quality of the results. Using Hermite polynomials in Section 4.1 was the natural choice given that the fine-

scale inputs to the wing box were normally distributed. However, for cases where the fine-scale inputs do not follow

distributions belonging to the Askey scheme, as is the case in Section 4.2, the choice of which univariate orthogonal

polynomial to use as a basis can become significant and can affect the convergence of the results. This point is demonstrated

in Figure 11: the same non-Askey scheme inputs to the wing box model are used as in Section 4.2 and the results of the PDF

matching between the coarse-scale and fine-scale PDFs are plotted against the gPCE order, 𝑝, for different choices of

univariate orthogonal polynomials. The orthonormal bases used are those in Table 2. The KS distance is again used to quantify

the PDF matching.

In this particular case the choice of Laguerre polynomials led to faster convergence. Such differences in speed of convergence

may become significant if computational resources are limited: a judicious choice of univariate orthogonal polynomial can

reduce computation time by limiting the number of coefficients which must be found through optimisation to achieve a

satisfactory PDF matching.

[14]

Figure 11: Plots comparing the convergence of the statistical distance, as quantified by the Kolmogorov-Smirnov distance, for different orthonormal polynomial bases.

4. PDF matching through the construction of a high-order, low-fidelity model

When there are multiple variables common to the fine and coarse models PDF matching may be achieved through a multi-

objective optimisation, for instance through a multi-objective genetic algorithm. However, this approach may fail to match

all of the PDFs satisfactorily if the coarse-scale model does not have sufficient degrees of freedom to replicate the behaviour

of the fine-scale model. For instance, the coarse-scale model may filter dynamics captured by the fine model, making it

impossible to carry out a satisfactory PDF match. In this case a high-order, low-fidelity model must be constructed at the

coarse scale. This may be done through a Fourier or wavelet interpolation with 𝑛𝑓 terms. The gPCE representation of the

coarse input vector for the high-order, low-fidelity model will then be:

𝜂 ℎ = ∑ �⃗� 𝛼Ψ𝛼(𝑧 )𝑃ℎ

𝛼,|𝛼|=0 , (22)

where the number of terms is defined as 𝑃ℎ =(𝑛+𝑛𝑓+𝑝)!

(𝑛+𝑛𝑓)!𝑝!. The multi-objective optimisation problem to be solved is then

defined as:

�̂� = minp⃗⃗

[𝑑 (𝑓ℎ (𝑤1(𝜉 )) , 𝑓𝑐(𝑤1(𝜂 ℎ))) , 𝑑 (𝑓ℎ (𝑤2(𝜉 )) , 𝑓𝑐(𝑤2(𝜂 ℎ)))… , 𝑑 (𝑓ℎ (𝑤𝑁𝑐(𝜉 )) , 𝑓𝑐(𝑤𝑁𝑐(𝜂 ℎ)))], (23)

where 𝑤𝑖 represents the 𝑖𝑡ℎ component of the output vector �⃗⃗� ∈ ℝ𝑁𝑐 . Note that it may be possible to reduce the

dimensionality of the coarse input vector by considering the functional dependencies of the Fourier coefficients.

Alternatively such dependencies may be reflected as constraints in the multi-objective problem.

4.1. Double spring-mass damper

The motivation for the high-order, low-fidelity model is to create a non-intrusive coupling that allow the dynamics of a fine-

scale model to be up-scaled to a coarse-scale model which may lack the sufficient degrees of freedom or may have simplified

physics such that it cannot capture the fine scale dynamics on its own. In an inverse formulation it allow models at the scale

of a single component to account for non-linear interactions between components when a structure or machine is modelled

at a higher scale. As has been recently pointed out in Guinard et al (2018), the top-down modelling approach currently

favoured by commercial codes resolves the loads or displacements of an entire structure before imposing these as boundary

conditions on local models. While such a method can be used to concentrate computational resources on weak areas in the

structure the propagation of information is one way, from higher to lower scales. There is motivation then, in developing

[15]

methods of non-intrusive coupling which allows for the propagation of information back up the hierarchy of scales. In so

doing, the effect of local non-linarites on the global structure may be appreciated [46].

An example of such a case is demonstrated here for the suspension in a machine which is modelled as a spring-mass damper.

The suspension is modelled at two scales: at the coarse scale of the entire machine it is approximated as a single spring-mass

damper undergoing forcing from another component. At a lower, more fundamental scale, the suspension is modelled as a

double spring-mass damper system. Several works in the literature have used double spring-mass dampers in their models:

for instance in Hać and Youn (1991) and Havelka et al. (2012) double spring-mass dampers are used to model the suspension

in automobiles [47], [48].

The forcing due to the other components in the machine is applied to the fine scale model (in a similar manner the results

of a global analysis are applied to local finite-element models of a notched composite plate in Jrad et al. (2014) as boundary

conditions [49]) and the displacement of the centre of mass (𝑥𝐶𝑂𝑀) calculated. However, the single mass at the coarse-scale

does not have the degrees of freedom to mimic these dynamics, hence a high-order, low-fidelity model is used to incorporate

the dynamics into the coarse-scale model. Figure 12 illustrates the problem and Table 3 the stochastic parameters at each

scale. The goal of the uncertainty upscaling is to find PDFs for the material properties at the coarse scale (𝑘 and 𝛿 at the scale

of the entire machine) such that the first natural frequency and oscillations of 𝑥𝐶𝑂𝑀 under the periodic forcing 𝐹(𝑡) are up-

scaled accurately from the fine scale.

The dynamics of both the double spring-mass damper and the single spring-mass damper are described through the following

equations of motion. For the double spring case a pair of coupled ordinary differential equations (ODEs) describing the

displacement of each of the masses are taken from Fay and Graham (2003) [50]:

𝑚1𝑥1̈ = −𝛿1𝑥1̇ − 𝑘1𝑥1 − 𝑘2(𝑥1 − 𝑥2) 𝑚2𝑥2̈ = −𝛿2𝑥2̇ − 𝑘2(𝑥2 − 𝑥1) + 𝐹(𝑡), (24)

where 𝑚𝑖 and 𝑥𝑖 refer to the mass and displacement and 𝑘𝑖 and 𝛿𝑖 refer to the spring constant and damping of the 𝑖𝑡ℎ mass.

Through substitution a fourth-order ODE may be found for each displacement, which is solved numerically. The equation of

motion for the centre of mass of the single spring case is much simpler:

𝑚�̈� = −𝛿�̇� − 𝑘𝑥 + 𝐹(𝑡), (25)

where 𝑥 is the location of the single mass and 𝛿 and 𝑘 the damping and stiffness at the coarse scale. Equations 22 and 23

are also be used to derive characteristic equations for the natural frequency of the component. For simplicity, normal

distributions are used for the material properties at the fine-scale (see Table 3).

Scale Stochastic parameter Distribution

Coarse (single spring) 𝑘 PCE

𝛿 PCE

Fine (double spring) 𝑘1 𝑁(0.4, 0.1)

𝑘2 𝑁(0.8, 0.1)

𝛿1 𝑁(0.1, 0.02)

𝛿2 𝑁(0.2, 0.05)

Table 3: the stochastic material properties at each of the scales in the example

4.1.1. Up-scaling the natural frequency

As can be seen in Figure 13, it is possible to find PDFs of 𝑘 and 𝛿 such that the PDFs of the fist natural frequencies for the

component at each scale match. However, whilst it is possible to match the first natural frequency using the coarse-scale

[16]

model, the model lacks sufficient degrees of freedom to capture the motion of the centre of mass in the fine-scale model.

Thus a high order, low-fidelity model must be used.

Figure 12: A component may be modelled coarsely as a spring damper system but also as a double spring damper system at a finer scale. A high-order, low-fidelty model is used to up-scale both the natural frequency and periodic motion of the component under periodic forcing.

Figure 13: PDF matching results for the first natural frequency. As with the wing box example, PDFs for the material properties at the coarse-scale can be found such that probabilistic equivalence is possible.

[17]

4.1.2. Up-scaling the double mass-damper oscillations

Using the equations of motion in equation 24 the motion of the centre of mass as a function of time at the fine-scale may be

described as:

𝑥𝐶𝑂𝑀𝑓 (𝑡) =

𝑚1𝑥1(𝑡)+𝑚2𝑥2(𝑡)

(𝑚1+𝑚2). (26)

A high order, low-fidelity model is created by adding a Fourier series correction to the solution:

𝑥𝐶𝑂𝑀ℎ𝑜,𝑙𝑓(𝑡) =

𝑚1𝑥1(𝑡)+𝑚2𝑥2(𝑡)

(𝑚1+𝑚2)+ 𝑎0 + ∑ 𝑎𝑖 cos(𝑖𝑤𝑡) + 𝑏𝑖sin (𝑖𝑤𝑡)

𝑛𝑓

𝑖=1, (27)

where 𝑛𝑓 is the order of the Fourier series correction. A PCE representation must then be found for 𝑘, 𝛿, 𝑎0, 𝑎𝑖 {𝑖 = 1…𝑛𝑓}

and 𝑏𝑖 {𝑖 = 1…𝑛𝑓}. The stochastic dimension of the coarse-scale model is thus increased from 2 to 3 + 2𝑛𝑓. As can be seen

from Figure 14, creating a high-order, low-fidelity model allows the dynamics at the coarse-scale to better reflect those of

the fine scale.

Table 4 compares the statistical distances between the fine scale displacements and those of the coarse-scale and high-order,

low-fidelity model. The high-order, low-fidelity model offers a 72% improvement in the statistical distance between the time

series of the displacements, as quantified by the method of moment distance. The example presented here is a simplistic

one and so care must be taken to not over-generalise the results, however, the proposed high-order low fidelity model

approach could be a useful tool for applying the stochastic upscaling approach to real-world problems.

Model Method of moments distance from

the fine scale displacements (𝛾 = 1)

Coarse-scale (single-spring) 1.3837

High-order, low-fidelity (ho,lf) 0.3948

Table 4: A comparison of the statistical distance between the fine-scale model displacements (double spring) and those of the coarse-scale

and high-order, low-fidelity model. The statistical distance is quantified by the method of moments (see section 3.2).

Figure 14: A comparison plot of the displacements of the component centre of mass (𝑥𝐶𝑂𝑀) for the coarse-scale (single spring), fine-scale (double spring), and high-order, low-fidelity model. By adding a Fourier series correction, the dynamics of the coarse-scale model may

better reflect those of the component when it is modelled at a finer scale.

[18]

5. Conclusions

Creating a non-intrusive framework for the propagation of uncertainties through multiscale models is still an open question.

Such a framework would allow uncertainty quantification to be conducted on multiscale models and would represent a

potentially huge saving in both time and money spent on physical testing. A method has been presented here to carry out

stochastic upscaling through PDF matching.

Two aspects of the method give it its novelty: firstly, SAMBA is used to propagate uncertainty through each model in the

hierarchy, as opposed to Monte Carlo sampling, to make the resulting optimisation problem less computationally intensive

and hence more suitable for industrial applications.

Secondly, for cases in which the coarse model lacks the sufficient degrees of freedom for PDF matching a high-order, low-

fidelity model is created using Fourier or wavelet interpolation.

The method has been able to achieve satisfactory PDF matching for both Askey and non-Askey scheme input distributions.

Achieving a good matching for thick tailed distributions such as the Cauchy distribution is significant as these distributions

are used often in accident prediction. A technique has been proposed for constructing a high-order, low-fidelity model in

order to achieve PDF matching for cases when the coarse-scale model filters the dynamics of the fine-scale model. The

technique has been demonstrated for the simplistic example of a double spring-mass damper system.

As has been discussed above, future works on stochastic model upscaling should focus on propagating uncertainty through

more than just two scales in case studies that are more complicated. An error analysis is needed in order to determine

whether the use of probabilistic equivalence more than once in the course of this roll-up, and in doing so achieving

probabilistic equivalence for quantities at the lower scales which are not quantities of interest at the highest scale, introduces

significant model error.

Funding data

This work was jointly funded by Airbus and EPSRC through iCASE voucher number 17000099

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