Sensitivity analysis based dimension reduction of multiscale models

Journal Pre-proof Sensitivity analysis based dimension reduction of multiscale models Anna Nikishova, Giovanni Eugenio Comi, Alfons G. Hoekstra

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S0378-4754(19)30316-7 https://doi.org/10.1016/j.matcom.2019.10.013 MATCOM 4884

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Mathematics and Computers in Simulation

Received date : 4 April 2019 Revised date : 6 October 2019 Accepted date : 21 October 2019 Please cite this article as: A. Nikishova, G.E. Comi and A.G. Hoekstra, Sensitivity analysis based dimension reduction of multiscale models, Mathematics and Computers in Simulation (2019), doi: https://doi.org/10.1016/j.matcom.2019.10.013. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Published by Elsevier B.V. on behalf of International Association for Mathematics and ⃝ Computers in Simulation (IMACS).

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Sensitivity analysis based dimension reduction of multiscale models Anna Nikishovaa1 , Giovanni Eugenio Comib , Alfons G. Hoekstraa a

Computational Science Lab, Institute for Informatics, Faculty of Science, University of Amsterdam2 b Scuola Normale Superiore di Pisa3

Abstract In this paper, the sensitivity analysis of a single scale model is employed in

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order to reduce the input dimensionality of the related multiscale model, in

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this way, improving the efficiency of its uncertainty estimation. The approach is illustrated with two examples: a reaction model and the standard OrnsteinUhlenbeck process. Additionally, a counterexample shows that an uncertain input should not be excluded from uncertainty quantification without estimating

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the response sensitivity to this parameter. In particular, an analysis of the function defining the relation between single scale components is required to understand whether single scale sensitivity analysis can be used to reduce the dimensionality of the overall multiscale model input space.

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Keywords: Dimension reduction; Sensitivity Analysis; Uncertainty Quantification; Multiscale model; Coupled models; Sobol Sensitivity Indices

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1. Introduction

Results of computational models should be supported by uncertainty esti-

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mates whenever precise values of their inputs are not available [1–3]. This is

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usually the case since measurements of inputs rarely can be made exactly, or

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1 Corresponding

author: Anna Nikishova ([email protected]) Address: Room C3.156, Science Park 904, 1098 XH Amsterdam, The Netherlands Park 904, 1098 XH Amsterdam, The Netherlands 3 Piazza Cavalieri 7, 56126 Pisa, Italy 2 Science

Preprint submitted to Mathematics and Computers in Simulation

October 6, 2019

Journal Pre-proof

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inputs may include aleatory uncertainty [4, 5]. Uncertainty Quantification (UQ)

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of a complex model usually requires powerful computational resources. More-

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over, the cost of some UQ methods increases exponentially with the number of

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uncertain inputs.

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Sensitivity analysis (SA) identifies the effects of uncertainty in a model input

9

or group of inputs to the model response. In 1990, Sobol introduced sensitivity

10

indices to measure the effect of input uncertainty on the model output variance

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[6, 7]. In [8, 9], Sobol employs SA in order to fix uncertain parameters with low

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total sensitivity indices and reduce the model dimensionality. Here such application of SA to multiscale models is considered. A multiscale

14

model is defined as a collection of single scale models that are coupled using scale

15

bridging methods [10]. The approach proposed here consists in examining the

16

type of function coupling the single scale components, followed by estimating

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the sensitivity of the response of a single scale model. This paper demonstrates

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that estimates of the single scale model sensitivity can be used to assess the

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sensitivity of the overall multiscale model response for some classes of multiscale

20

model functions. However, this is not always possible, as will be shown by a

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counterexample.

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Sobol’s variance based approach is the preferred method to measure model

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output sensitivity [11–14]. Even though it is important to note that variance

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is not always the most representative measure of model response uncertainty

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[15, 16], it is assumed to be so in this work. The proposed approach is based

26

on exploring the coupled structure of multiscale models, allowing to analyse

27

independently the single scale models. Therefore, the second assumption is

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that SA can be performed on the multiscale model components. Additionally,

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it is assumed that the multiscale model parameters are uncorrelated.

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In Section 2, a brief description of multiscale models is given. Section 3

is devoted to SA, and its application to dimensionality reduction of a multi-

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scale model is discussed in subsection 3.1. Together with some examples of

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the sensitivity analysis for multiscale models (subsections 3.1.1 and 3.1.2), a

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counterexample is considered in subsection 3.1.3 in order to illustrate that, even

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2

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though it is tempting to employ the SA result of single scale models to the

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response of the overall multiscale model, this is not always allowed. Section 4

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summarizes the results and includes a note on the application of the proposed

38

approaches to some real-world models. Some other cases of multiscale models

39

for which the proposed method on dimension reduction can be applied are in the

40

Appendix. In particular, in the Appendix D an upper bound for the sensitivity

41

of model output for a general class of coupling function is obtained.

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2. Multiscale model Following the concept introduced in the Multiscale Modelling and Simula-

44

tion Framework (MMSF) [10, 17–19], multiscale models are considered as a set

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of single scale models coupled using scale bridging methods. The single scale

46

models represent processes that operate on well defined spatio-temporal scales.

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In MMSF, the single scale models are placed on a scale separation map (SSM),

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where axes indicate the spatial-temporal scales. An example of SSM with a

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multiscale model that consists of two single scale components is shown in Fig-

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ure 1. The directed edges between the single scale components indicate their

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interactions. In general, cyclic and acyclic coupling topologies are recognised:

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the cyclic one, as in Figure 1, assumes a feedback loop between the components,

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and in the acyclic one, no feedback is present. Here we rely on the assumptions

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of a component-based structure of the multiscale models as well as on a drastic

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difference in the computational cost of the single scale components.

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The overall multiscale model is denoted by a function g(x, ξ) = z such that g : Rn+m → Rq

with n, m, q ∈ N and E[|g|2 ] < ∞, which produces the Quantity of Interest (QoI) z. We introduce a function G : Rs+p → Rq , with s, p ∈ N, as a representation

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of g, which underlines the relationship between the micro model response and the remaining variables inside the macro model, denoted by the function f : g(x, ξ) = G(f (x), h(ξ)). 3

Spatial scale

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x

ξ

G(f, h)

z = g(x, ξ)

h

Temporal scale Figure 1: Scale separation map. The functions G(f, ·) and h are the macro and micro models with inputs x and ξ, respectively. The function G(f (x), h(ξ)) defines the relation between the response of the micro model and the rest of the macro model parameters denoted by f . The

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final multiscale model output z = g(x, ξ) = G(f (x), h(ξ)) is produced by the macro model.

Therefore, the function G(f (x), ·) represents the macro model for some f : Rn →

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Rs which depends on parameters x = (x1 , . . . , xn ). It is assumed that f can be

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executed in a relatively short computational time, that it has a finite non-zero

59

variance, i.e. E[|f |2 ] < ∞ and f is not constant, and that it is possible to obtain

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its output sensitivity.

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The micro scale component is defined by a function h : Rm → Rp which

62

satisfies E[|h|2 ] < ∞. The sets of variables on which the function h depends4

63

are of the form ξ = (ξ1 , . . . , ξm ).

Without loss of generality, later in the text it is assumed that the uncertain

65

inputs x and ξ follow uniform distributions U([0, 1]n ) and U([0, 1]m ), respec-

66

tively.

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h may depend on the macro model response. When this is the case, the

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4 Additionally,

micro model function is denoted by h(x, ξ) or h(x), meaning that it depends on the same uncertain inputs as the macro model function f . This is a relevant feature of the method

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presented here.

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3. Sensitivity analysis Sensitivity analysis identifies the effect of uncertainty in the model input parameters on the model response [20]. The Sobol sensitivity indices [6, 11] (SIs) are widely used to measure the response sensitivity. The total SI of an input xi for the results of the multiscale model function g(x, ξ) = z is given by Var(z) − Varx∼i ,ξ (Exi [z|x∼i , ξ]) Var(z) R R R |g(x, ξ)|2 dxdξ − | g(x, ξ)dxi |2 dx∼i dξ R , = |g(x, ξ)|2 dxdξ − |g0 |2

STgx = i

(3.1)

where g0 = E[g(x, ξ)], and the notation x∼i = (x1 , . . . , xi−1 , xi+1 , . . . , xn ) is employed [21]. In [6, 9], the total SIs were employed to identifying the effective

f

dimensions of a model function and to fixing unessential variables. In particular,

satisfies

    1 δ(x0i ) < 1 + STgx > 1 − ε i ε

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it was shown that, when fixing xi to a value x0i in [0, 1], the error defined by R g(x, ξ) − g(x∼i , x0 , ξ) 2 dxdξ i 0 δ(xi ) = Var(z) P

(3.2)

for any ε > 0. This result is applied in this work, meaning that we expect with

70

high confidence that fixing an input with a low total sensitivity index does not

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produce a large error in the estimates of uncertainty. Then, this fact can be

72

employed to reduce input dimensionality, so that UQ can be performed more

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efficiently. However, sensitivity indices are usually not given in advance and

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their estimation can be a computationally expensive task as well.

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3.1. Sensitivity analysis of multiscale models

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In this work, it is proposed to evaluate the response sensitivity of the com-

putationally cheap single scale model f to estimate an upper bound of the

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sensitivity of the multiscale model output z. This approach can be highly com-

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putationally efficient; however, the method does not work in general.

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In order to fix uncertain inputs according to single scale model SA, it should

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be proved that the total sensitivity for an input xi remains small also for the 5

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output of the model g(x, ξ), i.e. STgx  1 given that STfx  1. This cannot be

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assumed in general, and it depends on the form of the model function G.

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The first step of the proposed approach is to analyse the multiscale model

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function G, as it is shown in the following sections. In the cases, in which our

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method applies, the next step is to estimate numerically STfx for i = 1, . . . , n by

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a black box method, for instance from [9]. Then, if it is found that STfx  1, it

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i

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shall follow automatically that STgx  1. Hence, according to (3.2), uncertainty

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can be estimated with fixed xi without producing a large error.

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While the results stated below hold also for vector valued functions, using the

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definition of total SI given in (3.1), we shall work mainly with scalar functions,

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in order to avoid a heavy notation.

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3.1.1. Case 1

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f

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We start by considering the homogeneous case: G : R2 → R, given by G(u, v) = uv.

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Theorem 3.1. Let g : (0, 1)m+n → R be a function in L2 ((0, 1)m+n ) such that g(x, ξ) = f (x)h(ξ),

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for some f : (0, 1)n → R and h : (0, 1)m → R satisfying f ∈ L2 ((0, 1)n ) and h ∈ L2 ((0, 1)m ). Then, we have

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In particular,

STgx = λf,h STfx , R

(3.3)

i

i

λf,h = R

f (x)2 dx − f02

f 2 (x)dx −

2 2 R f0 h0 h2 (ξ)dξ

.

STgx ≤ STfx , i

and

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where

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96

STgx ≥ i



1− R

(3.4)

i

f02 f 2 (x)dx

6



STfx . i

(3.5)

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Proof. The total SI of the input xi for the results of the model g(x, ξ) is equal to

RR R f 2 (x)h2 (ξ)dxdξ − ( f (x)h(ξ)dxi )2 dx∼i dξ RR f 2 (x)h2 (ξ)dxdξ − (f0 h0 )2 R 2 R R f (x)dx − ( f (x)dxi )2 dx∼i = R f 2 h20 f 2 (x)dx − R h20(ξ)dξ R R 2 R R f (x)2 dx − f02 f (x)dx − ( f (x)dxi )2 dx∼i R =R 2 2 f h0 f 2 (x)dx − f02 f 2 (x)dx − R h20(ξ)dξ R f (x)2 dx − f02 =R STfx , f02 h20 i 2 R f (x)dx − h2 (ξ)dξ

STgx = i

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RR

from which (3.3) follows.

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By the Cauchy-Schwarz inequality, Z h20 ≤ h2 (ξ)dξ.

Therefore, λf,h ≤ 1, and (3.4) is obtained. In addition, again by CauchySchwarz inequality, we get R 101

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f (x)2 dx − f02 f2 R =1− R 2 0 >0 f 2 (x)dx f (x)dx

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λf,h ≥

for any h ∈ L2 ((0, 1)m ). Hence, (3.5) is obtained.

Therefore, if a low sensitivity to the parameter xi is identified by computing

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STfx , i

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On the other hand, inequality (3.5) means that we have a lower bound for the

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total SI of the input xi for the model g(x, ξ) = f (x)h(ξ), which is independent

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from the choice of the function h(ξ). In particular, if xi is an important variable

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for the model f (x), then (3.5) implies that it cannot loose dramatically its

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importance in the model given by g.

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this parameter can be excluded from UQ of the whole multiscale model.

Example 3.2 (Reaction equation). An example of Case 1 can be a reaction equation presented by an acyclic model [22] with initial conditions provided by

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some function f (x):

∂z(t, x, ξ) = −ψ(ξ)z(t, x, ξ), ∂t z(0, x, ξ) = f (x),

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where x and ξ are uncertain model inputs. The analytical solution of the equation is z(t, x, ξ) = f (x)e−tψ(ξ) . Therefore, if we define ht (ξ) = e−tψ(ξ) , we get z(t, x, ξ) = f (x)ht (ξ), 109

and Theorem 3.1 can be applied. Since the proposed approach is applicable to multiscale models regardless of the complexity of f and h, in the example, these model components are represented by the following equations:

f

ψ(ξ) = ξ12 − ξ2 ,

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f (x) = x21 + x1 x2 x3 + x33 − x1 x3 , 110

where uncertain parameters x have uniform distribution U(0.9, 1.1), ξ1 is uni-

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formly distributed on [0.07, 0.09], and ξ2 on [0.05, 0.09].

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Sensitivity analysis of the function f results in: STfx ≈ 2.9 · 10−1 , 1

STfx ≈ 7.2 · 10−2 , 2

STfx 3

≈ 6.5 · 10−1 ,

suggesting that the parameter x2 does not significantly affect the output of the

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function f . Therefore, by Theorem 3.1, the value of this parameter can be

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equated to its mean when estimating uncertainty of the overall model response

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z.

Figure 2 (a) illustrates a satisfactory match between the mean values and

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standard deviations obtained by sampling the results varying all the uncertain

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inputs and keeping the input x2 equal to its mean value. Figure 2 (c) shows that

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the relative error in the standard deviation does not exceed 3.5% at any simu-

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lation time. Moreover, the resulting p-value of Levene’s test [23] is about 0.84.

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Therefore, the null hypothesis that the samples are obtained from distributions

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with equal variances cannot be rejected.

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Figure 2: (a) Comparison of the estimated mean and standard deviation of the model response z t using the original sample and the sample with the unimportant parameter x2 equal to its mean value (reduced); (b) and (d) Comparison of the probability density functions and the cumulative distribution functions at the final simulation time Tend = 100; (c) Relative error

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in the estimated mean and standard deviation using the samples with the reduced number of

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uncertain input.

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Figures 2 (b) and (d) show the probability density functions (PDFs) and the

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cumulative distribution functions (CDFs) of the uncertain model output z at the

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final simulation time obtained using these two samples. There is a good match

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in the PDFs and CDFs with Kolmogorov–Smirnov (K-S) two sample test shows

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the K-S distance nearly 3.6 · 10−4 and p-value larger than 0.5, therefore, the

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hypothesis that the two samples are drawn from the same distributions cannot

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be rejected5 .

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3.1.2. Case 2

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We consider the linear case, where the sampling function G : R2 → R is given by G(u, v) = u + v.

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Theorem 3.3. Let g : (0, 1)n+m → R be a function in L2 ((0, 1)n+m ) such that g(x, ξ) = f (x) + h(ξ),

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for some f : (0, 1)n → R and h : (0, 1)m → R satisfying f ∈ L2 ((0, 1)n ) and h ∈ L2 ((0, 1)m ). Then, we have

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STgx = µf,h STfx , i

where

µf,h :=

i

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Proof. The total SI of the input xi for the results of the model g is equal to R R R (f (x) + h(ξ))2 dxdξ − ( f (x) + h(ξ)dxi )2 dx∼i dξ g R STx = i (f (x) + h(ξ))2 dxdξ − (f0 + h0 )2 R 2 R R f (x)dx − ( f (x)dxi )2 dx∼i R =R 2 , f (x)dx − f02 + h2 (ξ)dξ − h20

from which we get (3.6) by dividing by Var(f ) numerator and denominator. Clearly, µf,h ∈ (0, 1], and so we conclude that STgx ≤ STfx .

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.

In particular, STgx ≤ STfx . i

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1 Var(h) Var(f )

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1+

(3.6)

i

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i

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5 This

conclusion also applies to the other simulation times (data not shown).

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Therefore, if the parameter xi is unimportant for f , it can be equated to its mean value in the uncertainty estimation of the model g. Example 3.4 (Standard Ornstein-Uhlenbeck process). An example of Case 2 can be a multiscale model whose micro scale dynamics does not depend on the macro scale response. Let us consider the system (Figure 3 (a)) [24, 25]: ∂z = v + f (x), ∂t ∂v 1 1 ˙ =− v+ √ W t, ∂t   f (x) = −x1 + (x22 x3 + x4 ), where z simulates the slow processes with z(t = 0) = 1, v is the fast process

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˙ t is a white noise with unite variance. The fast with v(t = 0) = 1,  = 10−2 , W

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dynamics is the standard Ornstein-Uhlenbeck process. At any simulation time

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˙ t plays the role of ξ in Theorem 3.3. The macro model uncertain parameters t, W

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x = (x1 , x2 , x3 , x4 ) follow normal distribution, such that x1 ∼ N (0, 10−4 ), x2 ∼

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N (0, 2.5 · 10−4 ), x3 ∼ N (0, 2.5 · 10−6 ), x4 ∼ N (0, 2.5 · 10−6 ). The system is

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simulated using the forward Euler method with the macro time step ∆tM = 1

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and the micro time step ∆tµ = 10−2 .

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Sensitivity analysis of the function f (x) yields STfx ≈ 7.7 · 10−1 ,

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1

STfx ≈ 2.6 · 10−4 , 2

STfx 3

≈ 3.9 · 10−4 ,

STfx ≈ 2.0 · 10−1 . 4

At any simulation time, the inputs x2 and x3 do not influence significantly

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the output of the function f . Therefore, they can be equated to their mean

151

values without a substantial loss of accuracy of the uncertainty estimate as a

152

consequence of Theorem 3.3. The uncertainty estimation results of z are presented in Figure 3 (b). As it is

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proven analytically, the estimates obtained by sampling the model results with

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uncertain parameters x2 and x3 equal to their mean values are close to those 11

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Figure 3: (a) Standard Ornstein-Uhlenbeck process; (b) Comparison of UQ result using the original sample and the sample obtained with values of the unimportant parameters x2 and x3 equal to their mean (reduced); (c) and (e) Comparison of the PDF and CDF at the final

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time step; (d) Relative error in the estimation of the mean and standard deviation.

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resulting from samples where all the uncertain inputs vary. At any simulation

157

time, the relative error between these estimates of the standard deviation does

158

not exceed 1.1% (Figure 3 (d)). Additionally, Levene’s test shows p-value about

159

0.66, therefore, we cannot reject the hypothesis that the two samples are drawn

160

from distributions with the same variance.

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The PDFs and CDFs for the model result at the final time point obtained

162

from these two samples are in Figure 3 (c) and (e). There is a good match of

163

the PDFs and CDFs obtained from these two samples, and K-S test produces

164

the distance about 0.01 and p-value about 0.47, therefore, the hypothesis that

165

the two samples are drawn from the same distributions cannot be rejected. Some additional cases of the function G for which the method of eliminating

167

unimportant parameters to reduce the input dimensionality is valid are pre-

168

sented in the Appendix.

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3.1.3. Counterexample

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In this section, the importance of the examination of properties of the func-

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170

171

tion G is demonstrated. The counterexample illustrates that low sensitivity

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to a parameter of the response of a function f does not necessarily imply low

173

sensitivity to this parameter of a response of the function g.

1 and i = 2,

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Example 3.5 (Total sensitivity indices of composite functions). Let n = 2, m =

5 ∂z 1 =− √ |x1 + x2 + ξ|− 4 , ∂x1 44β

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for (x1 , x2 , ξ) ∈ (0, 1)3 , with z(0, x2 , ξ) = √ 4

1 β|x2 +ξ|

(3.7) and β > 0 some fixed

parameter. The solution to equation (3.7) can be represented using the following

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system

u = f (x) = x1 + βx2 , v = h(x1 , ξ) = (1 − β)x1 − βξ, G(u, v) = p 4 13

1

|u − v|

,

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so that 1 1 √ z(x, ξ) = g(x, ξ) = √ . 4 β 4 x1 + x2 + ξ Let us now directly obtain sensitivity indices of the function f (x) for the parameter x2 : STfx =

1 3

β2 3

+

2

1 3

+

+ β2 3

β 2

−

+

β 2



1 3

−

+ 

β2 4

β+1 2

+ β2 2



=

β2 12 β 2 +1

=

12

β2 . 1 + β2

Note that STfx can be made arbitrarily small as β → 0: for instance, by choosing
2 1 β ∈ 0, 10 , we get 1 STfx < , 2 100 so that x2 becomes an unimportant input for f .

f

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√1 β

R

1 dx1 dx2 dξ x1 +x2 +ξ

−

√1 β

R

R

1 dx2 x1 +x2 +ξ

2

dx1 dξ 2 1 1 √1 √ dx1 dx2 dξ − √1β (0,1)3 √ dx1 dx2 dξ 4 β (0,1)3 x1 +x2 +ξ x1 +x2 +ξ R 2 R R 1 1 √ dx1 dx2 dξ − (0,1)2 (0,1) √ dx2 dx1 dξ 4 (0,1)3 x1 +x2 +ξ x1 +x2 +ξ = R 2 . R 1 1 √ dx1 dx2 dξ − (0,1)3 √ dx1 dx2 dξ 4 (0,1)3 x +x +ξ x +x +ξ =

(0,1)3

√

R

rep

STgx 2

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On the other hand, sensitivity of the function g(x, ξ) does not depend on β:

1

(0,1)2

R

2

1

(0,1)

√ 4

2

In addition, since g is symmetrical,

STgx = STgx = STgξ . 1

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2

Hence, this proves that x2 is not an unimportant input for the function g, since

176

it must be as relevant as x1 and ξ. Therefore, in general, it is wrong to eliminate

177

an uncertain input from UQ only based on sensitivity analysis of a single scale

178

model without verifying that STgx ≤ λSTfx holds for some finite λ ≥ 0 as in

179

Theorem 3.1 and Theorem 3.3.

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4. Concluding remarks

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An application of sensitivity analysis to reduce dimensionality of multiscale

182

models in order to improve the performance of their uncertainty estimation is 14

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discussed in this paper. It has been shown that for some multiscale models,

184

the estimates of Sobol sensitivity indices of a single scale output can be used as

185

an estimate of the upper bound for the sensitivity of the output of the whole

186

multiscale model. In other words, knowledge on the importance of inputs from

187

single scale models can be used to find the effective dimensionality of the overall

188

multiscale model. Two classes of coupling function G (multiplicative, additive)

189

were considered, where the approach was demonstrated to work, based on The-

190

orems 3.1 and 3.3, and two examples. However, a counterexample was also

191

constructed, showing that the success of the method strongly depends on the

192

properties of the coupling function G. Obviously, this analysis only covers a

193

very small portion of possible coupling functions, and a more systematic or case

194

by case investigation would be warranted.

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The next step is to apply the proposed approach to real-world multiscale

196

applications, for instance, to a multiscale fusion model [26] and to a coupled

197

human heart model [27]. Uncertainty quantification applied to these models is

198

computationally expensive due to the high dimension of the model parameters.

199

Therefore, the SA analysis on single scale models to reduce the dimensionality

200

of the overall multiscale model input can be one of the possible ways to improve

201

the efficiency of the model uncertainty quantification.

202

Funding

lP

rep

195

This work is a part of the eMUSC (Enhancing Multiscale Computing with

204

Sensitivity Analysis and Uncertainty Quantification) project. A. N. and A. H.

205

gratefully acknowledge financial support from the Netherlands eScience Center.

206

This project has received funding from the European Union Horizon 2020 re-

207

search and innovation programme under grant agreement #800925 (VECMA

208

project).

The authors would like to thank the anonymous reviewers for their valuable

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209

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203

210

comments and suggestions.

15

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211

Declarations of interest: none

212

None.

213

Appendices In this Appendix, additional cases of the function G are considered. In par-

215

ticular, relations between the function f and two or more functions representing

216

the micro model are investigated, in this way allowing for vector valued functions

217

h. Overall, our goal here is to show that the method presented in this work can

218

be applied to different types of functions of the multiscale model components.

219

Appendix A. Case 3

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220

f

214

Consider the affine linear case: G : R3 → R given by G(u, v1 , v2 ) = uv1 + v2 . Theorem A1. Let g : (0, 1)m+n+k → R be a function in L2 ((0, 1)m+n+k ) such

rep

that

g(x, ξ, η) = f (x)h1 (ξ) + h2 (x∼i , η),

221

222

for some f : (0, 1)n → R, h1 : (0, 1)m → R and h2 : (0, 1)k+n−1 → R satisfying

f ∈ L2 ((0, 1)n ), h1 ∈ L2 ((0, 1)m ), and h2 ∈ L2 ((0, 1)k+n−1 ). Then, STgx = γf,h1 ,h2 STfx ,

where

rna

f 2 (x)dx −

2 2 Rf0 2(h1 )0 h1 (ξ)dξ

+

R R

If, additionally, it is assumed that

R

f 2 (x)dx − f02 2 h22 (x∼i ∼i dη−(h2 )0 R ,η)dx h21 (ξ)dξ

(h1 )0 (f h2 )0 ≥ f0 (h1 )0 (h2 )0 ,

STgx ≤ STfx . i

i

16

(A.1)

i

then

Jo u

223

γf,h1 ,h2 := R

lP

i

0 (h2 )0 + 2(h1 )0 (f hR2 )h02−f (ξ)dξ 1

(A.2)

.

Journal Pre-proof

Proof. We compute Z Z Z Z Var(g)Txi = f 2 (x)h21 (ξ)dxdξ + h22 (x∼i , η)dx∼i dη + 2(h1 )0 (f h2 )0 + −

Var(g) =

Z Z Z

f (x)h1 (ξ)dxi

Z Z

− Z Z

2

dx∼i dξ+

h22 (x∼i , η)dx∼i dη − 2(h1 )0 (f h2 )0 , Z Z f 2 (x)h21 (ξ)dxdξ + h22 (x∼i , η)dx∼i dη + 2(h1 )0 (f h2 )0 +

− h20 f02 − (h2 )20 − 2f0 (h1 )0 (h2 )0 .

Thus, the total SI of the input xi for the results of the model g(x, ξ, η) is equal to Var(g)Txi Var(g)

=R

f

i

f 2 (x)dx −

R

2 2 Rf0 2(h1 )0 h1 (ξ)dξ

roo

STgx =

R R f 2 (x)dx − ( f (x)dxi )2 dx∼i +

R R

2 h22 (x∼i ∼i dη−(h2 )0 R ,η)dx h21 (ξ)dξ

0 (h2 )0 + 2(h1 )0 (f hR2 )h02−f (ξ)dξ

,

1

rep

from which (A.1) follows. By Cauchy-Schwarz inequality, we have Z (h1 )20 ≤ h21 (ξ)dξ, Z Z (h2 )20 ≤ h22 (x∼i , η)dx∼i dη, which imply

lP

γf,h1 ,h2 ≤

Var(f )

0 (h2 )0 Var(f ) + 2(h1 )0 (f hR2 )h02−f (ξ)dξ

.

1

224

225

227

γf,h1 ,h2 ≤ 1,

and the result follows. Note that, in the previous theorem, h2 can be independent of more than one

input xj , however, it is crucial to assume the independence from the unimpor-

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226

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To estimate the last term at the denominator, (A.2) is employed, yielding

tant parameters which we want to exclude from uncertainty quantification.

17

Journal Pre-proof

Remark A2. It is noticed that condition (A.2) is equivalent to assume that E(h1 )Cov(f, h2 ) ≥ 0, since Cov(f, h2 ) = (f h2 )0 − f0 (h2 )0 . Under the same assumption, one can get the following lower bound on γf,h1 ,h2 : R 2 f (x)dx − f02 R R γf,h1 ,h2 ≥ R . 2 2 h2 (x∼i ∼i dη−(h2 )0 0 (h2 )0 R ,η)dx f 2 (x)dx + + 2(h1 )0 (f hR2 )h02−f h2 (ξ)dξ (ξ)dξ 1

1

On the other hand, if E(h1 )Cov(f, h2 ) ≤ 0; that is, (h1 )0 (f h2 )0 ≤ f0 (h1 )0 (h2 )0 , then γf,h1 ,h2 ≥ R

R

f 2 (x)dx +

f 2 (x)dx − f02 R R

2 h22 (x∼i ∼i dη−(h2 )0 R ,η)dx h21 (ξ)dξ

.

In addition, if it is assumed that (h1 )0 (f h2 )0 ≤ f0 (h1 )0 (h2 )0 and that

f

Var(h2 ) + Cov(f, h2 ) ≥ 0, Var(f ) + R 2 h1 (ξ)dξ

γf,h1 ,h2 ≤

1

2 h22 (x∼i ,η)dx R ∼i dη−(h2 )0 Var(f ) h21 (ξ)dξ

Appendix B. Case 4

(f h2 )0 −f R 0 (h2 )0 + 2(h1 )0 Var(f ) h2 (ξ)dξ

.

1

rep

228

1+

R R

roo

we obtain the following upper bound for γf,h1 ,h2 :

229

A variant of the linear case G(u, v) = u + v is considered. The difference

230

with Case 2 of Theorem 3.3 is that now the functions f and h depend on the

231

same set of variables.

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Theorem B1. Let g : (0, 1)n → R be a function in L2 ((0, 1)n ) such that g(x) = f (x) + h(x),

232

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for some f, h : (0, 1)n → R, f, h ∈ L2 ((0, 1)n ). Then, if Cov(f, h) = (f h)0 − f0 h0 ≥ 0,

we have

STgx ≤

234

STfx Var(f ) + i

2 q SThx Var(h) i

Var(f ) + Var(h)

,

(B.1)

and so

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233

i

q

STgx ≤ 2 max{STfx , SThx }, i

i

where the factor 2 is sharp. 18

i

(B.2)

Journal Pre-proof

Proof. By a simple computation, it follows that R R R (f + h)2 dx − ( (f + h)dxi )2 dx∼i R STgx = i (f + h)2 dx − (f0 + h0 )2

235

236

237

R R R R Var(f )Txi + Var(h)Txi + 2 f hdx − 2 ( f dxi )( hdxi )dx∼i = , Var(f ) + Var(h) + 2Cov(f, h) R R R where Var(f )Txi = f 2 (x) dx − ( f (x) dxi )2 dx∼i , and Var(h)Txi is defined

analogously. Then, by applying the Cauchy-Schwarz inequality to the functions R R f (x) − f (x) dxi and h(x) − h(x) dxi , we get Z q  Z  Z Z f hdx − ≤ Var(f )Tx Var(h)Tx . f dx (B.3) hdx dx i i ∼i i i

f

Thus, if Cov(f, h) ≥ 0, by (B.3) we obtain

STfx Var(f ) + SThx Var(h) + 2 i

i

STfx Var(f )SThx Var(h) i

i

Var(f ) + Var(h)

,

from which (B.1) immediately follows. Finally, we show that √ √ ( ay + bz)2 ≤ 2 max{y, z} a+b

rep

238

≤

q

roo

STgx i

(B.4)

lP

for any a, b, y, z > 0. Indeed, without loss of generality, let y > z, and recall √ that 2 ab ≤ a + b: then, √ √ √ √ √ ( ay + bz)2 ( a + b)2 a + b + 2 ab ≤y =y ≤ 2y. a+b a+b a+b

239

rna

Moreover, the factor 2 is sharp: if y = z and a = b, √ √ ( ay + bz)2 4a =y = 2y = 2 max{y, z}. a+b 2a Therefore, inequality (B.4) shows that (B.1) implies (B.2). The bound given by (B.1) means that the total sensitivity index STgx for the i

Jo u

function g of the input xi is controlled by STfx and SThx . It is clear that this i

i

result can be applied also to a function g of the form g(x) = f (x) + h1 (x) + h2 (x) + · · · + hk (x), 19

Journal Pre-proof

for any k ≥ 1. Indeed, it is enough to proceed by iteration: at first, we let h(x) = h1 (x) + h2 (x) + · · · + hk (x), then (B.1) is applied to SThx , by seeing h as i

˜ 2 (x), h(x) = h1 (x) + h where ˜ 2 (x) = h2 (x) + · · · + hk (x). h By applying this procedure k times, the desired result is obtained. However, since the factor 2 in (B.2) is sharp, in general we cannot hope to obtain a better

f

control than

roo

STgx ≤ 2k max{STfx , SThx1 , SThx2 , . . . , SThxk }, where the factor 2 is again sharp.

241

Appendix C. Case 5

243

i

i

rep

240

242

i

i

i

k

A variant of Case 3 (Theorem A1), G(u, v1 , v2 ) = uv1 + v2 is considered. This time, we assume dependence of h2 also on the input xi . Theorem C1. Let g : (0, 1)n+m+k → R be a function in L2 ((0, 1)n+m+k ) such

lP

that

g(x, ξ, η) = f (x)h1 (ξ) + h2 (x, η)

245

for some f ∈ L2 ((0, 1)n ), h1 ∈ L2 ((0, 1)m ) and h2 ∈ L2 ((0, 1)n+k ). Then, if

E(h1 )Cov(f, h2 ) ≥ 0,

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244

q R STfx ( h21 (ξ) dξ)Var(f ) + SThx2 Var(h2 ) + 2(h1 )0 STfx Var(f )SThx2 Var(h2 ) i i i i R ≤ . ( h21 (ξ) dξ)Var(f ) + f02 Var(h1 ) + Var(h2 ) (C.1)

Jo u

STgx i

20

Journal Pre-proof

Proof. It is enough to evaluate STgx . We have i

Z Z

Z

g 2 (x, ξ, η) dx dξ dη − g(x, ξ, η) dxi Z Z = (f (x)h1 (ξ) + h2 (x, η))2 dx dξ dη+

2

dx∼i dξ dη

2 dx∼i dξ dη (f (x)h1 (ξ) + h2 (x, η)) dxi Z  h21 (ξ) dξ Var(f )Txi + Var(h2 )Txi + = Z  Z   Z Z + 2(h1 )0 f (x)h2 (x, η) dx dη − 2 f (x) dxi h2 (x, η) dxi dx∼i dη Z  q ≤ h21 (ξ) dξ Var(f )Txi + Var(h2 )Txi + 2(h1 )0 Var(f )Txi Var(h2 )Txi , Z Z Z

f

−

roo

by (B.3). On the other hand, we get Z Z Z Z g 2 (x, ξ, η) dx dξ dη − g02 = (f (x)h1 (ξ) + h(x, η))2 dx dξ dη − (f h1 + h2 )20 Z  = h21 dξ Var(f ) + f02 Var(h1 ) + Var(h2 )+

rep

+ 2(h1 )0 Cov(f, h2 ). Z  ≥ h21 dξ Var(f ) + f02 Var(h1 ) + Var(h2 ),

since (h1 )0 Cov(f, h2 ) ≥ 0. Then, it follows that

247

If h1 ≡ 1 and k = 0, there is no dependence on ξ and η, and Theorem C1

implies Theorem B1 for the functions f and h2 .

Jo u

248

which is (C.1).

rna

246

i

lP

STgx

q
h21 (ξ) dξ Var(f )Txi + Var(h2 )Txi + 2(h1 )0 Var(f )Txi Var(h2 )Txi R 2
≤ h1 dξ Var(f ) + f02 Var(h1 ) + Var(h2 ) q R STfx ( h21 dξ)Var(f ) + SThx2 Var(h2 ) + 2(h1 )0 STfx Var(f )SThx2 Var(h2 ) i i i i R = , ( h21 dξ)Var(f ) + f02 Var(h1 ) + Var(h2 ) R

21

Journal Pre-proof

249

Appendix D. An estimate on a general class of model functions Let G : R2 → R be such that there exist L ≥ c > 0 satisfying |G(u, v) − G(u0 , v)| ≤ L|u − u0 |, p |G(u, v)| ≥ c u2 + v 2

(D.1) (D.2)

250

for any u, u0 , v ∈ R, which means that G is Lipschitz in u, uniformly in v, and

251

that it is a coercive function.

252

Theorem D1. Let g : (0, 1)n+m → R be a function in L2 ((0, 1)n+m ) such that

253

g0 = 0 and

255

for some functions f : (0, 1)n → R and h : (0, 1)n+m−1 → R satisfying f ∈

roo

254

(D.3)

f

g(x, ξ) = G(f (x), h(x∼i , ξ)) L2 ((0, 1)n ) and h ∈ L2 ((0, 1)n+m−1 ). Then, STgx ≤ 2 i

L2 Var(f ) Sf . c2 Var(f ) + Var(h) Txi

(D.4)

(D.2) implies Var(g) =

Z

rep

Proof. By (D.1) and (D.3), it follows that g(x, ξ) ∈ L2 ((0, 1)n+m ). Since g0 = 0,

g 2 (x, ξ) dx dξ

(0,1)n+m

2

2

f (x) dx +

(0,1)n

lP

≥c

Z

Z

(0,1)n+m−1

2

h (x∼i , ξ) dx∼i dξ

!

≥ c2 (Var(f ) + Var(h)) .

Jo u

rna

We further notice that, by Jensen inequality combined with (D.1) and (D.3),

22

Journal Pre-proof

we get Z

(0,1)n+m

≤ =

Z



g(xi , x∼i , ξ) −

(0,1)n+m

Z

Z

Z

(0,1)n−1

2

= 2L

Z

258

0

2

dxi dx∼i dξ

Z

1

Z

1

0

0

|f (xi , x∼i ) − f (˜ xi , x∼i )|2 d˜ xi dxi dx∼i

2

f (x) dx −

Z

(0,1)n−1

Z

1

0

f (xi , x∼i ) dxi

2

dx∼i

!

.

f

Therefore, combining these two inequalities, (D.4) is obtained. We notice that we can replace the assumption g0 = 0 in Theorem D1 with

roo

257

g(˜ xi , x∼i , ξ) d˜ xi

|G(f (xi , x∼i ), h(x∼i , ξ)) − G(f (˜ xi , x∼i ), h(x∼i , ξ))|2 d˜ xi dxi dx∼i dξ

(0,1)n

256

1

|g(xi , x∼i , ξ) − g(˜ xi , x∼i , ξ)|2 d˜ xi dxi dx∼i dξ

0

(0,1)n+m+1

≤ L2

1

Z

a weaker one.

Corollary D2. Let g : (0, 1)m+n → R be a function in L2 ((0, 1)m+n ) as in (D.3), with f ∈ L2 ((0, 1)n ), h ∈ L2 ((0, 1)n+m−1 ). Then, if

rep

c2 (Var(f ) + Var(h)) − g02 ≥ 0,

we have

STgx ≤ 2 i

L2 Var(f ) Sf . c2 (Var(f ) + Var(h)) − g02 Txi

Proof. The proof is the same of Theorem D1, one needs just to subtract the

260

term g02 at the denominator.

lP

259

Remark D3. It is not difficult to see that we could restate Theorem D1 and Corollary D2 for a function G : R × Rl → R; that is, allowing v to be a

rna

vector (v1 , v2 , . . . , vl ) in Rl . The Lipschitz condition would not change, while the coercivity condition (D.2) would become q p |G(u, v)| ≥ c u2 + |v|2 = c u2 + v12 + v22 + · · · + vl2 .

Jo u

This would allow to have not only one function h, but a family of l different functions h1 , h2 , . . . , hl , which could be seen as a vector valued function h = (h1 , h2 , . . . , hl ) : (0, 1)n+m−1 → Rl , 23

Journal Pre-proof

satisfying Var(h) = Var(h1 ) + Var(h2 ) + · · · + Var(hl ). 261

262

The next example illustrates that the admissible function G for Theorem D1 and Corollary D2 can be very nonlinear. Example D4. Let G(u, v) = a|u| + b|v| + arctan



 |u| + |v| , 1 + v2

for some a, b > 0. Then, G is Lipschitz in u uniformly in v, since   ∂G(u, v) (1 + v 2 ) = a+ sgn(u), ∂u (1 + v 2 )2 + u2 + v 2 + 2|u||v|

roo

f

which is a bounded function. Thus, G satisfies condition (D.1), with ∂G(u, v) . L = sup ∂u u,v

As for the coercivity condition (D.2), it is easy to see that G(u, v) ≥ 0 and

rep

G(u, v) ≥ min{a, b}(|u| + |v|) ≥ min{a, b}

p u2 + v 2 ,

so that we have c = min{a, b}. It is clear that, since G(u, v) ≥ 0, any g(x, ξ) = G(f (x), h(x∼i , ξ)) cannot satisfy g0 = 0, unless f = h = 0. Hence, in general, we can apply Corollary D2 only if we ensure that

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