# Stochastic elliptic operators defined by non-gaussian random fields
with uncertain spectrum

C. Soize<EMAIL_ADDRESS>Université Gustave Eiffel, MSME UMR
8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France

###### Abstract

This paper present a construction and the analysis of a class of non-Gaussian
positive-definite matrix-valued homogeneous random fields with uncertain
spectral measure for stochastic elliptic operators. Then the stochastic
elliptic boundary value problem in a bounded domain of the 3D-space is
introduced and analyzed for stochastic homogenization.

###### keywords:

Non-Gaussian random field , uncertain spectral measure , stochastic elliptic
boundary value problem , stochastic homogenization , random effective
elasticity tensor

††journal: arXiv

##  1 Introduction

Random fields theory has extensively been developed [1, 2, 3, 4, 5], in
particular in the context of continuum physics [6, 7, 8]. The framework of
this paper is that of the analysis of the stochastic homogenization of a
3D-linear anisotropic elastic random medium. The elasticity field is modeled
by a Non-Gaussian positive-definite fourth-order tensor-valued homogeneous
random field. This paper present an extension of the works [9, 10, 11, 7]
devoted to random field representations for stochastic elliptic boundary value
problems and stochastic homogenization. We propose a novel probabilistic
modeling to take into account uncertainties in the spectral measure of the
elasticity random field and we analyze the stochastic elliptic boundary value
problem (BVP) that has to be solved to perform the stochastic homogenization.

Notations

The following notations are used:
$x$: lower-case Latin or Greek letters are deterministic real variables.
${\bm{x}}$: boldface lower-case Latin, Greek, and calligraphic letters are
deterministic vectors.
$X$: upper-case Latin, Greek, and calligraphic letters are real-valued random
variables.
${\bm{X}}$: boldface upper-case Latin or Greek letters are vector-valued
random variables.
$[x]$: lower-case Latin of Greek letters between brackets are deterministic
matrices.
$[{\bm{X}}]$: boldface upper-case letters between brackets are matrix-valued
random variables.
${\mathcal{C}}_{w},{\mathcal{C}}_{y},{\mathcal{C}}_{z},{\mathcal{C}}_{\xi},{\mathcal{C}}_{\varphi},{\mathcal{C}}_{\mathcal{S}}$:
set of values for
${\bm{w}},[y],{\bm{z}},{\bm{\xi}},{\bm{\varphi}},{\bm{{\mathcal{S}}}}$.
${\bm{{\mathcal{S}}}}$: vector-valued deterministic parameter that controls
spectrum uncertainties.
${\mathbb{C}}$: fourth-order tensor-valued random field.
${\mathbb{N}}$, ${\mathbb{R}}$: set of all the integers $\\{0,1,2,\ldots\\}$,
set of all the real numbers.
${\mathbb{R}}^{n}$: Euclidean vector space on ${\mathbb{R}}$ of dimension $n$.
${\mathbb{M}}_{n,m}$: set of all the $(n\times m)$ real matrices.
${\mathbb{M}}_{n}$: set of all the square $(n\times n)$ real matrices.
${\mathbb{M}}_{n}^{S}$: set of all the symmetric $(n\times n)$ real matrices.
${\mathbb{M}}_{n}^{+}$: set of all the positive-definite symmetric $(n\times
n)$ real matrices.
$[I_{n}]$: identity matrix in ${\mathbb{M}}_{n}$.
${\bm{x}}=(x_{1},\ldots,x_{n})$: point in ${\mathbb{R}}^{n}$.
$\langle{\bm{x}},{\bm{y}}\rangle_{2}=x_{1}y_{1}+\ldots+x_{n}y_{n}$: inner
product in ${\mathbb{R}}^{n}$.
$\|{\bm{x}}\|_{2}$: norm in ${\mathbb{R}}^{n}$ such that
$\|{\bm{x}}\|^{2}=\langle{\bm{x}},{\bm{x}}\rangle_{2}$.
$\|\,[a]\,\|_{2}\,=\,\sup_{\bm{x}}\\{\|[a]{\bm{x}}\|_{2}/\|{\bm{x}}\|_{2}\\}$
for $[a]\in{\mathbb{M}}_{n}$ and ${\bm{x}}\in{\mathbb{R}}^{n}$.
$[x]^{T}$: transpose of matrix $[x]$.
${\hbox{{{t}r}}}\\{[x]\\}$: trace of the square matrix $[x]$.
$\langle[x],[y]\rangle_{F}={\hbox{{{t}r}}}\\{[x]^{T}\,[y]\\}$, inner product
of matrices $[x]$ and $[y]$ in ${\mathbb{M}}_{n,m}$.
$\|\,[x]\,\|_{F}$: Frobenius norm of matrix $[x]$ such that
$\|[x]\|^{2}_{F}=\langle[x],[x]\rangle_{F}$.
${\mathonebb{1}}_{B}$: indicatrix function of set $B$.
$\iota$: imaginary unit.
$\delta_{kk^{\prime}}$: Kronecker’s symbol.
$\delta_{{\bm{x}}_{0}}$: Dirac measure at point ${\bm{x}}_{0}$.
$a.s$: almost surely.
$E$: mathematical expectation.

##  2 Non-Gaussian random field with uncertain spectral measure

Physical framework of the considered random fields class. For stochastic
homogenization of linear elastic heterogeneous media presented in Section 5,
the physical space ${\mathbb{R}}^{3}$ is referred to a Cartesian reference
system for which the generic point is ${\bm{x}}=(x_{1},x_{2},x_{3})$.
Nevertheless, all the developments presented in Sections 2 to 4, can easily be
adapted to any finite dimension greater or equal to $1$. We consider a linear
elastic heterogeneous medium for which the elasticity field is a non-Gaussian
fourth-order tensor-valued random field
$\widetilde{\mathbb{C}}=\\{\widetilde{\mathbb{C}}_{ijpq}\\}_{ijpq}$ with $i$,
$j$, $p$, and $q$ in $\\{1,2,3\\}$. A general probabilistic construction has
been proposed in [12, 13, 7] in order to take into account the material
symmetry in a given symmetry class for the mean value of the elasticity random
field and considering the statistical fluctuations either in the same symmetry
class, or in another symmetry class, or in a mixture of two symmetry classes.
In this paper, we start the construction of the random field with the initial
formulation proposed in [9]. It is thus assumed that the mean value of the
elasticity random field $\underline{\widetilde{\mathbb{C}}}$ is independent of
${\bm{x}}$ and belongs to any symmetry class. The statistical fluctuations of
$\widetilde{\mathbb{C}}$ around $\underline{\widetilde{\mathbb{C}}}$ are
assumed to be anisotropic and statistically homogeneous in ${\mathbb{R}}^{3}$
(it should be noted that the developments presented could be extended to a
more general case of material symmetry for the statistical fluctuations but
would greatly complicate the presentation). An important quantity that
controls the statistical fluctuations is the spectral measure that allows the
spatial correlation structure to be described (see for instance [14, 15, 4,
16, 17]) and that we will assumed to be uncertain in this paper.

Principle of construction of the uncertain spectral measure. The uncertainties
are modeled using the probability theory. A parameterization of the spectral
measure, involving a parameter ${\bm{{\mathcal{S}}}}$, is introduced. The
uncertain spectral measure is obtained by modeling ${\bm{{\mathcal{S}}}}$ by a
random variable ${\bm{S}}$. We then construct a non-Gaussian positive-definite
fourth-order tensor-valued homogeneous random field
${\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})$ parameterized with
${\bm{{\mathcal{S}}}}$ such that
$\widetilde{\mathbb{C}}={\mathbb{C}}(\cdot;{\bm{S}})$. Throughout the paper
the quantities surmounted by a tilde correspond to the case of uncertain
spectral measure modeled by a random spectral measure.

Non-Gaussian positive-definite matrix-valued homogeneous random fields with
uncertain spectral measure. For all ${\bm{x}}$ fixed in ${\mathbb{R}}^{3}$,
the fourth-order random tensor ${\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})$
verifies the usual symmetry and positiveness properties. Let
$\textbf{i}=(i,j)$ with $1\leq i\leq j\leq 3$ and $\textbf{j}=(p,q)$ with
$1\leq p\leq q\leq 3$ be the indices with values in $\\{1,\ldots,6\\}$, which
allow for defining the ${\mathbb{M}}^{+}_{6}$-valued random matrix
$[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ such that
$[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{\textbf{i}\textbf{j}}={\mathbb{C}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}})$
(use of the representation in Voigt notation for the constitutive equation).

Random effective elasticity matrix. For fixed ${\bm{{\mathcal{S}}}}$, the
parameterized effective elasticity matrix $[{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})]$ is a random matrix in ${\mathbb{M}}^{+}_{6}$,
which is obtained by stochastic homogenization solving a stochastic elliptic
BVP on a bounded domain $\Omega$ of ${\mathbb{R}}^{3}$. The random effective
elasticity matrix $[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}]$,
corresponding to the elasticity random field with uncertain spectral measure,
is then given by $[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]=[{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}({\bm{S}})]$.

######  Definition 1 (Non-Gaussian homogeneous random field
$\hbox{[}{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}$ given
${\bm{{\mathcal{S}}}}$)

Let $[\underline{{\mathbb{C}}}]$ be a given matrix in ${\mathbb{M}}^{+}_{6}$
independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$. We define
$\\{[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})],{\bm{x}}\in{\mathbb{R}}^{3}\\}$
as a non-Gaussian ${\mathbb{M}}^{+}_{6}$-valued second-order random field, on
a probability space $(\Theta,{\mathcal{T}},{\mathcal{P}})$, indexed by
${\mathbb{R}}^{3}$, homogeneous, mean-square continuous, whose mean value is
$[\underline{{\mathbb{C}}}]=E\\{[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}$
that is therefore independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$. We
have

${\hbox{{{t}r}}}\,[\underline{{\mathbb{C}}}]=\underline{c}_{1}\quad,\quad\langle[\underline{{\mathbb{C}}}]{\bm{\omega}},{\bm{\omega}}\rangle_{2}\,\,\,\geq\,\,\underline{c}_{0}\,\|{\bm{\omega}}\|_{2}^{2}\,\,,\,\,\forall\,{\bm{\omega}}\in{\mathbb{R}}^{6}\,,$
(2.1)

in which $\underline{c}_{0}$ and $\underline{c}_{1}$ are two positive finite
constants.

With the construction proposed in this paper,
$E\\{[\widetilde{\mathbb{C}}({\bm{x}})]\\}=E\\{[{\mathbb{C}}({\bm{x}};{\bm{S}})]\\}$
will not be equal to $[\underline{{\mathbb{C}}}]$ (that is not a difficulty).
However, we will see that
$E\\{[\widetilde{\mathbb{C}}({\bm{x}})]\\}\simeq[\underline{{\mathbb{C}}}]$.

######  Lemma 1 (Normalization of random field
$\hbox{[}{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}$ given
${\bm{{\mathcal{S}}}}$)

Let $[\underline{{\mathbb{L}}}]$ be the upper triangular $(6\times 6)$ real
matrix such that
$[\underline{{\mathbb{C}}}]=[\underline{{\mathbb{L}}}]^{T}\,[\underline{{\mathbb{L}}}]$.
For fixed ${\bm{{\mathcal{S}}}}$, the normalized representation of
$[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ is written as,

$[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]=\frac{1}{1+\epsilon}\,[\underline{{\mathbb{L}}}]^{T}(\epsilon\,[I_{6}]+[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})])[\underline{{\mathbb{L}}}]\,,$
(2.2)

in which $\epsilon>0$ is given and where
$\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})],{\bm{x}}\in{\mathbb{R}}^{3}\\}$
is a ${\mathbb{M}}^{+}_{6}$-valued random field (by construction), defined on
$(\Theta,{\mathcal{T}},{\mathcal{P}})$, indexed by ${\mathbb{R}}^{3}$. Then
$[{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is homogeneous, mean-square
continuous, and such that

$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}=[I_{6}]\quad,\quad\forall\,{\bm{x}}\in{\mathbb{R}}^{3}\,.$
(2.3)

###### Proof 1

(Lemma 1). Under the hypotheses introduced in Definition 1, it is easy to see
that $[{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is a second-order, homogeneous,
mean-square continuous random field, and satisfies Eq. (2.3).

It should be noted that the lower bound
$\epsilon\,[\underline{{\mathbb{C}}}]/(1+\epsilon)$ used in Eq. (2.2) could be
replaced by a more general lower bound
$[{\mathbb{C}}_{b}]\in{\mathbb{M}}_{6}^{+}$ as proposed in [7, 11]. Note also
that, as previously, introducing
$[\widetilde{\bm{C}}({\bm{x}})\\}=[{\bm{C}}({\bm{x}};{\bm{S}})]$,
$E\\{[\widetilde{\bm{C}}({\bm{x}})]\\}$ will not be equal to $[I_{6}]$ and we
will see that $E\\{[\widetilde{\bm{C}}({\bm{x}})]\\}\simeq[I_{6}]$.

######  Hypothesis 1 (Principle of construction of random field
$\hbox{[}{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}$ given
${\bm{{\mathcal{S}}}}$)

By construction (see Lemma 1), $[{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is a
${\mathbb{M}}^{+}_{6}$-valued random field indexed by ${\mathbb{R}}^{3}$ and
homogeneous. For all ${\bm{x}}$ fixed in ${\mathbb{R}}^{3}$, the
${\mathbb{M}}^{+}_{6}$-valued random variable
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ is constructed by using the
Maximum Entropy Principle under the following available information,

$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}=[I_{6}]\quad,\quad
E\\{\log(\det[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})])\\}=b_{c}\,,$ (2.4)

in which $b_{c}$ is independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$ and
such that $|b_{c}|<+\infty$. The second equality is introduced in order that
the random matrix $[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]^{-1}$ (that
exists almost surely) be a second-order random variable:
$E\\{\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]^{-1}\|_{2}^{2}\\}\leq
E\\{\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]^{-1}\|_{F}^{2}\\}<+\infty$.
With such a construction, $[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ will
appear as a nonlinear transformation of $6\times(6+1)/2=21$ independent
normalized Gaussian real-valued random variables
$\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}}),1\leq m\leq n\leq 6\\}$, such that

$E\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0\quad,\quad
E\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}=1\,.$ (2.5)

The spatial correlation structure of random field
$\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})],{\bm{x}}\in{\mathbb{R}}^{3}\\}$
is introduced by considering $21$ independent real-valued random fields
$\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$ for
$1\leq m\leq n\leq 6$, corresponding to $21$ independent copies of a unique
normalized Gaussian homogeneous mean-square continuous real-valued random
field $\\{G({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$
given its normalized spectral measure parameterized by ${\bm{{\mathcal{S}}}}$.
Note that the Gaussian random field $G(\cdot;{\bm{{\mathcal{S}}}})$ is
entirely defined by its normalized spectral measure (parameterized by
${\bm{{\mathcal{S}}}}$) because, $\forall{\bm{x}}\in{\mathbb{R}}^{3}$,
$E\\{G({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0$ and
$E\\{G({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}=1$. The constant $b_{c}$ is
eliminated in favor of a hyperparameter $\delta_{c}$, which allows for
controlling the level of statistical fluctuations of
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$, defined by
$\delta_{c}=(E\\{\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]-[I_{6}]\|_{F}^{2}\\}/6)^{1/2}$,
independent of ${\bm{x}}$ and chosen independent of ${\bm{{\mathcal{S}}}}$.

######  Proposition 1 (Random field
$\hbox{[}{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}$)

Let us assume Hypothesis 1.

(i) Let $d^{S}C=2^{15/2}\Pi_{1\leq m\leq n\leq 6}\,dC_{mn}$ be the volume
element on Euclidean space ${\mathbb{M}}^{S}_{6}$ in which $dC_{mn}$ is the
Lebesgue measure on ${\mathbb{R}}$. For all ${\bm{x}}$ fixed in
${\mathbb{R}}^{3}$, the probability measure
$P_{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]}(d^{S}C)$ of the
${\mathbb{M}}^{+}_{6}$-valued random variable
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ constructed with the Maximum
Entropy Principle under the constraints defined by Eq. (2.4), is independent
of ${\bm{x}}$ (homogeneous random field), independent of
${\bm{{\mathcal{S}}}}$ (this marginal probability measure does not depend of
the correlation structure), and written as
$P_{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]}(d^{S}C)=p_{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]}([C])\,d^{S}C$
in which the probability density function is written as
$p_{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]}([C])={\mathonebb{1}}_{{\mathbb{M}}^{+}_{6}}([C])\,c_{c}(\det[C])^{7(1-\delta_{c}^{2})/(2\delta_{c}^{2})}\exp(-7{\hbox{{{t}r}}}\\{[C]\\}/(2\delta_{c}^{2}))$
with $c_{c}$ the normalization constant and where hyperparameter $\delta_{c}$
must belong to the real interval $]0\,,\sqrt{7/11}[$.

(ii) For all ${\bm{x}}$ fixed in ${\mathbb{R}}^{3}$, random matrix
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ is written as

$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]=[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]^{T}\,[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,,$
(2.6)

in which $[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ is an upper triangular
random matrix in ${\mathbb{M}}_{6}$ such that

1) the $21$ random variables
$\\{[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn},1\leq m\leq n\leq 6\\}$ are
mutually independent.

2) for $1\leq m<n\leq 6$,
$[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}=\sigma_{c}\,G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})$
with $\sigma_{c}=\delta_{c}/\sqrt{7}$ and where
$G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})$ is a normalized Gaussian real-valued
random variable (see Eq. (2.5)).

3) for $1\leq m=n\leq 6$,
$[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mm}=\sigma_{c}\,\sqrt{2\,h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})}$
with $\alpha_{m}=1/(2\sigma_{c}^{2})+(1-m)/2$ such that
$\alpha_{1}>\ldots>\alpha_{6}>3$ and where
$G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}})$ is a normalized Gaussian real-valued
random variable (see Eq. (2.5)). The function $b\mapsto h(b;\alpha)$ is such
that ${\mathbb{\Gamma}}_{\alpha}=h({\mathcal{N}};\alpha)$ is a Gamma random
variable with parameter $\alpha$ when ${\mathcal{N}}$ is the normalized
Gaussian real-valued random variable.

(iii) The $21$ random fields
$\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$ for
$1\leq m\leq n\leq 6$ are $21$ independent copies of a normalized Gaussian
homogeneous mean-square continuous real-valued random field
$\\{G({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$,

$E\\{G({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0\quad,\quad
E\\{G({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}=1\quad,\quad\forall{\bm{x}}\in{\mathbb{R}}^{3}\,,$
(2.7)

and which will be defined in Section 3 for imposing its spatial correlation
structure via its spectral measure. parameterized by ${\bm{{\mathcal{S}}}}$.

###### Proof 2

(Proposition 1). We refer the reader to [18] for the construction using the
Maximum Entropy Principle and to [9, 11, 7] for the representation defined by
Eq. (2.6). However, we have to prove the properties that yield
$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}=[I_{6}]$, this proof being
used in Remark 1. For $1\leq m\leq n\leq 6$,
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}=\sum_{\ell=1}^{m}[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{\ell
m}\,[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{\ell n}$. For $m=n$,
$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mm}=2\sigma_{c}^{2}\,E\\{h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})\\}+\sigma_{c}^{2}\sum_{\ell<m}E\\{G_{\ell
m}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}$. Eq. (2.5) yields
$\sum_{\ell<m}E\\{G_{\ell m}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}$
$=\sum_{\ell<m}1=m-1$ and
$E\\{h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})\\}=E\\{{\mathbb{\Gamma}}_{\alpha_{m}}\\}=\alpha_{m}=1/(2\sigma_{c}^{2})+(1-m)/2$.
Therefore, $E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mm}\\}=1$. For
$1\leq m<n\leq 6$, we have
$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}=\sigma_{c}^{2}\,E\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})$
$\sqrt{2\,h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})}\\}+\sigma_{c}^{2}\sum_{\ell<m}E\\{G_{\ell
m}({\bm{x}};{\bm{{\mathcal{S}}}})\,$ $G_{\ell
n}({\bm{x}};{\bm{{\mathcal{S}}}})\\}$. Proposition 1-(iii) and Eq. (2.5) yield
$E\\{G_{\ell m}({\bm{x}};{\bm{{\mathcal{S}}}})$ $\,G_{\ell
n}({\bm{x}};{\bm{{\mathcal{S}}}})\\}\\!=\\!E\\{G_{\ell
m}({\bm{x}};{\bm{{\mathcal{S}}}})\\}\times E\\{G_{\ell
n}({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0$ and
$E\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})\sqrt{2\,h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})}\\}$
$=E\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})\\}\times
E\\{\sqrt{2\,h(G_{mm}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})}\\}\\!=\\!0$.
Therefore, $E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}\\}=0$. We thus
obtain the first equation in Eq. (2.4).

######  Lemma 2 (Properties of function $h$)

(i) Let be $\alpha>3$. Function $b\mapsto
h(b;\alpha):{\mathbb{R}}\mapsto]0\,,+\infty[$ defined in Proposition 1 is
written as $h(b;\alpha)=F_{\alpha}^{-1}(F(b))$ in which
$F(b)=\int_{-\infty}^{b}(2\pi)^{-1/2}e^{-t^{2}/2}\,dt$ and where
$F_{\alpha}^{-1}$ is the reciprocical function of
$F_{\alpha}({\mathbb{\gamma}})={\mathbb{\Gamma}}(\alpha)^{-1}\times$
$\int_{0}^{\mathbb{\gamma}}t^{\alpha-1}e^{-t}\,dt$ for ${\mathbb{\gamma}}\geq
0$ with ${\mathbb{\Gamma}}(\alpha)=\int_{0}^{+\infty}t^{\alpha-1}e^{-t}\,dt$.
For all $\alpha>3$ and for all $b\in{\mathbb{R}}$, $h(b;\alpha)\leq
2\alpha+b^{2}$.

(ii) If ${\mathcal{N}}$ is Gaussian with $E\\{{\mathcal{N}}\\}=0$ and
$E\\{{\mathcal{N}}^{2}\\}=1$, then $E\\{h({\mathcal{N}};\alpha)\\}=\alpha$.

(iii) If ${\mathcal{G}}$ is non-Gaussian with $E\\{{\mathcal{G}}\\}=0$ and
$E\\{{\mathcal{G}}^{2}\\}=1$, we have
$E\\{h({\mathcal{G}};\alpha)\\}\not=\alpha$, but for
$\alpha\rightarrow+\infty$, $E\\{h({\mathcal{G}};\alpha)\\}\rightarrow\alpha$.

###### Proof 3

(Lemma 2). (i) Function $h$ defined in Proposition 1 shows that
$F_{\alpha}(h(b;\alpha))=F(b)$ in which $F(b)$ is the c.d.f of ${\mathcal{N}}$
and $F_{\alpha}({\mathbb{\gamma}})$ is the c.d.f of the
${\mathbb{\Gamma}}_{\alpha}$ random variable such that
$E\\{{\mathbb{\Gamma}}_{\alpha}\\}=\alpha$. We then deduce that
$h(b;\alpha)=F_{\alpha}^{-1}(F(b))$. In order to prove that $h(b;\alpha)\leq
2\alpha+b^{2}$, since $u\mapsto F_{\alpha}^{-1}(u)$ is a strictly increasing
function from $[0,1[$ into ${\mathbb{R}}^{+}$, we have to prove that, for all
$\alpha>3$ and for all $b\in{\mathbb{R}}$, we have $J_{\alpha}(b)\geq 0$ in
which

$J_{\alpha}(b)=F_{\alpha}(2\alpha+b^{2})-F(b)=\frac{1}{{\mathbb{\Gamma}}(\alpha)}\int_{0}^{2\alpha+b^{2}}t^{\alpha-1}e^{-t}\,dt-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{b}e^{-t^{2}/2}\,dt\,,$
(2.8)

which can be rewritten as

$J_{\alpha}(b)=P_{\alpha}+\frac{1}{{\mathbb{\Gamma}}(\alpha)}\int_{2\alpha}^{2\alpha+b^{2}}t^{\alpha-1}e^{-t}\,dt-\frac{\hbox{{\Ppcarac
sgn}}\,(b)}{\sqrt{2\pi}}\int_{0}^{|b|}e^{-t^{2}/2}\,dt\,,$ (2.9)

in which $P_{\alpha}=F_{\alpha}(2\alpha)-1/2$ that is such that,
$\forall\alpha>3$, $P_{\alpha}>F_{3}(6)-1/2\simeq 0.438$, and sgn is the sign
function. Equation (2.9) shows that $\forall\alpha>3$ and $\forall\,b<0$,
$J_{\alpha}(b)>0$ because $J_{\alpha}(b)$ is the sum of three positive terms.
For $b=0$ and $\forall\alpha>3$, Eq . (2.9) shows that
$J_{\alpha}(0)=P_{\alpha}>0$. For $b>0$, we use Eq . (2.8). Let
$J^{\prime}_{\alpha}(b)=dJ_{\alpha}(b)/db$ be such that
$J^{\prime}_{\alpha}(b)=(2\pi)^{-1/2}e^{-b^{2}/2}(a_{\alpha}b\,(1+{b^{2}}/{(2\alpha)})^{\alpha-1}e^{-b^{2}/2}-1)$,
in which
$a_{\alpha}=2\sqrt{2\pi}e^{-2\alpha}(2\alpha)^{\alpha-1}/{\mathbb{\Gamma}}(\alpha)$
that is such that, $\forall\alpha>3$, $a_{\alpha}\leq 0.223679$. From Eq.
(2.8), it can be seen that, $\forall\alpha>3$, for $b\rightarrow+\infty$,
$J_{\alpha}(b)\rightarrow 0$ and
$J^{\prime}_{\alpha}(b)\sim-e^{-b^{2}/2}/\sqrt{2\pi}$. Consequently,
$J_{\alpha}\rightarrow 0_{+}$. Since $J_{\alpha}(0)>0$, we will have
$J_{\alpha}(b)>0$ for $b>0$ if $J^{\prime}_{\alpha}(b)<0$, that is to say if
$a_{\alpha}b\,(1+\frac{b^{2}}{2\alpha})^{\alpha-1}<e^{b^{2}/2}$, which is true
for $\alpha>3$ and $b>0$. (ii) If ${\mathcal{N}}$ is normalized and Gaussian,
then $h({\mathcal{N}};\alpha)={\mathbb{\Gamma}}_{\alpha}$ and hence
$E\\{h({\mathcal{N}};\alpha)\\}=\alpha$. (iii) Let $P(x^{2}|\widehat{\alpha})$
be defined by
$P(x^{2}|\widehat{\alpha})=(2^{\widehat{\alpha}/2}{\mathbb{\Gamma}}(\widehat{\alpha}/2))^{-1}\int_{0}^{x^{2}}v^{(\widehat{\alpha}/2)-1}e^{-v/2}\,dv$.
The change of variable $v=2t$ yields
$P(x^{2}|\widehat{\alpha})=({\mathbb{\Gamma}}(\widehat{\alpha}/2))^{-1}\int_{0}^{x^{2}/2}t^{(\widehat{\alpha}/2)-1}e^{-t}\,dt$.
Taking $\widehat{\alpha}=2\alpha$ yields
$P(x^{2}|\widehat{\alpha})=F_{\alpha}(x^{2}/2)$. For
$\widehat{\alpha}\rightarrow+\infty$, $P(x^{2}|\widehat{\alpha})\sim F(y)$
with $y=(x^{2}-\widehat{\alpha})/\sqrt{2\widehat{\alpha}}$ and hence
$F_{\alpha}(x^{2}/2)\sim F(y)$. Since $F_{\alpha}(h(b;\alpha))=F(b)$, taking
$x^{2}/2=h(b;\alpha)$ yields $F(b)\sim F(y)$ that is to say $b\sim
y=(2h(b;\alpha)-2\alpha)/\sqrt{4\alpha}$, which shows that
$h(b;\alpha)\sim\alpha+b\sqrt{\alpha}$. If ${\mathcal{G}}$ is a non-Gaussian
random variable such that $E\\{{\mathcal{G}}\\}=0$, then
$E\\{h({\mathcal{G}};\alpha)\\}\sim\alpha+\sqrt{\alpha}\,E\\{{\mathcal{G}}\\}=\alpha$.

###### Remark 1

The analysis of the proof of Proposition 1 shows that the property
$E\\{[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}=[I_{6}]$ holds for a fixed
value of ${\bm{{\mathcal{S}}}}$. When ${\bm{{\mathcal{S}}}}$ will be modeled
by a random variable ${\bm{S}}$ in order to take into account uncertainties in
the spectral measure (see Section 4 and as we have previously explained at the
beginning of Section 2) the random field
$\\{[\widetilde{\bm{C}}({\bm{x}})],{\bm{x}}\in{\mathbb{R}}^{3}\\}$ such that
$[\widetilde{\bm{C}}({\bm{x}})]=[{\bm{C}}({\bm{x}};{\bm{S}})]$ will then
depend on copies of the random field
$\\{G({\bm{x}};{\bm{S}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$ that will always
satisfy $E\\{G({\bm{x}};{\bm{S}})\\}=0$ and
$E\\{G({\bm{x}};{\bm{S}})^{2}\\}=1$, but which will no longer be Gaussian. By
examining the proof of Proposition 1, it can be seen that we will always have
$E\\{[\widetilde{\bm{C}}({\bm{x}})]_{mn}\\}=0$ for $m\not=n$ but that
$E\\{[\widetilde{\bm{C}}({\bm{x}})]_{mm}\\}=2\,\sigma_{c}^{2}\,E\\{h(G({\bm{x}};{\bm{S}});\alpha_{m})\\}+\sigma_{c}^{2}(m-1)\not=1$.
Nevertheless, from Proposition 1, since $\alpha_{m}>3$ and from Lemma 2-(iii),
taking ${\mathcal{G}}=G({\bm{x}};{\bm{S}})$ yields
$E\\{[\widetilde{\bm{C}}({\bm{x}})]_{mm}\\}\simeq 1$. It has numerically been
verified that, if ${\mathcal{G}}$ is a uniform random variable (that will not
be the case for $G({\bm{x}};{\bm{S}})$) such that $E\\{{\mathcal{G}}\\}=0$ and
$E\\{{\mathcal{G}}^{2}\\}=1$ (that will be the case for
$G({\bm{x}};{\bm{S}})$), then we have
$E\\{h({\mathcal{G}};\alpha_{m})\\}\simeq\alpha_{m}$ with an error of $5\times
10^{-4}$ for all $1\leq m\leq 6$.

##  3 Construction and analysis of the Gaussian random field
$G(\cdot;{\bm{{\mathcal{S}}}})$ with uncertain spectral measure parameterized
by ${\bm{{\mathcal{S}}}}$

We start by constructing a normalized Gaussian, homogeneous, second-order,
mean-square continuous random field
$\\{G({\bm{x}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$. This field corresponds to
$G(.;\underline{{\bm{{\mathcal{S}}}}})$ for which its spectral measure is
given and represented by a given value $\underline{{\bm{{\mathcal{S}}}}}$ of
${\bm{{\mathcal{S}}}}$, that is to say,
$G=G(\cdot;\underline{{\bm{{\mathcal{S}}}}})$. Therefore, there exists a
positive bounded spectral measure $m_{G}(d{\bm{k}})$ on ${\mathbb{R}}^{3}$
such that the correlation function $\rho_{G}$ of $G$ is written, for all
${\bm{x}}$ and ${\bm{\zeta}}$ in ${\mathbb{R}}^{3}$, as

$\rho_{G}({\bm{\zeta}})=E\\{G({\bm{x}}+{\bm{\zeta}})\,G({\bm{x}})\\}=\int_{{\mathbb{R}}^{3}}e^{\iota\,{\bm{k}}\cdot{\bm{\zeta}}}m_{G}(d{\bm{k}})=\int_{{\mathbb{R}}^{3}}\cos({\bm{k}}\cdot{\bm{\zeta}})\,m_{G}(d{\bm{k}})\,,$
(3.1)

in which $\iota=\sqrt{-1}$,
${\bm{k}}\cdot{\bm{\zeta}}=\sum_{j=1}^{3}k_{j}\zeta_{j}$,
$d{\bm{k}}=dk_{1}dk_{2}dk_{3}$. In addition, it is assumed that
$m_{G}(d{\bm{k}})=s({\bm{k}})\,d{\bm{k}}$ admits a spectral density function
(s.d.f) ${\bm{k}}\mapsto
s({\bm{k}}):{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{+}$. Equations (2.7) and
(3.1) yield $\rho_{G}(0)=E\\{G({\bm{x}})^{2}\\}=1$, and consequently,

$\int_{{\mathbb{R}}^{3}}s({\bm{k}})\,d{\bm{k}}=1\,.$ (3.2)

In this section, we begin with the analysis of the spectral measure and the
modeling of uncertainties. Then we introduce a finite representation of
$G(\cdot;{\bm{{\mathcal{S}}}})$ and we study its properties. Note that a
dimensionless s.d.f, $\chi$, of $s$ will be introduced.

######  Hypothesis 2 (Spectral density function $s$ and spatial correlation
length of $G$)

It is assumed that $s$ has a compact support
$\overline{\mathbb{K}}=\partial{\mathbb{K}}\cup{\mathbb{K}}$ with
${\mathbb{K}}=\Pi_{j=1}^{3}\,]-K_{j},K_{j}[$ in which
$K_{j}\in[K_{j}^{\hbox{{\ppppcarac min}}},K_{j}^{\hbox{{\ppppcarac max}}}]$
with $0<K_{j}^{\hbox{{\ppppcarac min}}}<K_{j}^{\hbox{{\ppppcarac
max}}}<+\infty$. It is assumed that $s$ is a continuous function on
${\mathbb{R}}^{3}$. Since supp $s=\overline{\mathbb{K}}$, we must have
$s({\bm{k}})=0$, $\forall{\bm{k}}\in\partial{\mathbb{K}}$, and thus

$\int_{{\mathbb{R}}^{3}}s({\bm{k}})\,d{\bm{k}}=\int_{\overline{\mathbb{K}}}s({\bm{k}})\,d{\bm{k}}=1\,.$
(3.3)

Since $G$ is real, we have $s(-{\bm{k}})=s({\bm{k}})$ for all
${\bm{k}}\in{\mathbb{R}}^{3}$. In addition to this symmetry property, we
assume that $s$ satisfies the following quadrant symmetry [4]: defining
${\bm{k}}_{\\{-j\\}}$ as vector ${\bm{k}}$ for which its component $k_{j}$ is
replaced by $-k_{j}$, then $s({\bm{k}}_{\\{-j\\}})=s({\bm{k}})$ for $j=1,2,3$
and $\forall{\bm{k}}\in{\mathbb{R}}^{3}$. The spatial correlation length for
coordinate $\zeta_{j}$ is defined by

$L_{cj}=\int_{0}^{+\infty}|\rho_{j}(0,\ldots,\zeta_{j},\ldots,0)|\,d\zeta_{j}\,,$
(3.4)

and is assumed to be finite.

######  Definition 2 (Spectral domain sampling)

Let $\nu_{s}$ be a given even integer. For $j\in\\{1,2,3\\}$, we define
$\Delta_{j}=2\,K_{j}/\nu_{s}$ as the sampling step of interval
$[-K_{j},K_{j}]$ and $k_{j\beta_{j}}=-K_{j}+(\beta_{j}-1/2)\,\Delta_{j}$ for
$\beta_{j}=1,\ldots,\nu_{s}$ as its spectral sampling points. Let
${\mathcal{B}}=\\{{\bm{\beta}}=(\beta_{1},\beta_{2},\beta_{3})\,,\beta_{j}=1,\ldots,\nu_{s}\\}$
be the finite subset of ${\mathbb{N}}^{3}$. We define $\Delta$, $K$, $\nu$,
and ${\bm{k}}_{\bm{\beta}}$ such that:
$\Delta=\Delta_{1}\Delta_{2}\Delta_{3}$, $K=K_{1}K_{2}K_{3}$,
$\nu=(\nu_{s})^{3}$, and for all $\beta\in{\mathcal{B}}$,
${\bm{k}}_{\beta}=(k_{1\beta_{1}},k_{2\beta_{2}},k_{3\beta_{3}})\in{\mathbb{K}}\subset{\mathbb{R}}^{3}$.

######  Lemma 3 (Discretization of the spectral measure and convergence
properties)

Let
$\delta_{{\bm{k}}_{\bm{\beta}}}({\bm{k}})=\otimes_{j=1}^{3}\delta_{k_{j\beta_{j}}}(k_{j})$
be the Dirac measure on ${\mathbb{R}}^{3}$ at sampling point
${\bm{k}}_{\bm{\beta}}\in{\mathbb{K}}\subset{\mathbb{R}}^{3}$ defined in
Definition 2. Let $m_{G}^{\nu}(d{\bm{k}})$ be the positive bounded measure on
${\mathbb{R}}^{3}$ defined by

$m_{G}^{\nu}(d{\bm{k}})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}s_{\bm{\beta}}^{\Delta}\,\delta_{{\bm{k}}_{\bm{\beta}}}({\bm{k}})\quad,\quad
s_{\bm{\beta}}^{\Delta}=\Delta\,s({\bm{k}}_{\bm{\beta}})\,,$ (3.5)

which is such that
$m_{G}^{\nu}({\mathbb{R}}^{3})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}s_{\bm{\beta}}^{\Delta}=\eta_{\nu}$
with $\eta_{\nu}>0$. The sequence of measures
$\\{m_{G}^{\nu}(d{\bm{k}})\\}_{\nu}$ converges narrowly towards the measure
$m_{G}(d{\bm{k}})$ and the positive sequence $\\{\eta_{\nu}\\}_{\nu}$
converges towards $1$.

###### Proof 4

(Lemma 3). We have to prove that $\forall f\in C^{0}(\overline{\mathbb{K}})$,
the sequence $m_{G}^{\nu}(f)=\int_{\overline{\mathbb{K}}}f({\bm{k}})\,$
$m_{G}^{\nu}(d{\bm{k}})=\Delta\sum_{{\bm{\beta}}\in{\mathcal{B}}}f({\bm{k}}_{\bm{\beta}})\,s({\bm{k}}_{\bm{\beta}})$
converges towards
$m_{G}(f)=\int_{\overline{\mathbb{K}}}f({\bm{k}})\,m_{G}(d{\bm{k}})=\int_{\overline{\mathbb{K}}}f({\bm{k}})$
$s({\bm{k}})\,d{\bm{k}}$. Since the function ${\bm{k}}\mapsto
f({\bm{k}})\,s({\bm{k}})$ is continuous on $\overline{\mathbb{K}}$, it is
known that for $\nu_{s}\rightarrow+\infty$ (that is to say for
$\nu\rightarrow+\infty$),
$\Delta\sum_{{\bm{\beta}}\in{\mathcal{B}}}f({\bm{k}}_{\bm{\beta}})\,s({\bm{k}}_{\bm{\beta}})\rightarrow\int_{\overline{\mathbb{K}}}f({\bm{k}})\,s({\bm{k}})\,d{\bm{k}}$.
Taking $f({\bm{k}})=1$ for all ${\bm{k}}\in\overline{\mathbb{K}}$ yields
$m_{G}(1)=\int_{\overline{\mathbb{K}}}s({\bm{k}})\,d{\bm{k}}=1$ and
$m_{G}^{\nu}(1)=\Delta\sum_{{\bm{\beta}}\in{\mathcal{B}}}s({\bm{k}}_{\bm{\beta}})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}s_{\bm{\beta}}^{\Delta}=\eta_{\nu}$.
Therefore, $\\{\eta_{\nu}\\}_{\nu}$ converges towards $1$.

######  Hypothesis 3 (Choice of $\nu=(\nu_{s})^{3}$)

Let $s$ be the s.d.f satisfying Hypothesis 2. Let us consider the spectral
domain sampling introduced in Definition 2. Using Lemma 3, we will assume that
$\nu$ is chosen sufficiently large in order that
$|\sum_{{\bm{\beta}}\in{\mathcal{B}}}s_{\bm{\beta}}^{\Delta}-1|\leq\epsilon_{s}\ll
1$ and consequently, we will write
$\sum_{{\bm{\beta}}\in{\mathcal{B}}}s_{\bm{\beta}}^{\Delta}\simeq 1$.

######  Definition 3 (Dimensionless spectral density function)

The spectral density function ${\bm{k}}\mapsto s({\bm{k}})$, which verifies
Hypothesis 2, is written for all ${\bm{k}}$ in ${\mathbb{R}}^{3}$ as
$s(k_{1},k_{2},k_{3})=(K_{1}K_{2}K_{3})^{-1}\chi(k_{1}/K_{1},k_{2}/K_{2},k_{3}/K_{3})$
in which $\chi$ is a given function
${\bm{\tau}}=(\tau_{1},\tau_{2},\tau_{3})\mapsto\chi(\tau_{1},\tau_{2},\tau_{3}):{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{+}$
with compact support $[-1,1]^{3}$.

Function $\chi$ has the same properties as $s$:
$\chi(-{\bm{\tau}})=\chi({\bm{\tau}})$, quadrant symmetry, and continuity. For
$j=1,2,3$, the change of variable $\tau_{j}=k_{j}/K_{j}$ yields
$s({\bm{k}})\,d{\bm{k}}=\chi({\bm{\tau}})\ d{\bm{\tau}}$ and thus
$m_{G}(d{\bm{k}})=\mu_{G}(d{\bm{\tau}})$ with
$\mu_{G}(d{\bm{\tau}})=\chi({\bm{\tau}})\,d{\bm{\tau}}$, and consequently, Eq.
(3.3) yields

$\int_{{\mathbb{R}}^{3}}\chi({\bm{\tau}})\,d{\bm{\tau}}=\int_{[-1,1]^{3}}\chi({\bm{\tau}})\,d{\bm{\tau}}=1\,.$
(3.6)

The dimensionless spectral domain sampling is directly deduced from Definition
2,

$\\{{\bm{\tau}}_{\bm{\beta}}=(\tau_{\beta_{1}},\tau_{\beta_{2}},\tau_{\beta_{3}}),{\bm{\beta}}\in{\mathcal{B}}\\}\,\,,\,\,\tau_{\beta_{j}}=-1+(\beta_{j}-\frac{1}{2})\,\frac{2}{\nu_{s}}\,\,,\,\,j\in\\{1,2,3\\}\,.$
(3.7)

The discretization $\mu_{G}^{\nu}(d{\bm{\tau}})$ of $\mu_{G}(d{\bm{\tau}})$,
such that $\mu_{G}^{\nu}(d{\bm{\tau}})=m_{G}^{\nu}(d{\bm{k}})$, is written as

$\mu_{G}^{\nu}(d{\bm{\tau}})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\chi^{\Delta}_{\bm{\beta}}\,\,\delta_{{\bm{\tau}}_{\bm{\beta}}}({\bm{\tau}})\quad,\quad\chi^{\Delta}_{\bm{\beta}}=({2}/{\nu_{s}})^{3}\,\chi({\bm{\tau}}_{\bm{\beta}})\,,$
(3.8)

in which
$\delta_{{\bm{\tau}}_{\bm{\beta}}}=\otimes_{j=1}^{3}\delta_{\tau_{\beta_{j}}}(\tau_{j})$
and where, from Hypothesis 3,

$\sum_{{\bm{\beta}}\in{\mathcal{B}}}\,\chi_{\bm{\beta}}^{\Delta}\simeq 1\,.$
(3.9)

Definition 3 implies that measure $\mu_{G}^{\nu}(d{\bm{\tau}})$ is independent
of $K_{1}$, $K_{2}$, and $K_{3}$. In order to introduce the probability model
of the spectral measure, we start by defining an adapted parameterization
$[y]$ that takes into account quadrant symmetry.

######  Definition 4 (Parameterization of the discretized dimensionless
spectral measure)

Let $\widehat{\nu}_{s}=\nu_{s}/2$ ($\nu_{s}$ is even). Let ${\mathcal{C}}_{y}$
be the subset of ${\mathbb{M}}_{3,\widehat{\nu}_{s}}$ defined by

${\mathcal{C}}_{y}=\\{\,[y]\in{\mathbb{M}}_{3,\widehat{\nu}_{s}}\,\,,\,[y]_{j\widehat{\beta}}\in[0,1]\,\,\,\hbox{for}\,\,\,j=1,2,3\,\,\,\hbox{and}\,\,\,\widehat{\beta}=1,\ldots,\widehat{\nu}_{s}\\}\,.$
(3.10)

Let $[\underline{y}]$ be in ${\mathcal{C}}_{y}$ such that
$[\underline{y}]_{j\widehat{\beta}}=1/2$ for $j=1,2,3$ and
$\widehat{\beta}=1,\ldots,\widehat{\nu}_{s}$.

Let
$\widehat{\mathcal{B}}=\\{\widehat{\bm{\beta}}=(\widehat{\beta}_{1},\widehat{\beta}_{2},\widehat{\beta}_{3}),\widehat{\beta}_{j}=1,\ldots,\widehat{\nu}_{s}\\}\subset{\mathcal{B}}$
be the set of $\widehat{\nu}=(\widehat{\nu}_{s})^{3}=\nu/8$ elements. We
define the finite family of functions $[y]\mapsto
a_{\widehat{\bm{\beta}}}([y]):{\mathcal{C}}_{y}\rightarrow{\mathbb{R}}$ such
that
$a_{\widehat{\bm{\beta}}}([y])=\sqrt{\chi_{\widehat{\bm{\beta}}}^{\Delta}}\,q_{\widehat{\bm{\beta}}}([y];\delta_{s})$,
in which $\delta_{s}>0$ is a hyperparameter that will allow the level of
spectrum uncertainties to be controlled and where $[y]\mapsto
q_{\widehat{\bm{\beta}}}([y];\delta_{s})$ is any given continuous real
function on ${\mathcal{C}}_{y}$ such that
$q_{\widehat{\bm{\beta}}}([\underline{y}];\delta_{s})=1$. For all
$[y]\in{\mathcal{C}}_{y}$, let
$\\{a_{\bm{\beta}}([y]),{\bm{\beta}}\in{\mathcal{B}}\\}$ be the $\nu$ real
numbers that are directly constructed from
$\\{a_{\widehat{\bm{\beta}}}([y]),\widehat{\bm{\beta}}\in\widehat{\mathcal{B}}\\}$
using the quadrant symmetry (see Hypothesis 2); an example of such a
construction is given in Example 1-(iv). For all
${\bm{\beta}}\in{\mathcal{B}}$, we define the function
$[y]\mapsto\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y]):{\mathcal{C}}_{y}\rightarrow{\mathbb{R}}^{+}$
such that

$\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])=a_{\bm{\beta}}([y])^{2}(\sum_{{\bm{\beta}}^{\prime}\in{\mathcal{B}}}a_{{\bm{\beta}}^{\prime}}([y])^{2})^{-1}\,.$
(3.11)

The dimensionless spectral measure
$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])$ for $[y]$ given in
${\mathcal{C}}_{y}$ is then defined by

$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])\,\delta_{{\bm{\tau}}_{\bm{\beta}}}({\bm{\tau}})\,.$
(3.12)

Note that $[y]$ is one parameter of ${\bm{{\mathcal{S}}}}$ and
$[\underline{y}]$ is the corresponding parameter of
$\underline{{\bm{{\mathcal{S}}}}}$.

######  Proposition 2 (Random discretized dimensionless spectral measure)

Let us consider Definitions 3 and 4. It is assumed that Eq. (3.9) holds.

(i) $\forall{\bm{\beta}}\in{\mathcal{B}}$, function
$[y]\mapsto\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])$ is continuous on
${\mathcal{C}}_{y}$ (and thus bounded on ${\mathcal{C}}_{y}$), is such that
$\widetilde{\chi}_{\bm{\beta}}^{\Delta}([\underline{y}])\simeq\chi_{\bm{\beta}}^{\Delta}$,
and $\forall\,[y]\in{\mathcal{C}}_{y}$,
$\sum_{{\bm{\beta}}\in{\mathcal{B}}}\,\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])=1$.

(ii) Let $[{\bm{Y}}]$ be the ${\mathbb{M}}_{3,\widehat{\nu}_{s}}$-valued
random variable, defined on $(\Theta,{\mathcal{T}},{\mathcal{P}})$, whose
support of its probability measure is
${\mathcal{C}}_{y}\subset{\mathbb{M}}_{3,\widehat{\nu}_{s}}$, and such that
$\\{[{\bm{Y}}]_{j\widehat{\beta}}\,,\,j\in\\{1,2,3\\}\,,\,\widehat{\beta}\in\\{1,\ldots,\widehat{\nu}_{s}\\}\\}$
are $3\,\widehat{\nu}_{s}$ independent uniform random variables on $[0,1]$.
Its mean value is
$E\\{[{\bm{Y}}]\\}=\int_{{\mathcal{C}}_{y}}[y]\,P_{[{\bm{Y}}]}(dy)=\int_{{\mathcal{C}}_{y}}[y]\,dy=[\underline{y}]$.
For all $\widehat{\bm{\beta}}\in\widehat{\mathcal{B}}$,
$A_{\widehat{\bm{\beta}}}=a_{\widehat{\bm{\beta}}}([{\bm{Y}}])$ is a second-
order real-valued random variable.

(iii) $\forall{\bm{\beta}}\in{\mathcal{B}}$,
$\widetilde{\chi}_{\bm{\beta}}^{\Delta}([{\bm{Y}}])$ is a second-order
positive-valued random variable, defined on
$(\Theta,{\mathcal{T}},{\mathcal{P}})$ such that
$\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{\chi}_{\bm{\beta}}^{\Delta}([{\bm{Y}}])=1$
almost surely.

(iv) The dimensionless spectral measure
$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])$ for given $[y]$ in
${\mathcal{C}}_{y}$, is a bounded positive measure on ${\mathbb{R}}^{3}$ and
is such that
$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[\underline{y}])\simeq\mu_{G}^{\nu}(d{\bm{\tau}})$.
For all $[y]\in{\mathcal{C}}_{y}$,
$\widetilde{\mu}_{G}^{\nu}({\mathbb{R}}^{3};[\underline{y}])=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\,\widetilde{\chi}_{\bm{\beta}}^{\Delta}([\underline{y}])=1$.

###### Proof 5

(Proposition 2). This proposition is easy to prove and is left to the reader.

######  Example 1 (Illustration of a construction for a separable spatial
correlation structure)

(i) Spectral density function.
$\forall{\bm{k}}=(k_{1},k_{2},k_{3})\in{\mathbb{R}}^{3}$,
$s({\bm{k}})=\Pi_{j=1}^{3}s_{j}(k_{j})$. For $j=1,2,3$,
$s_{j}(k_{j})=K_{j}^{-1}\,(1-|k_{j}|/K_{j})\,{\mathonebb{1}}_{[-K_{j},K_{j}]}(k_{j})$
and thus, supp $s_{j}=[-K_{j},K_{j}]$, $s_{j}(-k_{j})=s_{j}(k_{j})$ (yielding
$s(-{\bm{k}})=s({\bm{k}})$ and the quadrant symmetry), and
$\int_{[-K_{j},K_{j}]}s_{j}(k_{j})\,dk_{j}=1$.

(ii) Correlation function and spatial correlation length. For all
${\bm{\zeta}}=(\zeta_{1},\zeta_{2},\zeta_{3})\in{\mathbb{R}}^{3}$,
$\rho_{G}({\bm{\zeta}})=\Pi_{j=1}^{3}\rho_{j}(\zeta_{j})$ and for $j=1,2,3$,
$\rho_{j}(\zeta_{j})=\int_{\mathbb{R}}e^{\iota\,k_{j}\zeta_{j}}\,s_{j}(k_{j})\,dk_{j}$,
$\rho_{j}(0)=1$, and the spatial correlation length is
$L_{cj}=\int_{0}^{+\infty}|\rho_{j}(\zeta_{j})|\,d\zeta_{j}=\pi\,s_{j}(0)=\pi/K_{j}$.

(iii) Dimensionless spectral density function and spectral sampling.
$\forall{\bm{\tau}}=(\tau_{1},\tau_{2},\tau_{3})$,
$\chi(\tau)=\Pi_{j=1}^{3}\,\chi_{j}(\tau_{j})$. For $j=1,2,3$,
$\chi_{j}(\tau_{j})=(1-|\tau_{j}|)\,{\mathonebb{1}}_{[-1,1]}(\tau_{j})$ and
therefore, supp $\chi_{j}=[-1,1]$, $\chi_{j}(-\tau_{j})=\chi_{j}(\tau_{j})$,
and $\int_{\mathbb{R}}\chi_{j}(\tau_{j})\,d\tau_{j}=1$. For all
${\bm{\beta}}=(\beta_{1},\beta_{2},\beta_{3})\in{\mathcal{B}}$,
$\chi_{\bm{\beta}}^{\Delta}=\Pi_{j=1}^{3}\chi_{j\beta_{j}}^{\Delta}$ with
$\chi_{j\beta_{j}}^{\Delta}=(2/\nu_{s})\,\chi_{j}(\tau_{\beta_{j}})$.

(iv) Construction of $a_{\bm{\beta}}([y])$. For $j\in\\{1,2,3\\}$,
$\widehat{\beta}_{j}\in\\{1,\ldots,\widehat{\nu}_{s}\\}$,
$\widehat{\bm{\beta}}=(\widehat{\beta}_{1},\widehat{\beta}_{2},\widehat{\beta}_{3})$,
and $\forall[y]\in{\mathcal{C}}_{y}$,
$q_{\widehat{\bm{\beta}}}([y];\delta_{s})=\Pi_{j=1}^{3}q_{j\widehat{\beta}_{j}}([y];\delta_{j})$
in which
$q_{j\widehat{\beta}_{j}}([y];\delta_{j})=1+\sqrt{12}\,\delta_{j}\,([y]_{j\widehat{\beta}_{j}}-1/2)$
with $\delta_{j}>0$ the hyperparameter. We thus have
$a_{\widehat{\bm{\beta}}}([y])=\Pi_{j=1}^{3}a_{j\widehat{\beta}_{j}}([y])$ in
which
$a_{j\widehat{\beta}_{j}}([y])=\sqrt{\chi^{\Delta}_{j\widehat{\beta}_{j}}}\,q_{j\widehat{\beta}_{j}}([y],\delta_{j})$
and for
$\beta_{j}\in\\{\widehat{\nu}_{s}\\!+\\!1,\ldots,2\,\widehat{\nu}_{s}\\}$,
$a_{j\beta_{j}}([y])=a_{j,(2\,\widehat{\nu}_{s}+1-\beta_{j})}([y])$.

(v) Random variable $A_{\widehat{\bm{\beta}}}$ and hyperparameter
$\delta_{s}$. For $j\in\\{1,2,3\\}$ and
$\widehat{\beta}_{j}\in\\{1,\ldots,\widehat{\nu}_{s}\\}$, the mean value and
the second-order moment of random variable
$A_{j\widehat{\beta}_{j}}=a_{j\widehat{\beta}_{j}}([{\bm{Y}}])$ are
$E\\{A_{j\widehat{\beta}_{j}}\\}=\sqrt{\chi^{\Delta}_{j\widehat{\beta}_{j}}}$
and
$E\\{A_{j\widehat{\beta}_{j}}^{2}\\}=\chi^{\Delta}_{j\widehat{\beta}_{j}}\,(1+\delta_{j}^{2})$.
Since the random variables
$\\{A_{j\widehat{\beta}_{j}}\\}_{j,\widehat{\beta}_{j}}$ are independent, the
mean value and the second-order moment of the random variable
$A_{\widehat{\bm{\beta}}}=a_{\widehat{\bm{\beta}}}([{\bm{Y}}])=\Pi_{j=1}^{3}A_{j\widehat{\beta}_{j}}$
are
$E\\{A_{\widehat{\bm{\beta}}}\\}=\sqrt{\chi^{\Delta}_{\widehat{\bm{\beta}}}}$
and
$E\\{A_{\widehat{\bm{\beta}}}^{2}\\}=\chi^{\Delta}_{\widehat{\bm{\beta}}}\,\Pi_{j=1}^{3}(1+\delta_{j}^{2})$.
Defining the hyperparameter $\delta_{s}$ as
$\delta_{s}^{2}=E\\{(A_{\widehat{\bm{\beta}}}-\sqrt{\chi^{\Delta}_{\widehat{\bm{\beta}}}})^{2}\\}/\chi^{\Delta}_{\widehat{\bm{\beta}}}$,
it can be seen that we have
$\delta_{s}^{2}=(\Pi_{j=1}^{3}(1+\delta_{j}^{2}))-1>0$, which is independent
of $\widehat{\bm{\beta}}$.

(vi) Discretized dimensionless spectral measure. Eq. (3.11) yields,
$\forall{\bm{\beta}}=(\beta_{1},\beta_{2},\beta_{3})\in{\mathcal{B}}$,
$\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])=\Pi_{j=1}^{3}\,\widetilde{\chi}_{j\beta_{j}}^{\Delta}([y])$
in which
$\widetilde{\chi}_{j\beta_{j}}^{\Delta}([y])=a_{j\beta_{j}}([y])^{2}(\sum_{{\bm{\beta}}^{\prime}\in{\mathcal{B}}}a_{{\bm{\beta}}^{\prime}}([y])^{2})^{-1/3}$.

######  Definition 5 (Spectrum parameters ${\bm{w}}$ and
${\bm{{\mathcal{S}}}}$ and their probabilistic models ${\bm{W}}$ and
${\bm{S}}$)

Let ${\bm{w}}=(w_{1},w_{2},w_{3})$ in which $w_{j}=\pi/K_{j}$ (this parameter
allows the support of the spectral measure to be controlled). Let
${\mathcal{C}}_{w}=\\{{\bm{w}}\in{\mathbb{R}}^{3};w_{j}\in[w_{j}^{\hbox{{\ppppcarac
min}}},w_{j}^{\hbox{{\ppppcarac max}}}]$ for $j=1,2,3\\}$ be the compact
subset of ${\mathbb{R}}^{3}$, in which $0<w_{j}^{\hbox{{\ppppcarac
min}}}=\pi/K_{j}^{\hbox{{\ppppcarac max}}}<w_{j}^{\hbox{{\ppppcarac
max}}}=\pi/K_{j}^{\hbox{{\ppppcarac min}}}<+\infty$ (see Hypothesis 2).
Parameter ${\bm{w}}$ is modeled by a ${\mathbb{R}}^{3}$-valued random variable
${\bm{W}}$, defined on $(\Theta,{\mathcal{T}},{\mathcal{P}})$, independent of
$[{\bm{Y}}]$, whose support of its given probability measure
$P_{\bm{W}}(d{\bm{w}})$ is ${\mathcal{C}}_{w}$. We define the parameter
${\bm{{\mathcal{S}}}}$ as $\\{{\bm{w}},[y]\\}$, which takes its values in the
subset ${\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$
of ${\mathbb{R}}^{3}\times{\mathbb{M}}_{3,\widehat{\nu}_{s}}$. The
probabilistic model of ${\bm{{\mathcal{S}}}}$ is the
${\mathbb{R}}^{3}\times{\mathbb{M}}_{3,\widehat{\nu}_{s}}$\- valued random
variable ${\bm{S}}=\\{{\bm{W}},[{\bm{Y}}]\\}$ whose probability measure is the
product of measures $P_{\bm{S}}=P_{\bm{W}}(d{\bm{w}})\otimes
P_{[{\bm{Y}}]}(dy)$ whose compact support is ${\mathcal{C}}_{\mathcal{S}}$.

######  Definition 6 (Normalized Gaussian random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ given
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},\hbox{[}y\hbox{]}\\}\in{\mathcal{C}}_{\mathcal{S}}=$
${\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$ )

Let $\nu=(\nu_{s})^{3}$ be fixed. Let
$\\{Z_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$ and
$\\{\Phi_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$ be $2\,\nu$ independent
random variables on $(\Theta,{\mathcal{T}},{\mathcal{P}})$, which are
independent of ${\bm{W}}$ and $[{\bm{Y}}]$. For all
${\bm{\beta}}\in{\mathcal{B}}$, $Z_{\bm{\beta}}=\sqrt{-log\Psi_{\bm{\beta}}}$
in which $\Psi_{\bm{\beta}}$ is uniform of $[0,1]$ and $\Phi_{\bm{\beta}}$ is
uniform on $[0,2\,\pi]$. Let $P_{\bm{Z}}(d{\bm{z}})$ and
$P_{\bm{\Phi}}(d{\bm{\varphi}})$ be the probability measures on
${\mathbb{R}}^{\nu}$ of the ${\mathbb{R}}^{\nu}$-valued random variables
${\bm{Z}}=\\{Z_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$ and
${\bm{\Phi}}=\\{\Phi_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$. The
unbounded support ${\mathcal{C}}_{z}$ of $P_{\bm{Z}}(d{\bm{z}})$ is
${\mathcal{C}}_{z}=\\{{\bm{z}}=\\{z_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\},z_{\bm{\beta}}>0\\}\subset{\mathbb{R}}^{\nu}$
and the compact support ${\mathcal{C}}_{\varphi}$ of
$P_{\bm{\Phi}}(d{\bm{\varphi}})$ is
${\mathcal{C}}_{\varphi}=\\{{\bm{\varphi}}=\\{\varphi_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\},\varphi_{\bm{\beta}}\in[0,2\,\pi]\\}\subset{\mathbb{R}}^{\nu}$.
Let ${\bm{x}}\mapsto
g^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{z}},{\bm{\varphi}}):{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$
be such that, for all
$\\{{\bm{{\mathcal{S}}}},{\bm{z}},{\bm{\varphi}}\\}\in{\mathcal{C}}_{\mathcal{S}}\times{\mathcal{C}}_{z}\times{\mathcal{C}}_{\varphi}$,

$g^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{z}},{\bm{\varphi}})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\sqrt{2\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])}\,z_{\bm{\beta}}\,\cos(\varphi_{\bm{\beta}}+\sum_{j=1}^{3}\frac{\pi}{w_{j}}\tau_{\beta_{j}}x_{j})\,.$
(3.13)

For all ${\bm{w}}\in{\mathcal{C}}_{w}$ and $[y]\in{\mathcal{C}}_{y}$, we
define the real-valued random field
$\\{G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$
with
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$,
such that

$G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})=g^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{Z}},{\bm{\Phi}})\,.$
(3.14)

Equation (3.14) with Eq. (3.13) corresponds to a finite discretization of the
stochastic integral representation with a stochastic spectral measure for
homogeneous second-order mean-square continuous random fields [19, 20, 15].

######  Proposition 3 (Properties of random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$)

For all
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$,
the real-valued random field
$\\{G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$ is
Gaussian, homogeneous, second-order, mean-square continuous, and normalized,

$E\\{G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0\quad,\quad
E\\{G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}=1\quad,\quad\forall{\bm{x}}\in{\mathbb{R}}^{3}\,.$
(3.15)

Its dimensionless spectral measure
$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])$, expressed with the
dimensionless spectral variable ${\bm{\tau}}=(\tau_{1},\tau_{2},\tau_{3})$
with $\tau_{j}=k_{j}/K_{j}$, is the spectral measure defined by Eq. (3.12).

###### Proof 6

(Proposition 3). Since $\\{Z_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$ and
$\\{\Phi_{\bm{\beta}},{\bm{\beta}}\in{\mathcal{B}}\\}$ are $2\,\nu$
independent random variables, it can easily be proven that
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ is centered (first equation in Eq.
(3.15)) and,
$\forall{\bm{\zeta}}=(\zeta_{1},\zeta_{2},\zeta_{3})\in{\mathbb{R}}^{3}$ and
$\forall{\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$,

$\rho_{G}^{\nu}({\bm{\zeta}};{\bm{{\mathcal{S}}}})=E\\{G^{\nu}({\bm{x}}\\!+\\!{\bm{\zeta}};{\bm{{\mathcal{S}}}})\,G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})\\}=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])\,\cos(\sum_{j=1}^{3}\frac{\pi}{w_{j}}\tau_{\beta_{j}}\zeta_{j})\,.$
(3.16)

Using Proposition 2-(i) yields the second equation in Eq. (3.15). For all
${\bm{\beta}}\in{\mathcal{B}}$, the random variable
$Z_{\bm{\beta}}\,\cos(\Phi_{\bm{\beta}}+\sum_{j=1}^{3}\frac{\pi}{w_{j}}\tau_{\beta_{j}}x_{j})$
is Gaussian and consequently, random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ is Gaussian. Since
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ is a Gaussian random field with zero
mean function and a correlation function that depends only on ${\bm{\zeta}}$,
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ is homogeneous on ${\mathbb{R}}^{3}$.
Since ${\bm{\zeta}}\mapsto\rho_{G}^{\nu}({\bm{\zeta}};{\bm{{\mathcal{S}}}})$
defined by Eq. (3.16) is continuous on ${\mathbb{R}}^{3}$,
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ is mean-square continuous on
${\mathbb{R}}^{3}$ and thus there exists a spectral measure given by Eq.
(3.12). Note that the spectral measure in ${\bm{k}}=(k_{1},k_{2},k_{3})$ is
such that
$\widetilde{m}_{G}^{\nu}(d{\bm{k}};{\bm{{\mathcal{S}}}})=\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])$
with
$\widetilde{m}_{G}^{\nu}(d{\bm{k}};{\bm{{\mathcal{S}}}})=\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{s}_{\bm{\beta}}^{\Delta}({\bm{w}},[y])\,\delta_{{\bm{k}}_{\bm{\beta}}}({\bm{k}})$
in which
$\widetilde{s}_{\bm{\beta}}^{\Delta}({\bm{w}},[y])=\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])\,\Pi_{j=1}^{3}(\pi/w_{j})$
with ${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}$.

##  4 Non-Gaussian random field
$\hbox{[}{\mathbb{C}}(.;{\bm{{\mathcal{S}}}})\hbox{]}$ parameterized by
${\bm{{\mathcal{S}}}}$ and random field
$\hbox{[}\widetilde{\mathbb{C}}\hbox{]}$ with uncertain spectral measure

In the construction of $G$ with an uncertain spectrum, the spectral measure
$m_{G}(d{\bm{k}})$ of $G$ is given (see Hypothesis 2). This is the reason why
the convergence of the sequence $\\{m_{G}^{\nu}(d{\bm{k}})\\}_{\nu}$ of
measures towards $m_{G}(d{\bm{k}})$ has been studied (see Lemma 3). The
uncertain dimensionless spectrum, represented by
$\widetilde{\mu}_{G}^{\nu}(d{\bm{\tau}};[y])$ for $[y]$ given in
${\mathcal{C}}_{y}$, is constructed from
$\mu_{G}^{\nu}(d{\bm{\tau}})=m_{G}^{\nu}(d{\bm{k}})$ and constitutes the
uncertain spectral measure of random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ given
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$.
Although a limit $G^{\infty}(\cdot;{\bm{{\mathcal{S}}}})$ of random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$ exits for $\nu\rightarrow+\infty$ (see
[21]), a convergence analysis is not useful for the probabilistic construction
that is proposed because the limit is not given (unknown). The value of
$\nu=(\nu_{s})^{3}$ (see Definition 2) is chosen sufficiently large in order
that Hypothesis 3 be verified. Proposition 3 shows that, for all
${\bm{{\mathcal{S}}}}\in{\mathcal{C}}_{\mathcal{S}}$, the random field
$G^{\nu}(\cdot;{\bm{{\mathcal{S}}}})$, defined by Eq. (3.14), satisfies all
the required properties (Gaussian, homogeneous, mean-square continuous, and
normalization). We are therefore led to introduce the following definition in
coherence with Proposition 1, Lemma 2, and Remark 1.

######  Definition 7 (Random field
$\hbox{[}{\bm{C}}(\cdot,{\bm{{\mathcal{S}}}})\hbox{]}$ given
${\bm{{\mathcal{S}}}}$)

We assume that $\nu$ is fixed and satisfies Hypothesis 3. The non-Gaussian
random field ${\bm{C}}(\cdot,{\bm{{\mathcal{S}}}})$ given
${\bm{{\mathcal{S}}}}\in{\mathcal{C}}_{\mathcal{S}}$ is defined by Eq. (2.6)
in which the $21$ Gaussian random fields
$\\{G_{mn}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}_{1\leq
m\leq n\leq 6}$ are replaced by $21$ independent copies of the Gaussian real-
valued random field
$\\{G^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}$
defined by Eq. (3.14), and denoted by
$\\{G^{\nu}_{mn}({\bm{x}};{\bm{{\mathcal{S}}}}),{\bm{x}}\in{\mathbb{R}}^{3}\\}_{1\leq
m\leq n\leq 6}$. For all ${\bm{{\mathcal{S}}}}\in{\mathcal{C}}_{\mathcal{S}}$,
${\bm{x}}\in{\mathbb{R}}^{3}$, and for $1\leq m\leq n\leq 6$, using Eq. (3.14)
yields

$G^{\nu}_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})=g^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{Z}}^{mn},{\bm{\Phi}}^{mn})\,.$
(4.1)

in which $\\{{\bm{Z}}^{mn},{\bm{\Phi}}^{mn}\\}_{1\leq m\leq n\leq 6}$ are $21$
independent copies of ${\mathbb{R}}^{\nu}$-valued random variables ${\bm{Z}}$
and ${\bm{\Phi}}$ (see Definition 6), and we have

$E\\{G^{\nu}_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})\\}=0\quad,\quad
E\\{G^{\nu}_{mn}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\\}=1\,.$ (4.2)

######  Proposition 4 (Properties of the non-Gaussian
${\mathbb{M}}^{+}_{6}$-valued random field
$\\{\hbox{[}{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}\\}$)

The non-Gaussian random field $[{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})]$,
defined in Definition 7 for a given uncertain spectral measure parameterized
by ${\bm{{\mathcal{S}}}}$, is a second-order random field such that

$\|\,[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\,\,\leq\,\,\Gamma_{C}\,\,\,a.s.\quad,\quad\forall{\bm{x}}\in{\mathbb{R}}^{3}\,,$
(4.3)

in which $\Gamma_{C}$ is a second-order positive-valued random variable,
independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$, such that

$E\\{\Gamma_{C}^{2}\\}=\underline{\gamma}_{2,C}^{2}<+\infty\quad,\quad
E\\{\Gamma_{C}^{4}\\}=\underline{\gamma}_{4,C}^{4}<+\infty\,.$ (4.4)

For all ${\bm{\omega}}$ and ${\bm{\omega}}^{\prime}$ in ${\mathbb{R}}^{6}$ and
for all ${\bm{x}}$ in ${\mathbb{R}}^{3}$,

$|\langle[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}}\,,{\bm{\omega}}^{\prime}\rangle_{2}|\,\,\,\leq\,\,\Gamma_{C}\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}\,\,\,a.s.$
(4.5)

###### Proof 7

(Proposition 4). For all ${\bm{x}}\in{\mathbb{R}}^{3}$ and
${\bm{{\mathcal{S}}}}\in{\mathcal{C}}_{\mathcal{S}}$, Proposition 1 and
Definition 7 yield
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\!=\\![{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]^{T}[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]$
in which
$[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}\\!=\\!\sigma_{c}\,G_{mn}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})$
for $1\\!\leq\\!m\\!<\\!n\\!\leq\\!6$ and
$[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mm}\\!=\\!\sigma_{c}\,\sqrt{2h(G_{mm}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}});\alpha_{m})}$
for $1\\!\leq\\!m\\!=\\!n\\!\leq\\!6$. We thus have, almost surely,
$\|\,[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\,\leq\,\,\|[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]\|_{F}^{2}=\sum_{m}[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mm}^{2}+\sum_{m<n}[{\bm{L}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{mn}^{2}=\sigma_{c}^{2}(\sum_{m}2\,h(G_{mm}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}});$
$\alpha_{m})+\sum_{m<n}G_{mn}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})^{2})$.
Using Lemma 2-(i) yields

$\|\,[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\,\leq\,\,\sigma_{c}^{2}\,(4\sum_{m}\alpha_{m}+2\sum_{m}G_{mm}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}+\sum_{m<n}G_{mn}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})^{2})\,.$
(4.6)

For all
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$,
${\bm{z}}\in{\mathcal{C}}_{z}$, and
${\bm{\varphi}}\in{\mathcal{C}}_{\varphi}$, Eq. (3.13) allows for writing
$|g^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{z}},{\bm{\varphi}})|\leq\sum_{{\bm{\beta}}\in{\mathcal{B}}}\sqrt{2\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])}\,$
$z_{\bm{\beta}}\leq\sqrt{2\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])}\,\sqrt{\sum_{{\bm{\beta}}\in{\mathcal{B}}}z_{\bm{\beta}}^{2}}\leq\sqrt{2\sum_{{\bm{\beta}}\in{\mathcal{B}}}z_{\bm{\beta}}^{2}}$
because, from Proposition 2-(i),
$\sum_{{\bm{\beta}}\in{\mathcal{B}}}\widetilde{\chi}_{\bm{\beta}}^{\Delta}([y])=1$.
For all ${\bm{x}}\in{\mathbb{R}}^{3}$, using Eq. (4.1) yields
$G_{mn}^{\nu}({\bm{x}};{\bm{{\mathcal{S}}}})^{2}\leq
2\sum_{{\bm{\beta}}\in{\mathcal{B}}}(Z_{\bm{\beta}}^{mn})^{2}$ almost surely.
Using Eq. (4.6), we obtain Eq. (4.3) in which
$\Gamma_{C}=\sigma_{c}^{2}\,(4\sum_{m}\alpha_{m}+4\sum_{m}\sum_{\bm{\beta}}(Z_{\bm{\beta}}^{mm})^{2}+2\sum_{m<n}\sum_{\bm{\beta}}(Z_{\bm{\beta}}^{mn})^{2})$
that is independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$. The
$21\times\nu$ random variables $\\{Z_{\bm{\beta}}^{mn}\\}_{mn,{\bm{\beta}}}$
are independent copies of random variable $Z=\sqrt{-log\Psi}$ whose
probability measure is
$P_{Z}(dz)={\mathonebb{1}}_{{\mathbb{R}}^{+}}(z)\,2z\,\exp(-z^{2})\,dz$. We
have $E\\{Z\\}=\sqrt{\pi}/2$ and $E\\{Z^{2}\\}=1$. Hence,
$E\\{\Gamma_{C}\\}=\underline{\gamma}_{1,C}$ with
$\underline{\gamma}_{1,C}=\sigma_{c}^{2}(4\sum_{m}\alpha_{m}+54\,\nu)<+\infty$.
For $p=2$ or $4$,
$E\\{\Gamma_{C}^{p}\\}=2(2\sigma_{c}^{2})^{p}\int_{0}^{+\infty}(2\sum_{m}\alpha_{m}+27\nu
z^{2})^{p}\,z\,\exp(-z^{2})\,dz=\underline{\gamma}_{p,C}^{p}<+\infty$ that
yields Eq. (4.4) (note that $E\\{\Gamma_{C}^{4}\\}<+\infty$ implies
$E\\{\Gamma_{C}\\}<+\infty$ and $E\\{\Gamma_{C}^{2}\\}<+\infty$). Since
$E\\{\|\,[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}^{2}\\}\leq
E\\{\Gamma_{C}^{2}\\}<+\infty$, $[{\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is a
second-order random field. Finally, for all ${\bm{x}}\in{\mathbb{R}}^{3}$,
${\bm{\omega}}$ and ${\bm{\omega}}^{\prime}$ in ${\mathbb{R}}^{6}$,
$|\langle[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}}\,,{\bm{\omega}}^{\prime}\rangle_{2}|\,\,\leq\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\|_{2}\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}\leq\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\|_{F}\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}$,
which yields Eq. (4.5) using Eq. (4.3).

######  Corollary 1 (Properties of the non-Gaussian
${\mathbb{M}}^{+}_{6}$-valued random field
$\hbox{[}{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})\hbox{]}$)

Let ${\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})$ be the non-Gaussian second-
order random field defined by Eq. (2.2) in which
${\bm{C}}(\cdot;{\bm{{\mathcal{S}}}})$ satisfies the properties given in
Proposition 4. For all ${\bm{\omega}}\in{\mathbb{R}}^{6}\backslash\\{0\\}$,

$\frac{\langle[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}},{\bm{\omega}}\rangle_{2}}{\|{\bm{\omega}}\|_{2}^{2}}\leq\|\,[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\,\,\leq\,\,\Gamma_{\mathbb{C}}\,\,\,a.s.\quad,\quad\forall{\bm{x}}\in{\mathbb{R}}^{3}\,,$
(4.7)

with $\Gamma_{\mathbb{C}}$ a second-order ${\mathbb{R}}^{+}$-valued random
variable, independent of ${\bm{x}}$ and ${\bm{{\mathcal{S}}}}$, such that

$E\\{\Gamma_{\mathbb{C}}^{2}\\}=\underline{\gamma}_{2,{\mathbb{C}}}^{2}<+\infty\quad,\quad
E\\{\Gamma_{\mathbb{C}}^{4}\\}=\underline{\gamma}_{4,{\mathbb{C}}}^{4}<+\infty\,.$
(4.8)

For all ${\bm{\omega}}$ and ${\bm{\omega}}^{\prime}$ in ${\mathbb{R}}^{6}$ and
for all ${\bm{x}}$ in ${\mathbb{R}}^{3}$,

$\displaystyle|\langle[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}},{\bm{\omega}}^{\prime}\rangle_{2}|\,\,\,$
$\displaystyle\leq\,\,\Gamma_{\mathbb{C}}\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}\,\,\,a.s.\,,$
(4.9)
$\displaystyle\langle[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}},{\bm{\omega}}\rangle_{2}\,\,\,$
$\displaystyle\geq\,\,\underline{c}_{\epsilon}\,\|{\bm{\omega}}\|_{2}^{2}\,\,\,a.s.$
(4.10)

in which $\underline{c}_{\epsilon}=\underline{c}_{0}\,\epsilon/(1+\epsilon)$
is a finite positive constant independent of ${\bm{x}}$ and
${\bm{{\mathcal{S}}}}$.

###### Proof 8

(Corollary 1). Equation (2.2) yields
$\|\,[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\leq(1+\epsilon)^{-1}\|[\underline{{\mathbb{L}}}]^{T}\|_{F}\,\|[\underline{{\mathbb{L}}}]\|_{F}$
$(\epsilon\,\|[I_{6}]\|_{F}+\|[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})\|_{F})$.
We have
$\|[\underline{{\mathbb{L}}}]^{T}\|_{F}=\|[\underline{{\mathbb{L}}}]\|_{F}=({\hbox{{{t}r}}}[\underline{{\mathbb{C}}}])^{1/2}$
and $\|[I_{6}]\|_{F}=\sqrt{6}$. Eqs. (2.1) and (4.3) yield
$\|\,[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F}\,\,\leq\,\underline{c}_{1}(1+\epsilon)^{-1}(\epsilon\sqrt{6}+\|\,[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,\|_{F})\leq\,\Gamma_{\mathbb{C}}$
almost surely with
$\Gamma_{\mathbb{C}}=\underline{c}_{1}(1+\epsilon)^{-1}(\epsilon\sqrt{6}+\Gamma_{C})$,
which is the second part of Eq. (4.7). For all ${\bm{\omega}}$ and
${\bm{\omega}}^{\prime}$ in ${\mathbb{R}}^{6}$, we have
$|\langle[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}},{\bm{\omega}}^{\prime}\rangle_{2}|\,\,\leq\,\|[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\|_{F}\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}$.
Taking
${\bm{\omega}}^{\prime}={\bm{\omega}}\in{\mathbb{R}}^{6}\backslash\\{0\\}$
yields the first part of Eq. (4.7). Then using the second part of Eq. (4.7)
yields Eq. (4.9). From Eq. (2.2), it can be deduced that
$\langle[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,{\bm{\omega}}\,,{\bm{\omega}}\rangle_{2}=(\epsilon\langle[\underline{{\mathbb{C}}}]\,{\bm{\omega}},{\bm{\omega}}\rangle_{2}+\langle[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\,[\underline{{\mathbb{L}}}]\,{\bm{\omega}},[\underline{{\mathbb{L}}}]{\bm{\omega}}\rangle_{2})/(1+\epsilon)$.
From Eq. (2.1) and since $[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ is a
${\mathbb{M}}^{+}_{6}$-valued random variable, we obtain Eq. (4.10).

######  Definition 8 (Random field $\hbox{[}\widetilde{\mathbb{C}}\hbox{]}$
with uncertain spectral measure)

We assume that $\nu$ is fixed and satisfies Hypothesis 3. The random field
$\\{[\widetilde{\mathbb{C}}({\bm{x}})]\in{\mathbb{R}}^{3}\\}$ with uncertain
spectral measure is defined by

$[\widetilde{\mathbb{C}}({\bm{x}})]=[{\mathbb{C}}({\bm{x}};{\bm{S}})]\quad,\quad\forall{\bm{x}}\in{\mathbb{R}}^{3}\,,$
(4.11)

in which ${\bm{S}}=\\{{\bm{W}},[{\bm{Y}}]\\}$ is the
${\mathbb{R}}^{3}\times{\mathbb{M}}_{3,\widehat{\nu}_{s}}$\- valued random
variable defined by Definition 5.

###### Remark 2

Random field $\\{[\widetilde{\mathbb{C}}({\bm{x}})]\in{\mathbb{R}}^{3}\\}$
defined by Eq. (4.11) is the random field with uncertain spectral measure. As
explained in Remark 1,
$E\\{[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]\\}=[\underline{{\mathbb{C}}}]$
for all ${\bm{x}}\in{\mathbb{R}}^{3}$, we have not
$E\\{[\widetilde{\mathbb{C}}({\bm{x}})]\\}=[\underline{{\mathbb{C}}}]$, but we
have the approximation
$E\\{[\widetilde{\mathbb{C}}({\bm{x}})]\\}\simeq[\underline{{\mathbb{C}}}]$.

##  5 Stochastic elliptic boundary value problem for stochastic homogenization

We consider a heterogeneous complex elastic microstructure occupying domain
$\Omega$, which by definition is a microstructure that cannot be described in
terms of its constituents at the microscale. This is typically the case of
live tissues. In such a case, the stochastic model of the apparent elasticity
field can be constructed at the mesoscale that corresponds to the scale of the
spatial correlation length of the microstructure $\Omega$ as proposed in [9,
22, 7]. The stochastic homogenization from the mesoscale to the macroscale
allows the effective elasticity tensor
$\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}$ to be constructed. The
study of the statistical properties of
$\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}$ allows for analyzing the
scale separation. The separation is obtained if the statistical fluctuations
of $\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}$ are sufficiently small
and, in this case, $\Omega$ is a representative volume element (RVE) [23, 24,
25, 22]. Such a separation occurs if the spatial correlation length at the
mesoscale is sufficiently small with respect to the characteristic geometrical
dimension of $\Omega$. If not, $\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}$ exhibits significant statistical fluctuations and therefore, $\Omega$
is not a RVE.

The deterministic part of the formulation used in Sections 5.1 to 5.3 to write
the problem of homogenization on $\Omega$ is that proposed in [26] for
homogeneous deformations on the boundary $\partial\Omega$. We use the
convention for summations over repeated Latin indices $j$, $p$, and $q$ taking
values in $\\{1,2,3\\}$.

###  5.1 Definition of the stochastic boundary value problem (BVP)

Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{3}$ with a
sufficiently regular boundary $\partial\Omega$. Let ${\bm{{\mathcal{S}}}}$ be
fixed in ${\mathcal{C}}_{\mathcal{S}}$. For all $\ell$ and $r$ in
$\\{1,2,3\\}$, we have to find the ${\mathbb{R}}^{3}$-valued random field
$\\{\widetilde{\bm{U}}^{\ell r}({\bm{x}})=(\widetilde{U}_{1}^{\ell
r}({\bm{x}}),\widetilde{U}_{2}^{\ell r}({\bm{x}}),\widetilde{U}_{3}^{\ell
r}({\bm{x}})),{\bm{x}}\in\overline{\Omega}\\}$, defined on
$(\Theta,{\mathcal{T}},{\mathcal{P}})$, indexed by $\overline{\Omega}$, such
that almost surely,

$\displaystyle-\frac{\partial}{\partial
x_{j}}({\mathbb{C}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}})\,\varepsilon_{pq}(\widetilde{\bm{U}}^{\ell
r}({\bm{x}}))=0$ $\displaystyle\quad,\quad\forall{\bm{x}}\in\Omega\quad,\quad
i=1,2,3\,,$ (5.1) $\displaystyle\widetilde{\bm{U}}^{\ell
r}({\bm{x}})=\widetilde{\bm{u}}_{0}^{\ell r}({\bm{x}})$
$\displaystyle\quad,\quad\forall{\bm{x}}\in\partial\Omega\,,$ (5.2)

in which $\varepsilon_{pq}({\bm{u}})=(\partial u_{p}/\partial x_{q}+\partial
u_{q}/\partial x_{p})/2$ for ${\bm{u}}=(u_{1},u_{2},u_{3})$ and where for all
${\bm{x}}\in\partial\Omega$, $\widetilde{\bm{u}}_{0}^{\ell
r}({\bm{x}})=(\widetilde{u}_{0,1}^{\ell r}({\bm{x}}),\widetilde{u}_{0,2}^{\ell
r}({\bm{x}}),\widetilde{u}_{0,3}^{\ell r}({\bm{x}}))$ is defined by
$\widetilde{u}_{0,j}^{\ell
r}({\bm{x}})=(\delta_{j\ell}x_{r}+\delta_{jr}x_{\ell})/2$ with
$\delta_{j\ell}$ the Kronecker symbol. The fourth-order tensor-valued random
field $\\{{\mathbb{C}}_{ijpq}(\cdot;{\bm{{\mathcal{S}}}})\\}_{ijpq}$ is such
that
${\mathbb{C}}_{ijpq}={\mathbb{C}}_{jipq}={\mathbb{C}}_{ijqp}={\mathbb{C}}_{pqij}$
for $i$, $j$, $p$, and $q$ in $\\{1,2,3\\}$ and is such that
${\mathbb{C}}_{ijpq}(\cdot;{\bm{{\mathcal{S}}}})=[{\mathbb{C}}(\cdot,{\bm{{\mathcal{S}}}})]_{\textbf{i}\textbf{j}}$
in which $\textbf{i}=(i,j)$ with $1\leq i\leq j\leq 3$ and $\textbf{j}=(p,q)$
with $1\leq p\leq q\leq 3$ are indices with values in $\\{1,\ldots,6\\}$, and
where the ${\mathbb{M}}^{+}_{6}$-valued random field
$[{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is the one constructed in Section
4 and whose properties are given by Corollary 1.

###  5.2 Random effective tensor from stochastic homogenization and its random
eigenvalues

For ${\bm{{\mathcal{S}}}}$ fixed in ${\mathcal{C}}_{\mathcal{S}}$, for $i$,
$j$, $\ell$, and $r$ in $\\{1,2,3\\}$ the component ${\mathbb{C}}_{ij\ell
r}^{\hbox{{\ppppcarac eff}}}({\bm{{\mathcal{S}}}})$ of the random fourth-order
effective tensor ${\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})$ is defined by

${\mathbb{C}}_{ij\ell r}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})=\frac{1}{|\Omega|}\int_{\Omega}{\mathbb{C}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}})\,\varepsilon_{pq}(\widetilde{\bm{U}}^{\ell
r}({\bm{x}}))\,d{\bm{x}}\,,$ (5.3)

in which $\widetilde{\bm{U}}^{\ell r}$ is the ${\mathbb{R}}^{3}$-valued random
field that satisfies Eqs. (5.1) and (5.2) and where
$|\Omega|=\int_{\Omega}d{\bm{x}}$. The fourth-order effective tensor
${\mathbb{C}}^{\hbox{{\ppppcarac eff}}}({\bm{{\mathcal{S}}}})$ satisfies the
symmetry and positive-definiteness properties [26]. We can thus define the
effective ${\mathbb{M}}^{+}_{6}$-valued random matrix
$[{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}({\bm{{\mathcal{S}}}})]$ associated
with random tensor ${\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})$, which is such that
$[{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})]_{\textbf{i}\textbf{j}}={\mathbb{C}}_{ij\ell
r}^{\hbox{{\ppppcarac eff}}}({\bm{{\mathcal{S}}}})$ in which
$\textbf{i}=(i,j)$ with $1\leq i\leq j\leq 3$ and $\textbf{j}=(\ell,r)$ with
$1\leq\ell\leq r\leq 3$.

###  5.3 Transforming the nonhomogeneous Dirichlet BVP in a homogeneous
Dirichlet BVP

For fixed $\ell$ and $r$, since ${\bm{x}}\mapsto\widetilde{\bm{u}}_{0}^{\ell
r}({\bm{x}})$ is a linear function in ${\bm{x}}$, we can perform the following
translation (without having to resort the trace theorem in Hilbert spaces),

$\widetilde{\bm{U}}^{\ell r}({\bm{x}})={\bm{U}}^{\ell
r}({\bm{x}})+\widetilde{\bm{u}}_{0}^{\ell
r}({\bm{x}})\quad,\quad\forall{\bm{x}}\in\overline{\Omega}\,.$ (5.4)

Since, $\forall{\bm{x}}\in\overline{\Omega}$,
$\varepsilon_{pq}(\widetilde{\bm{u}}_{0}^{\ell
r}({\bm{x}}))=(\delta_{p\ell}\delta_{qr}+\delta_{pr}\delta_{q\ell})/2$ and
${\mathbb{C}}_{ij\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}})={\mathbb{C}}_{ijr\ell}({\bm{x}};{\bm{{\mathcal{S}}}})$,
for all $\ell$ and $r$ in $\\{1,2,3\\}$, the nonhomogeneous Dirichlet BVP
defined by Eqs. (5.1) and (5.2) becomes the following homogeneous Dirichlet
BVP for the ${\mathbb{R}}^{3}$-valued random field $\\{{\bm{U}}^{\ell
r}({\bm{x}})=(U_{1}^{\ell r}({\bm{x}}),U_{2}^{\ell r}({\bm{x}}),U_{3}^{\ell
r}({\bm{x}})),{\bm{x}}\in\overline{\Omega}\\}$, defined on
$(\Theta,{\mathcal{T}},{\mathcal{P}})$, indexed by $\overline{\Omega}$, such
that almost surely,

$\displaystyle-\frac{\partial}{\partial
x_{j}}({\mathbb{C}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}})\,\varepsilon_{pq}({\bm{U}}^{\ell
r}({\bm{x}}))$ $\displaystyle=f_{i}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}})\quad,\quad\forall{\bm{x}}\in\Omega\quad,\quad
i=1,2,3\,,$ (5.5) $\displaystyle{\bm{U}}^{\ell r}({\bm{x}})$
$\displaystyle={\hbox{\bf 0}}\quad,\quad\forall{\bm{x}}\in\partial\Omega\,,$
(5.6)

in which $f_{i}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}})=\frac{\partial}{\partial
x_{j}}({\mathbb{C}}_{ij\ell r}({\bm{x}};{\bm{{\mathcal{S}}}})$ that, from
Proposition 1 and Eqs. (3.13), (3.14), and (4.1), exists almost surely.

###  5.4 Analysis of the stochastic homogeneous Dirichlet BVP

(i) Definition of random vector ${\bm{\Xi}}$. Let
${\bm{\Xi}}=\\{\\{{\bm{Z}}^{mn},$ $1\leq m\leq n\leq
6\\},\\{{\bm{\Phi}}^{mn},1\leq m\leq n\leq 6\\}\\}$ be the second-order random
variable on $(\Theta,{\mathcal{T}},{\mathcal{P}})$ with values in
${\mathbb{R}}^{n_{\xi}}$ with $n_{\xi}=2\times 21\times\nu$, whose probability
measure is
$P_{\bm{\Xi}}=(\otimes_{m,n}P_{{\bm{Z}}^{mn}})\otimes(\otimes_{m,n}P_{{\bm{\Phi}}^{mn}})$
in which $P_{{\bm{Z}}^{mn}}=P_{\bm{Z}}$ and
$P_{{\bm{\Phi}}^{mn}}=P_{\bm{\Phi}}$ for all $1\leq m\leq n\leq 6$ (see
Definitions 6 and 7). Let ${\mathcal{C}}_{\xi}\subset{\mathbb{R}}^{n_{\xi}}$
be the support of $P_{\bm{\Xi}}$, which is known and can easily be written.
Consequently, we have
$E\\{\|{\bm{\Xi}}\|_{2}^{2}\\}=\int_{{\mathbb{R}}^{n_{\xi}}}\|{\bm{\xi}}\|_{2}^{2}\,P_{\bm{\Xi}}(d{\bm{\xi}})=\int_{{\mathcal{C}}_{\xi}}\|{\bm{\xi}}\|_{2}^{2}\,P_{\bm{\Xi}}(d{\bm{\xi}})<+\infty$.

(ii) Definition of mappings
${\bm{\xi}}\mapsto{\mathbb{c}}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ and
${\bm{\xi}}\mapsto{\bm{u}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$.
For ${\bm{{\mathcal{S}}}}$ fixed in ${\mathcal{C}}_{\mathcal{S}}$, the fourth-
order tensor-valued random field ${\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})$ is
written as
${\mathbb{C}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}})=[{\mathbb{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]_{\textbf{i}\textbf{j}}$
in which $\textbf{i}=(i,j)$ with $i\leq j$ and $\textbf{j}=(p,q)$ with $p\leq
q$. Taking into account the construction presented in Sections 2 to 4, the
random field $[{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})]$ is defined by a
${\mathbb{M}}^{+}_{6}$-valued measurable mapping
${\bm{\xi}}\mapsto[{\mathbb{c}}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})]$ on
${\mathcal{C}}_{\xi}$ such that
$[{\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})]=[{\mathbb{c}}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\Xi}})]$.
Therefore, the fourth-order random field
${\mathbb{C}}(\cdot;{\bm{{\mathcal{S}}}})$ is defined by a measurable mapping
${\bm{\xi}}\mapsto{\mathbb{c}}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ on
${\mathcal{C}}_{\xi}$ such that

${\mathbb{C}}_{ijpq}(\cdot;{\bm{{\mathcal{S}}}})={\mathbb{c}}_{ijpq}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\Xi}})\quad,\quad
i,j,p,q\in\\{1,2,3\\}\,.$ (5.7)

Similarly, random field ${\bm{U}}^{\ell r}$ involved in the stochastic BVP,
defined by Eqs. (5.5) and (5.6), depends only on ${\bm{{\mathcal{S}}}}$ and
${\bm{\Xi}}$, and is defined by a measurable mapping
${\bm{\xi}}\mapsto{\bm{u}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ on
${\mathcal{C}}_{\xi}$ such that ${\bm{U}}^{\ell r}={\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\Xi}})$ for $\ell$ and $r$ in
$\\{1,2,3\\}$.

(iii) Definition of Hilbert space ${\mathbb{H}}$. Let
${\mathbb{H}}=\\{{\bm{v}}=(v_{1},v_{2},v_{3});v_{j}\in H^{1}(\Omega)$ for
$j=1,2,3;$ ${\bm{v}}={\hbox{\bf 0}}$ on $\partial\Omega\\}$ be the Hilbert
space equipped with the inner product and the associated norm,

$\langle{\bm{u}},{\bm{v}}\rangle_{\mathbb{H}}=\int_{\Omega}\varepsilon_{pq}({\bm{u}}({\bm{x}}))\,\varepsilon_{pq}({\bm{v}}({\bm{x}}))\,d{\bm{x}}\quad,\quad\|{\bm{v}}\|_{\mathbb{H}}=(\langle{\bm{v}},{\bm{v}}\rangle_{\mathbb{H}})^{1/2}\,.$
(5.8)

Note that $\|{\bm{v}}\|_{\mathbb{H}}$ is a norm on ${\mathbb{H}}$ due to the
Korn inequality and because ${\bm{v}}={\hbox{\bf 0}}$ on $\partial\Omega$ (see
for instance [27]). Introducing the matrix $[\varepsilon]$ such that
$[\varepsilon]_{pq}=\varepsilon_{pq}$, Eq. (5.8) can be rewritten as

$\langle{\bm{u}},{\bm{v}}\rangle_{\mathbb{H}}=\int_{\Omega}\langle[\varepsilon({\bm{u}}({\bm{x}}))]\,,[\varepsilon({\bm{v}}({\bm{x}}))]\rangle_{F}d{\bm{x}}\quad,\quad\|{\bm{v}}\|^{2}_{\mathbb{H}}=\int_{\Omega}\|\,[\varepsilon({\bm{v}}({\bm{x}}))]\,\|_{F}^{2}\,d{\bm{x}}\,.$
(5.9)

(iv) Definition of bilinear form
$b(\cdot,\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ and linear form
${\mathcal{L}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$. For
${\bm{{\mathcal{S}}}}$ fixed in ${\mathcal{C}}_{\mathcal{S}}$, for all
${\bm{\xi}}\in{\mathcal{C}}_{\xi}$, for $\ell$ and $r$ in $\\{1,2,3\\}$, and
using Eq. (5.7), we define the bilinear form $({\bm{u}},{\bm{v}})\mapsto
b({\bm{u}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}}):{\mathbb{H}}\times{\mathbb{H}}\rightarrow{\mathbb{R}}$
such that

$b({\bm{u}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})=\int_{\Omega}{\mathbb{c}}_{ijpq}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,\varepsilon_{pq}({\bm{u}}({\bm{x}}))\,\varepsilon_{ij}({\bm{v}}({\bm{x}}))\,d{\bm{x}}\,,$
(5.10)

and the linear form ${\bm{v}}\mapsto{\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}}):{\mathbb{H}}\rightarrow{\mathbb{R}}$
such that

${\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})=-\int_{\Omega}{\mathbb{c}}_{\ell
rij}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,\varepsilon_{ij}({\bm{v}}({\bm{x}}))\,d{\bm{x}}\,,$
(5.11)

whose right-hand side member in Eq. (5.11) is the transformation of
$\int_{\Omega}f_{i}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}})\,v_{i}({\bm{x}})\,d{\bm{x}}$ for which we
have used ${\bm{v}}={\hbox{\bf 0}}$ on $\partial\Omega$ and the symmetry
property ${\mathbb{c}}_{ij\ell r}={\mathbb{c}}_{ji\ell r}={\mathbb{c}}_{\ell
rij}$.

(v) Type of stochastic solution sought. From a computational point of view, a
solution of the stochastic BVP defined by Eqs. (5.5) and (5.6) will be
constructed by using the Monte Carlo simulation method. Consequently, we only
need to analyze the strong stochastic solution of the weak formulation of this
stochastic BVP and the weak stochastic solution is not useful. We then limit
Proposition 5 to the strong stochastic solution.

######  Proposition 5 (Weak formulation of the stochastic homogeneous
Dirichlet BVP and its strong stochastic solution)

(i) For ${\bm{{\mathcal{S}}}}$ fixed in ${\mathcal{C}}_{\mathcal{S}}$ and for
$1\leq\ell\leq r\leq 3$, the weak formulation of the stochastic BVP defined by
Eqs. (5.5) and (5.6) is: for $P_{\bm{\Xi}}$-almost all ${\bm{\xi}}$ in
${\mathcal{C}}_{\xi}\subset{\mathbb{R}}^{n_{\xi}}$, find ${\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ in ${\mathbb{H}}$ such that

$b({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}}),{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})={\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\quad,\quad\forall{\bm{v}}\in{\mathbb{H}}\,.$
(5.12)

(ii) For $1\leq\ell\leq r\leq 3$, there exists a unique solution
${\bm{u}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})\in{\mathbb{H}}$
(strong stochastic solution) such that Eq. (5.12) holds and
${\bm{u}}^{r\ell}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})={\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$.

(iii) The associated stochastic solution ${\bm{U}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}})={\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\Xi}})$ is of second-order,

$E\\{\|{\bm{U}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}})\|_{\mathbb{H}}^{2}\\}=\underline{\gamma}_{u}^{2}<+\infty\,.$
(5.13)

###### Proof 9

(Proposition 5).

(i) Using Eqs. (5.7) and (5.8) to (5.11), it is easy to prove that Eq. (5.12)
is the weak formulation of Eqs. (5.5) and (5.6).

(ii) Let $[{\mathbb{c}}^{\ell r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]$
be the $(3\times 3)$ real matrix such that $[{\mathbb{c}}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]_{ij}={\mathbb{c}}_{\ell
rij}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})$. Therefore Eq. (5.11) can be
rewritten as ${\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})=-\int_{\Omega}\langle[{\mathbb{c}}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,,[\varepsilon({\bm{v}}({\bm{x}}))]\rangle_{F}\,d{\bm{x}}$
and using Eq. (5.9) yields

$|{\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})|\leq(\int_{\Omega}\,\|\,[{\mathbb{c}}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,\|_{F}^{2}\,d{\bm{x}})^{1/2}\,\|{\bm{v}}\|_{\mathbb{H}}\,.$

For all $\ell$ and $r$ in $\\{1,2,3\\}$ and since ${\mathbb{c}}_{\ell
rij}={\mathbb{c}}_{\ell rji}$, we have

$\|\,[{\mathbb{c}}^{\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,\|_{F}^{2}=\sum_{i,j}{\mathbb{c}}_{\ell
rij}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})^{2}\leq
2\,\sum_{\ell^{\prime}\leq r^{\prime},i\leq
j}{\mathbb{c}}_{\ell^{\prime}r^{\prime}ij}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})^{2}=2\,\|\,[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,\|_{F}^{2}$

in which we have used the notation
$[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]_{\textbf{j}^{\prime}\textbf{i}}={\mathbb{c}}_{\ell^{\prime}r^{\prime}ij}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})$
in which $\textbf{j}^{\prime}=(\ell^{\prime},r^{\prime})$ with
$\ell^{\prime}\leq r^{\prime}$ and $\textbf{i}=(i,j)$ with $i\leq j$. Taking
into account Proposition 4 and its proof, Eqs. (4.7) and (4.8) of Corollary 1
with its proof, show that
$\Gamma_{\mathbb{C}}=\gamma_{\mathbb{C}}({\bm{\Xi}})$ in which
${\bm{\xi}}\mapsto\gamma_{\mathbb{C}}({\bm{\xi}})$ is a positive-valued
measurable mapping on ${\mathcal{C}}_{\xi}$, which is independent of
${\bm{{\mathcal{S}}}}$ and such that

$E\\{\Gamma_{\mathbb{C}}^{2}\\}=\int_{{\mathcal{C}}_{\xi}}\gamma_{\mathbb{C}}({\bm{\xi}})^{2}\,P_{\bm{\Xi}}(d{\bm{\xi}})=\underline{\gamma}_{\mathbb{C}}^{2}<+\infty\,.$
(5.14)

Equation (4.7) shows that

$\|\,[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,\|_{F}\,\,\leq\,\gamma_{\mathbb{C}}({\bm{\xi}})\,\,,\,\,\hbox{for}\,\,P_{\bm{\Xi}}\hbox{
- almost all}\,\,{\bm{\xi}}\,\,\hbox{in}\,\,{\mathcal{C}}_{\xi}\,.$ (5.15)

It can then be deduced that

$|{\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})|\,\,\leq\,\sqrt{2}\,|\Omega|^{1/2}\,\gamma_{\mathbb{C}}({\bm{\xi}})\,\|{\bm{v}}\|_{\mathbb{H}}\,,$
(5.16)

which shows that linear form ${\bm{v}}\mapsto{\mathcal{L}}^{\ell
r}({\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})$ is continuous on ${\mathbb{H}}$
for $P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in ${\mathcal{C}}_{\xi}$. Equation
(4.9) shows that, $\forall{\bm{\omega}}$ and ${\bm{\omega}}^{\prime}$ in
${\mathbb{R}}^{6}$,
$|\langle[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,{\bm{\omega}},{\bm{\omega}}^{\prime}\rangle_{2}|\,\leq\gamma_{\mathbb{C}}({\bm{\xi}})\,\|{\bm{\omega}}\|_{2}\,\|{\bm{\omega}}^{\prime}\|_{2}$
for $P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in ${\mathcal{C}}_{\xi}$. Using Eq.
(5.10) and taking into account the symmetry properties of $\varepsilon_{pq}$
and ${\mathbb{c}}_{ijpq}$ yield, for all ${\bm{u}}$ and ${\bm{v}}$ in
${\mathbb{H}}$,
$|b({\bm{u}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})|\,\,\leq\,2\,\gamma_{\mathbb{C}}({\bm{\xi}})\,\|{\bm{u}}\|_{\mathbb{H}}\,\|{\bm{v}}\|_{\mathbb{H}}$,
which shows that bilinear form $({\bm{u}},{\bm{v}})\mapsto
b({\bm{u}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})$ is continuous on
${\mathbb{H}}\times{\mathbb{H}}$ for $P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in
${\mathcal{C}}_{\xi}$. Equation (4.10) shows that,
$\forall{\bm{\omega}}\in{\mathbb{R}}^{6}$,
$\langle[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})]\,{\bm{\omega}},{\bm{\omega}}\rangle_{2}\,\,\,\geq\,\,\underline{c}_{\epsilon}\,\|{\bm{\omega}}\|_{2}^{2}$
for $P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in ${\mathcal{C}}_{\xi}$. From Eq.
(5.10), it can be deduced that, $\forall{\bm{v}}\in{\mathbb{H}}$,

$b({\bm{v}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,\,\geq\,\underline{c}_{\epsilon}\,\|{\bm{v}}\|_{\mathbb{H}}^{2}\,,$
(5.17)

which proves that bilinear form $({\bm{u}},{\bm{v}})\mapsto
b({\bm{u}},{\bm{v}};{\bm{{\mathcal{S}}}},{\bm{\xi}})$ is coercive for
$P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in ${\mathcal{C}}_{\xi}$. Due to the
continuity and coercivity of bilinear form
$b(\cdot,\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ and due to the continuity of
linear form ${\mathcal{L}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$
for $P_{\bm{\Xi}}$-almost ${\bm{\xi}}$ in ${\mathcal{C}}_{\xi}$, the use of
the Lax-Milgram theorem [28, 29] allows for proving (ii) of the Proposition.

(iii) Taking ${\bm{v}}={\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})$ in Eq. (5.12) yields
$b({\bm{u}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}}),{\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})$
$=|{\mathcal{L}}^{\ell r}({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})|$.
From Eq. (5.16), it can be deduced that $|{\mathcal{L}}^{\ell
r}({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})|\,\,\leq$
$\sqrt{2}\,|\Omega|^{1/2}\,\gamma_{\mathbb{C}}({\bm{\xi}})\,\|{\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{\mathbb{H}}$ and using Eq. (5.17)
yield $\underline{c}_{\epsilon}\,\|{\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{\mathbb{H}}^{2}\leq
b({\bm{u}}^{\ell r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}}),$ ${\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})$.
Consequently, we obtain

$\|{\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{\mathbb{H}}\,\,\,\leq\frac{\sqrt{2}\,|\Omega|^{1/2}}{\underline{c}_{\epsilon}}\,\gamma_{\mathbb{C}}({\bm{\xi}})\,.$
(5.18)

Finally,

$E\\{\|{\bm{U}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}})\|_{\mathbb{H}}^{2}\\}=\int_{{\mathcal{C}}_{\xi}}\|{\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{\mathbb{H}}^{2}\,P_{\bm{\Xi}}(d{\bm{\xi}})\leq{2\,|\Omega|}\,\underline{c}_{\epsilon}^{-2}\int_{{\mathcal{C}}_{\xi}}\gamma_{\mathbb{C}}({\bm{\xi}})^{2}\,P_{\bm{\Xi}}(d{\bm{\xi}})$

and using Eq. (5.14) yield $E\\{\|{\bm{U}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}})\|_{\mathbb{H}}^{2}\\}\\!=\\!2|\Omega|\underline{\gamma}_{\mathbb{C}}^{2}/\underline{c}_{\epsilon}^{2}$
that is Eq. (5.13) with
$\underline{\gamma}_{u}^{2}=2\,|\Omega|\underline{\gamma}_{\mathbb{C}}^{2}/\underline{c}_{\epsilon}^{2}$.

###  5.5 Random eigenvalues of the random effective elasticity matrix

For
${\bm{{\mathcal{S}}}}=\\{{\bm{w}},[y]\\}\in{\mathcal{C}}_{\mathcal{S}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$,
the random effective elasticity matrix $[{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})]$ defined in Section 5.2 can be written as
$[{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}})]=[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\Xi}})]$ in which
${\bm{\xi}}\mapsto[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\xi}})]$ is a ${\mathbb{M}}^{+}_{6}$-valued
measurable mapping on ${\mathcal{C}}_{\xi}\subset{\mathbb{R}}^{n_{\xi}}$. For
all ${\bm{{\mathcal{S}}}}\in{\mathcal{C}}_{\mathcal{S}}$ and
${\bm{\xi}}\in{\mathcal{C}}_{\xi}$, let
$\lambda_{1}({\bm{{\mathcal{S}}}},{\bm{\xi}})\geq\ldots\geq\lambda_{6}({\bm{{\mathcal{S}}}},{\bm{\xi}})>0$
be the eigenvalues of matrix $[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\xi}})]\in{\mathbb{M}}_{6}^{+}$ and let
${\bm{\lambda}}({\bm{{\mathcal{S}}}},{\bm{\xi}})=(\lambda_{1}({\bm{{\mathcal{S}}}},{\bm{\xi}}),\ldots,\lambda_{6}({\bm{{\mathcal{S}}}},{\bm{\xi}}))\in({\mathbb{R}}^{+*})^{6}$.
Let ${\bm{S}}=\\{{\bm{W}},[{\bm{Y}}]\\}$ be the
${\mathbb{R}}^{3}\times{\mathbb{M}}_{3,\widehat{\nu}_{s}}$-valued random
variable defined in Definition 5 and Proposition 2-(ii), for which the support
of its probability measure $P_{\bm{S}}=P_{\bm{W}}(d{\bm{w}})\otimes
P_{[{\bm{Y}}]}(dy)$ is subset
${\mathcal{C}}_{\bm{{\mathcal{S}}}}={\mathcal{C}}_{w}\times{\mathcal{C}}_{y}$
of ${\mathbb{R}}^{3}\times{\mathbb{M}}_{3,\widehat{\nu}_{s}}$. The random
effective elasticity matrix $[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]$, which corresponds to the elasticity random field
$\widetilde{\mathbb{C}}$ for which its spectral measure is uncertain, can then
be written as $[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]=[{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}({\bm{S}})]=[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{S}},{\bm{\Xi}})]$. Let
$\\{\widetilde{\Lambda}_{j}=\lambda_{j}({\bm{S}},{\bm{\Xi}}),j=1,\ldots,6\\}$
be the ordered (a.s) random eigenvalues of
$[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}]$. Let
$\widetilde{\bm{\Lambda}}=(\widetilde{\Lambda}_{1},\ldots,\widetilde{\Lambda}_{6})$
be the ${\mathbb{R}}^{6}$-valued random variable whose support of its
probability measure $P_{\widetilde{\bm{\Lambda}}}(d\widetilde{\bm{\lambda}})$
is $({\mathbb{R}}^{+*})^{6}$. The operator norm of
$[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}]$ is
$\|\,[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]\,\|_{2}=\widetilde{\Lambda}_{1}$.

######  Corollary 2 (Second-order properties of the random eigenvalues)

Under proposition 5, $\widetilde{\bm{\Lambda}}$ is a second-order
${\mathbb{R}}^{6}$-valued random variable,

$E\\{\|\widetilde{\bm{\Lambda}}\|_{2}^{2}\\}=\gamma_{\lambda}^{2}<+\infty\,.$
(5.19)

###### Proof 10

(Corollary 2). From Eqs. (5.3), (5.4), and (5.11), it can be deduced that

${\mathbb{c}}_{ij\ell r}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\xi}})=-|\Omega|^{-1}{\mathcal{L}}^{ij}({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})+|\Omega|^{-1}\int_{\Omega}{\mathbb{c}}_{ij\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,d{\bm{x}}\,.$

Since $(a+b)^{2}\leq 2(a^{2}+b^{2})$, we have

$\|{\bm{\lambda}}({\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{2}^{2}\,=\|\,[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\xi}})]\,\|_{F}^{2}=\sum_{i\leq j,\ell\leq
r}{\mathbb{c}}_{ij\ell r}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}},{\bm{\xi}})^{2}\leq 2|\Omega|^{-2}\sum_{i\leq
j,\ell\leq r}\left\\{|{\mathcal{L}}^{ij}({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})|^{2}+(\int_{\Omega}{\mathbb{c}}_{ij\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,d{\bm{x}})^{2}\right\\}\,.$

Using Eq. (5.15) allows us to write $\sum_{i\leq j,\ell\leq
r}(\int_{\Omega}{\mathbb{c}}_{ij\ell
r}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}})\,d{\bm{x}})^{2}\leq|\Omega|\int_{\Omega}\|\,[{\mathbb{c}}({\bm{x}};{\bm{{\mathcal{S}}}},{\bm{\xi}}))]\,\|_{F}^{2}\,d{\bm{x}}\leq|\Omega|^{2}\,\gamma_{\mathbb{C}}({\bm{\xi}})^{2}$.
Using Eq. (5.16) and (5.18) yields $\sum_{i\leq j,\ell\leq
r}|{\mathcal{L}}^{ij}({\bm{u}}^{\ell
r}(\cdot;{\bm{{\mathcal{S}}}},{\bm{\xi}});{\bm{{\mathcal{S}}}},{\bm{\xi}})|^{2}\leq
84\,\underline{c}_{\epsilon}^{-2}\,|\Omega|^{2}\,\gamma_{\mathbb{C}}({\bm{\xi}})^{4}$.
It can then be deduced the inequality
$\|{\bm{\lambda}}({\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{2}^{2}\,\,\leq
168\,\underline{c}_{\epsilon}^{-2}\,\gamma_{\mathbb{C}}({\bm{\xi}})^{4}+2\,\gamma_{\mathbb{C}}({\bm{\xi}})^{2}$
and consequently, we have
$E\\{\|\widetilde{\bm{\Lambda}}\|_{2}^{2}\\}=\int_{{\mathcal{C}}_{\mathcal{S}}}\int_{{\mathcal{C}}_{\xi}}\|{\bm{\lambda}}({\bm{{\mathcal{S}}}},{\bm{\xi}})\|_{2}^{2}\,P_{\bm{S}}(d{\bm{s}})\otimes
P_{\bm{\Xi}}(d{\bm{\xi}})\,\,\leq
168\,\underline{c}_{\epsilon}^{-2}\,E\\{\Gamma_{\mathbb{C}}^{4}\\}+2\,E\\{\Gamma_{\mathbb{C}}^{2}\\}$.
Using Eq. (4.8) yields Eq. (5.19) with
$\gamma_{\lambda}^{2}=168\,\underline{c}_{\epsilon}^{-2}\,\underline{\gamma}_{4,{\mathbb{C}}}^{4}+2\,\underline{\gamma}_{2,{\mathbb{C}}}^{2}$.

###  5.6 Brief comments about numerical aspects of stochastic solver

The Monte Carlo simulation method [30, 31] is used as stochastic solver. Let
$\\{({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa})\in{\mathcal{C}}_{\mathcal{S}}\times{\mathcal{C}}_{\xi},\,\kappa=1,\ldots,\kappa_{\hbox{{\ppppcarac
sim}}}\\}$ be $\kappa_{\hbox{{\ppppcarac sim}}}$ independent realizations of
random variables $({\bm{S}},{\bm{\Xi}})$ using the generator of probability
measures $P_{\bm{S}}$ (see Section 5.5) and $P_{\bm{\Xi}}$ (see Section
5.4-(i)). The spatial discretization of the weak formulation defined by Eq.
(5.12) of the stochastic BVP and the discretization of Eq. (5.3) with Eq.
(5.4) can be performed by the finite element method. Using Section 5.5,
$\lambda_{1}({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa})\geq\ldots\geq\lambda_{6}({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa})>0$
are computed as the eigenvalues of $[{\mathbb{c}}^{\hbox{{\ppppcarac
eff}}}({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa})]\in{\mathbb{M}}^{+}_{6}$.
Taking into account Eq. (5.19), the mean-square convergence of the random
eigenvalues can be analyzed with the convergence function

$\kappa_{\hbox{{\ppppcarac sim}}}\mapsto\hbox{{\pcarac
conv}}\,(\kappa_{\hbox{{\ppppcarac
sim}}})=\|\,[\underline{{\mathbb{C}}}]\,\|_{F}^{-1}\,(\kappa_{\hbox{{\ppppcarac
sim}}}^{-1}\sum_{\kappa=1}^{\kappa_{\hbox{{\pppppcarac
sim}}}}\|{\bm{\lambda}}({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa})\|^{2}_{2})^{1/2}\,.$
(5.20)

For a given tolerance of convergence, the probability density function (pdf)
$\widetilde{\bm{\lambda}}\mapsto
p_{\widetilde{\bm{\Lambda}}}(\widetilde{\bm{\lambda}})$ on ${\mathbb{R}}^{6}$
(with support $({\mathbb{R}}^{+*})^{6}$) with respect to the Lebesgue measure
$d\widetilde{\bm{\lambda}}$ can be estimated with the
$\kappa_{\hbox{{\ppppcarac sim}}}$ independent realizations
$\\{{\bm{\lambda}}({\bm{{\mathcal{S}}}}^{\kappa},{\bm{\xi}}^{\kappa}),\kappa=1,\ldots,\kappa_{\hbox{{\ppppcarac
sim}}}\\}$ using, for instance, the multidimensional Gaussian kernel-density
estimation method [32]. The pdf $\widetilde{\lambda}_{1}\mapsto
p_{\widetilde{\Lambda}_{1}}(\widetilde{\lambda}_{1})$ of
$\widetilde{\Lambda}_{1}$ can also be estimated yielding the pdf of the
operator norm $\|\,[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]\,\|_{2}=\widetilde{\Lambda}_{1}$.

##  6 Numerical illustration

(i) Mean model of the microstructure. Domain $\Omega=(]0,1[)^{3}$ and the mean
model of the elastic material is chosen in the orthotropic class with mean
Young moduli $\underline{E}_{1}=10^{10}$, $\underline{E}_{2}=0.5\times
10^{10}$, and $\underline{E}_{3}=0.1\times 10^{10}$, with mean Poisson
coefficients $\underline{\nu}_{23}=0.25$, $\underline{\nu}_{31}=0.15$, and
$\underline{\nu}_{12}=0.1$ (the International System of Units is used).

(ii) Elasticity random field. The hyperparameter $\delta_{c}$ that allows for
controlling the level of statistical fluctuations of
$[{\bm{C}}({\bm{x}};{\bm{{\mathcal{S}}}})]$ (see Hypothesis 1) is fixed to the
value $0.4$.

(iii) Uncertain spectral measure. The model of the spectral measure is the one
described in Example 1. The probability measure $P_{\bm{W}}(d{\bm{w}})$ of
${\bm{W}}$ (see Definition 5) is chosen as uniform on ${\mathcal{C}}_{w}$. For
all $j\in\\{1,2,3\\}$, the mean value of the random correlation length
$L_{cj}=\pi/K_{j}$ is $\underline{L}_{c}$ and its coefficient of variation
$\delta_{L_{cj}}$ is $\delta_{L_{c}}$, which are independent of $j$.
Consequently, the support $[w_{j}^{\hbox{{\ppppcarac
min}}},w_{j}^{\hbox{{\ppppcarac max}}}]$ of the probability measure of
$W_{j}=L_{cj}$ is such that $w_{j}^{\hbox{{\ppppcarac
min}}}=\underline{L}_{c}\,(1-\sqrt{3}\,\delta_{L_{c}})$ and
$w_{j}^{\hbox{{\ppppcarac max}}}=2\,\underline{L}_{c}-w_{j}^{\hbox{{\ppppcarac
min}}}$. The hyperparameter $\delta_{s}$ that controls the level of
uncertainties of the spectral measure, which is such that
$\delta_{s}^{2}=(\Pi_{j=1}^{3}(1+\delta_{j}^{2}))-1$, is generated with
$\delta_{1}=\delta_{2}=\delta_{3}=\delta_{\hbox{{\ppppcarac unc}}}$. A
sensitivity analysis with respect to the level of spectrum uncertainties will
be performed by considering $9$ values of the triplet of parameters
$(\underline{L}_{c},\delta_{L_{c}},\delta_{\hbox{{\ppppcarac unc}}})$ with
$\underline{L}_{c}\in\\{0.2,0.4,0.6\\}$ and
$\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}\in\\{0.2,0.3,0.4\\}$. For
this set of data, the minimum of correlation lengths is $0.06$ (obtained for
$\underline{L}_{c}=0.2$ with $\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac
unc}}}=0.4$) while the maximum is $1.01$ (obtained for $\underline{L}_{c}=0.6$
with $\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0.4$). The spectral
domain sampling for the discretization of the spectral measure is performed
with $\nu_{s}=8$ and thus $\nu=8^{3}=512$. With the quadrant symmetry, we have
$\widehat{\nu}_{s}=4$ yielding $\widehat{\nu}=64$.

(iv) Finite element discretization. The weak formulation defined by Eq. (5.12)
is discretized by the finite element method. The finite element mesh is made
up of $20\times 20\times 20=8\,000$ solid finite elements (8-nodes solid),
$9\,261$ nodes, and $27\,783$ degrees of freedom (dof). There are $2\,402$
nodes on the boundary and thus $7\,206$ zeros Dirichlet conditions. There are
$2^{3}$ integrations points in each finite element, which yields $64\,000$
integrations points. The spatial discretization of the
${\mathbb{M}}^{+}_{6}$-valued elasticity random field
$[{\bm{C}}(\cdot,{\bm{{\mathcal{S}}}})]$ yields $21\times 64\,000=1\,344\,000$
random terms (taking into account the symmetry).

(v) Stochastic solver. The Monte Carlo simulation method is performed for
$\kappa_{\hbox{{\ppppcarac sim}}}\in[1,2\,000]$. Figure 1-(left) displays the
convergence function $\kappa_{\hbox{{\ppppcarac sim}}}\mapsto\hbox{{\pcarac
conv}}\,(\kappa_{\hbox{{\ppppcarac sim}}})$ defined by Eq. (5.20) for
$\underline{L}_{c}=0.2$ and with no uncertainty in the spectral measure
($\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0$) and for
$\underline{L}_{c}\in\\{0.2,0.4,0.6\\}$ with the largest uncertainties in the
spectral measure, $\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0.4$, which
is the most unfavorable value with respect to convergence. It can be seen that
the mean-square convergence is obtained for $\kappa_{\hbox{{\ppppcarac
sim}}}=2\,000$.

Figure 1: Left figure: function $\kappa_{\hbox{{\ppppcarac
sim}}}\mapsto\hbox{{\pcarac conv}}\,(\kappa_{\hbox{{\ppppcarac sim}}})$ for
$\delta_{\hbox{{\ppppcarac unc}}}=0$ and $\underline{L}_{c}=0.2$ (dashed
line); $\delta_{\hbox{{\ppppcarac unc}}}=0.4$ and $\underline{L}_{c}=0.2$
(black thin line), $0.4$ (blue med line), and $0.6$ (red thick line). Right
figure: $E\\{\widetilde{\Lambda}_{1}\\}$ as a function of
$\delta_{\hbox{{\ppppcarac unc}}}$ for $\underline{L}_{c}=0.2$ (circle
marker), $0.4$ (square marker), and $0.6$ (triangle-up marker).

Figure 2: Sensitivity of the pdf $\lambda_{1}\mapsto
p_{\Lambda_{1}}(\lambda_{1})$ for $\underline{L}_{c}=0.2$ (top left figure),
$0.4$ (top right figure), and $0.6$ (down figure) as a function of the
spectral measure uncertainty level: $\delta_{\hbox{{\ppppcarac unc}}}=0$
(dashed line), $0.2$ (black thin line), $0.3$ (blue med line), $0.4$ (red
thick line).

(vi) Sensitivity of the probability density function of the normalized
operator norm of the random effective elasticity matrix as a function of the
uncertainty level of the spectral measure. Let
$\Lambda_{1}=\widetilde{\Lambda}_{1}/E\\{\widetilde{\Lambda}_{1}\\}$ be the
normalized operator norm $\|\,[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]\,\|/E\\{[\widetilde{\mathbb{C}}^{\hbox{{\ppppcarac
eff}}}]\,\|\\}=\Lambda_{1}$. For $\underline{L}_{c}=0.2$, $0.3$, $0.4$, Fig.
1-(right) displays the graph of $E\\{\widetilde{\Lambda}_{1}\\}$ as a function
of the level $\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}$ of the spectral
measure uncertainties. Figure 2 shows the pdf $\lambda_{1}\mapsto
p_{\Lambda_{1}}(\lambda_{1})$ of the normalized operator norm of the random
effective elasticity matrix for $\underline{L}_{c}=0.2$, $0.4$, and $0.6$,
with no uncertainty in the spectral measure
($\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0$) and with uncertainties
$\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0.2$, $0.3$, and $0.4$.

(vii) Sensitivity of the probabilistic analysis of the RVE size with respect
to the uncertainty level of the spectral measure. We analyze the random
largest eigenvalue $\Lambda_{1}$ (normalized operator norm). Let $\eta$ be a
positive real number and let $\eta\mapsto P(\eta)$ be the function from
$]0,1]$ into $[0,1]$ defined by

$P(\eta)=\hbox{Proba}\\{1-\eta<\Lambda_{1}\leq
1+\eta\\}=F_{\Lambda_{1}}(1+\eta)-F_{\Lambda_{1}}(1-\eta)\,,$ (6.1)

in which $F_{\Lambda_{1}}$ is the cumulative distribution function of
$\Lambda_{1}$. Figure 3 shows the sensitivity of the graph of function
$\eta\mapsto P(\eta)$ for $\underline{L}_{c}=0.2$, $0.4$, and $0.6$ with
respect to the level of uncertainties in the spectral measure for
$\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0$, $0.2$, $0.3$, and $0.4$.
Table 1 yields an extraction from Fig. 3 of the probability levels.

Figure 3: Sensitivity of the graph of function $\eta\mapsto P(\eta)$ for $\underline{L}_{c}=0.2$ (top left figure), $0.4$ (top right figure), and $0.6$ (down figure) with respect to the level of uncertainties in the spectral measure for $\delta_{\hbox{{\ppppcarac unc}}}=0$ (dashed line), $0.2$ (circle marker), $0.3$ (square marker), and $0.4$ (no marker). Table 1: Sensitivity of the probabilistic analysis of the RVE size with respect to the uncertainty level of the spectral measure. $\underline{L}_{c}$ | $\delta_{\hbox{{\ppppcarac unc}}}$ | $P\\{0.98<\Lambda_{1}\leq 1.02\\}$ | $P\\{0.96<\Lambda_{1}\leq 1.04\\}$ | $P\\{0.92<\Lambda_{1}\leq 1.08\\}$
---|---|---|---|---
0.2 | 0.0 | 0.365 | 0.655 | 0.942
| 0.2 | 0.385 | 0.683 | 0.945
| 0.3 | 0.430 | 0.721 | 0.948
| 0.4 | 0.475 | 0.705 | 0.954
0.4 | 0.0 | 0.171 | 0.332 | 0.610
| 0.2 | 0.175 | 0.344 | 0.625
| 0.3 | 0.185 | 0.360 | 0.650
| 0.4 | 0.202 | 0.395 | 0.700
0.6 | 0.0 | 0.108 | 0.230 | 0.442
| 0.2 | 0.125 | 0.240 | 0.465
| 0.3 | 0.130 | 0.260 | 0.488
| 0.4 | 0.145 | 0.280 | 0.526

(viii) Brief discussion. When there are no uncertainties in the spectral
measure ($\delta_{\hbox{{\ppppcarac unc}}}=0$), Figure 3 and Table 1 shows
that the condition to obtain a scale separation is not really obtained
because, for $\underline{L}_{c}=0.2$ and $\delta_{\hbox{{\ppppcarac unc}}}=0$
it can be seen that $\hbox{Proba}\\{0.98<\Lambda_{1}\leq 1.02\\}=0.365$ and
the probability becomes $0.942$ only for $\\{0.92<\Lambda_{1}\leq 1.08\\}$.
The results show in Table 1 shows that, for the specific case analyzed (in
particular choosing the same level of uncertainties for the spatial
correlation lengths and for the values of the spectral density function) and
contrary to what was expected, the introduction of spectral measure
uncertainties improves the scale separation from a probabilistic analysis
point of view. In fact, for $\underline{L}_{c}=0.2$ and for
$\delta_{L_{c}}=\delta_{\hbox{{\ppppcarac unc}}}=0.4$, the minimum value of
the realizations of the random correlation length is $0.06$, value less than
$10$ percent of the characteristic length of the dimensions of domain
$\Omega$, for which the scale separation can be obtained. It should be noted
that these results are presented as an illustration of the use of the proposed
mathematical construction of a random field with uncertainties in the spectral
measure in order to improve the probabilistic analysis of stochastic
homogenization of random elastic media. More advanced computational analyses
should be performed with this probabilistic model in order to deeply analyze
the role played by uncertain spectral measure for stochastic homogenization of
random elastic media, in particular in taking different values of
uncertainties for the spatial correlation lengths and for the spectral density
function.

## References

  * [1] M. I. Yadrenko, Spectral theory of random fields, Optimization Software, 1983.
  * [2] Y. Rozanov, Random Fields and Stochastic Partial Differential Equations, Kluwer Academic Publishers, 1998.
  * [3] R. Adler, The geometry of random fields, SIAM, 2010.
  * [4] E. Vanmarcke, Random Fields: Analysis and Synthesis, World Scientific, Singapore, 2010.
  * [5] M. Rosenblatt, Stationary sequences and random fields, Springer Science & Business Media, 2012.
  * [6] M. Ostoja-Starzewski, Random field models of heterogeneous materials, International Journal of Solids and Structures 35 (19) (1998) 2429–2455.  doi:10.1016/S0020-7683(97)00144-3.
  * [7] C. Soize, Uncertainty Quantification, Vol. 47 of Interdisciplinary Applied Mathematics, Springer, New York, 2017.  doi:10.1007/978-3-319-54339-0.
  * [8] A. Malyarenko, M. Ostoja-Starzewski, Tensor-valued random fields for continuum physics, Cambridge University Press, 2018.
  * [9] C. Soize, Non gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Computer Methods in Applied Mechanics and Engineering 195 (1-3) (2006) 26–64.  doi:10.1016/j.cma.2004.12.014.
  * [10] A. Nouy, C. Soize, Random field representations for stochastic elliptic boundary value problems and statistical inverse problems, European Journal of Applied Mathematics 25 (3) (2014) 339–373.  doi:10.1017/S0956792514000072.
  * [11] C. Soize, Random vectors and random fields in high dimension: parametric model-based representation, identification from data, and inverse problems, in: R. Ghanem, D. Higdon, H. Owhadi (Eds.), Handbook of Uncertainty Quantification, Vol. 2, Springer, Cham, Switzerland, 2017, Ch. 26, pp. 883–936.
  * [12] J. Guilleminot, C. Soize, On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, Journal of Elasticity 111 (2) (2013) 109–130.  doi:10.1007/s10659-012-9396-z.
  * [13] J. Guilleminot, C. Soize, Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media, Multiscale Modeling & Simulation (A SIAM Interdisciplinary Journal) 11 (3) (2013) 840–870.  doi:10.1137/120898346.
  * [14] A. V. Skorokhod, M. I. Yadrenko, On absolute continuity of measures corresponding to homogeneous gaussian fields, Theory of Probability & Its Applications 18 (1) (1973) 27–40.  doi:10.1137/1118002.
  * [15] P. Krée, C. Soize, Mathematics of Random Phenomena, Reidel Pub. Co, 1986, (first published by Bordas in 1983 and also published by Springer Science & Business Media in 2012).
  * [16] N. Leonenko, A. Olenko, Tauberian and abelian theorems for long-range dependent random fields, Methodology and Computing in Applied Probability 15 (4) (2013) 715–742.  doi:10.1007/s11009-012-9276-9.
  * [17] A. Malyarenko, M. Ostoja-Starzewski, Statistically isotropic tensor random fields: correlation structures, Mathematics and Mechanics of Complex Systems 2 (2) (2014) 209–231.  doi:10.2140/memocs.2014.2.209.
  * [18] C. Soize, A nonparametric model of random uncertainties on reduced matrix model in structural dynamics, Probabilistic Engineering Mechanics 15 (3) (2000) 277–294.  doi:10.1016/S0266-8920(99)00028-4.
  * [19] J. Doob, Stochastic processes, John Wiley & Sons, New York, 1953.
  * [20] I. I. Guikhman, A. Skorokhod, Introduction à la théorie des processus aléatoires, Edition Mir, 1980.
  * [21] F. Poirion, C. Soize, Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous gaussian vector fields, in: P. Krée, W. Wedig (Eds.), Probabilistic Methods in Applied Physics, Springer-Verlag, Berlin, 1995, pp. 17–53.  doi:10.1007/3-540-60214-3-50.
  * [22] C. Soize, Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size, Probabilistic Engineering Mechanics 23 (2-3) (2008) 307–323.  doi:10.1016/j.probengmech.2007.12.019.
  * [23] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, International Journal of solids and structures 40 (13-14) (2003) 3647–3679.  doi:10.1016/S0020-7683(03)00143-4.
  * [24] M. Ostoja-Starzewski, Material spatial randomness: From statistical to representative volume element, Probabilistic engineering mechanics 21 (2) (2006) 112–132.  doi:10.1016/j.probengmech.2005.07.007.
  * [25] M. Ostoja-Starzewski, X. Du, Z. Khisaeva, W. Li, Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures, International Journal for Multiscale Computational Engineering 5 (2) (2007) 73–82.  doi:10.1615/IntJMultCompEng.v5.i2.10.
  * [26] M. Bornert, T. Bretheau, P. Gilormini, Homogenization in Mechanics of Materials, ISTE Ltd and John Wiley and Sons, New York, 2008.
  * [27] R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer Science & Business Media, Berlin, 2013.
  * [28] P. Lax, A. Milgram, Parabolic equations: Contributions to the theory of partial differential equations, Annals of Mathematical Studies (33) (1954) 167–190.
  * [29] J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications: Vol. 1, Vol. 181, Springer Science & Business Media, 2012.  doi:10.1007/978-3-642-65161-8.
  * [30] C. Robert, G. Casella, Monte Carlo Statistical Methods, Springer Science & Business Media, 2005.  doi:10.1007/978-1-4757-4145-2.
  * [31] R. Rubinstein, D. Kroese, Simulation and the Monte Carlo Method, Second Edition, John Wiley & Sons, New York, 2008.
  * [32] A. Bowman, A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernel Approach With S-Plus Illustrations, Vol. 18, Oxford University Press, Oxford: Clarendon Press, New York, 1997.  doi:10.1007/s001800000033.