$\underline{{\bm{x}}}=(1/N_{d})\,\sum_{j=1}^{N_{d}}{\bm{x}}_{d}^{j}\in{\mathbb{R}}^{n_{x}}$.
Let
$[x_{\rm{ctr}}]=[{\bm{x}}_{\rm{ctr}}^{1}\ldots{\bm{x}}_{\rm{ctr}}^{N_{d}}]$ be
the matrix in ${\mathbb{M}}_{n_{x},\,N_{d}}$ and let
$[\Phi]\,[S]\,[\Psi]^{T}=[x_{\rm{ctr}}]$ be the thin SVD [90] (economy size
SVD) of matrix $[x_{\rm{ctr}}]$. The diagonal entries of diagonal matrix $[S]$
are the singular values $S_{1}\geq\ldots\geq S_{N_{d}-1}>S_{N_{d}}=0$ that are
in decreasing order and we have $S_{N_{d}}=0$. The matrix $[\Phi]$ is in
${\mathbb{M}}_{n_{x},\nu}$ with $\nu=N_{d}-1$ and
$[\Phi]^{T}\,[\Phi]=[I_{\nu}]$. Then random vector ${\bm{X}}$ can be written
as

${\bm{X}}=\underline{{\bm{x}}}+[\Phi]\,[\kappa]^{1/2}\,{\bm{H}}\,,$ (5.20)

in which $[\kappa]$ is the diagonal matrix such that
$\kappa_{\alpha}=[\kappa]_{\alpha\alpha}=S_{\alpha}^{2}/(N_{d}-1)$, and where
${\bm{H}}$ is the ${\mathbb{R}}^{\nu}$-valued random variable whose $N_{d}$
independent realizations are computed by

${\bm{\eta}}_{d}^{j}=[\kappa]^{-1/2}\,[\Phi]^{T}({\bm{x}}_{d}^{j}-\underline{{\bm{x}}})\quad,\quad
j=1,\ldots,N_{d}\,.$ (5.21)

Note that $\\{\kappa_{\alpha}\\}_{\alpha}$ are the eigenvalues of the
covariance matrix of ${\bm{X}}$ estimated with
$\\{{\bm{x}}_{d}^{1},\ldots,{\bm{x}}_{d}^{N_{d}}\\}$. Random vector ${\bm{H}}$
is then normalized. The empirical estimation of its mean value and its
covariance matrix are given by Eq. (3.1) (centered and identity matrix). From
Eq. (5.20), it can be deduced that, for $1\leq m\leq r\leq 3$, we have

$\displaystyle{\bm{\mathcal{Y}}}^{mr}=\underline{{\bm{y}}}^{mr}+[\Phi_{y}^{mr}]\,[\kappa]^{1/2}\,{\bm{H}}\quad$
$\displaystyle,\quad\underline{{\bm{y}}}^{mr}\in{\mathbb{R}}^{n_{y}}\quad,\quad[\Phi_{y}^{mr}]\in{\mathbb{M}}_{n_{y},\nu}\,,$
(5.22)
$\displaystyle{\bm{\mathcal{G}}}=\underline{{\bm{g}}}+[\Phi_{g}]\,\,[\kappa]^{1/2}\,{\bm{H}}\quad$
$\displaystyle,\quad\underline{{\bm{g}}}\in{\mathbb{R}}^{n_{g}}\quad,\quad[\Phi_{g}]\in{\mathbb{M}}_{n_{g},\nu}\,,$
(5.23)
$\displaystyle{\bm{W}}=\underline{{\bm{w}}}+[\Phi_{w}]\,\,[\kappa]^{1/2}\,{\bm{H}}\quad$
$\displaystyle,\quad\underline{{\bm{w}}}\in{\mathbb{R}}^{n_{w}}\quad,\quad[\Phi_{w}]\in{\mathbb{M}}_{n_{w},\nu}\,.$
(5.24)

The training set ${\mathcal{D}}_{d}$ introduced in Definition 1 is written as

${\mathcal{D}}_{d}=\\{{\bm{\eta}}_{d}^{1},\ldots,{\bm{\eta}}_{d}^{N_{d}}\\}\quad,\quad{\bm{\eta}}_{d}^{j}\in{\mathbb{R}}^{\nu}\,.$
(5.25)

Figure 1 displays the distribution of the eigenvalues $\kappa_{\alpha}$ for
the 3 cases, SC1, SC2, and SC3 of scale separation, computed with the training
set and the prior probability model.

Figure 1: For cases, SC1, SC2, and SC3, distribution of the eigenvalues
$\kappa_{\alpha}$.

###  5.7 Definition of the random normalized residue induced by the use of the
constrained learned set

The construction of the random normalized residue is based on a similar
approach of the one that has been used in [26]. Let us consider the
constrained learned set
${\mathcal{D}}_{{\bm{H}}_{{\bm{\lambda}}^{\,i}}}=\\{{\bm{\eta}}_{{\bm{\lambda}}^{\,i}}^{1},\ldots,{\bm{\eta}}_{{\bm{\lambda}}^{\,i}}^{N}\\}$
generated with Algorithm 1 for iteration $i$. Using Eqs. (5.22) to (5.24), for
$\ell\in\\{1,\ldots,N\\}$, the corresponding realizations
${\bm{y}}_{{\bm{\lambda}}^{\,i}}^{mr,\ell}\in{\mathbb{R}}^{n_{y}}$ for $1\leq
m\leq r\leq 3$,
${\bm{g}}_{{\bm{\lambda}}^{\,i}}^{\ell}\in{\mathbb{R}}^{n_{g}}$, and
${\bm{w}}_{{\bm{\lambda}}^{\,i}}^{\ell}\in{\mathbb{R}}^{n_{w}}$, are computed
by

$\displaystyle{\bm{y}}_{{\bm{\lambda}}^{\,i}}^{mr,\ell}$
$\displaystyle=\underline{{\bm{y}}}^{mr}+[\Phi_{y}^{mr}]\,[\kappa]^{1/2}\,{\bm{\eta}}_{{\bm{\lambda}}^{\,i}}^{\ell}\,,$
(5.26) $\displaystyle{\bm{g}}_{{\bm{\lambda}}^{\,i}}^{\ell}$
$\displaystyle=\underline{{\bm{g}}}+[\Phi_{g}]\,\,[\kappa]^{1/2}\,{\bm{\eta}}_{{\bm{\lambda}}^{\,i}}^{\ell}\,,$
(5.27) $\displaystyle{\bm{w}}_{{\bm{\lambda}}^{\,i}}^{\ell}$
$\displaystyle=\underline{{\bm{w}}}+[\Phi_{w}]\,\,[\kappa]^{1/2}\,{\bm{\eta}}_{{\bm{\lambda}}^{\,i}}^{\ell}\,.$
(5.28)

For $1\leq m\leq r\leq 3$ and for $\ell\in\\{1,\ldots,N\\}$ the realization
${\bm{\mathcal{R}}}_{{\bm{\lambda}}^{\,i}}^{\,mr,\ell}$ of the
${\mathbb{R}}^{n_{y}}$-valued random residue are computed using Eq. (5.9) and
Eqs. (5.26) to (5.28),
${\bm{\mathcal{R}}}_{{\bm{\lambda}}^{\,i}}^{\,mr,\ell}={\bm{\mathcal{N}}}^{mr}({\bm{y}}_{{\bm{\lambda}}^{\,i}}^{mr,\ell},{\bm{g}}_{{\bm{\lambda}}^{\,i}}^{\ell},{\bm{w}}_{{\bm{\lambda}}^{\,i}}^{\ell})$.
We define the realization $\hat{\rho}_{{\bm{\lambda}}^{\,i}}^{\ell}$ of the
random residue $\hat{\rho}_{{\bm{\lambda}}^{\,i}}$ by

$\hat{\rho}_{{\bm{\lambda}}^{\,i}}^{\ell}=\frac{1}{\sqrt{6\,n_{y}}}\left(\sum_{1\leq
m\leq r\leq
3}\|\,{\bm{\mathcal{R}}}_{{\bm{\lambda}}^{\,i}}^{\,mr,\ell}\,\|^{2}\right)^{1/2}\,.$
(5.29)

Finally, we define the realization $\rho_{{\bm{\lambda}}^{\,i}}^{\ell}$ of the
random normalized residue $\rho_{{\bm{\lambda}}^{\,i}}$ by

$\rho_{{\bm{\lambda}}^{\,i}}^{\ell}=\frac{\hat{\rho}_{{\bm{\lambda}}^{\,i}}^{\ell}}{\hat{\underline{\rho}}_{\,0}}\,,$
(5.30)

in which $\hat{\underline{\rho}}_{\,0}$ is the estimation of
$E\\{\,\rho_{{\bm{\lambda}}^{1}}\\}$ using the constrained learned set
${\mathcal{D}}_{{\bm{H}}_{{\bm{\lambda}}^{1}}}$ of the first iteration $i=1$.
Since for $i=1$, ${\bm{\lambda}}^{\,i-1}={\bm{\lambda}}^{\,0}={\hbox{\bf
0}}_{n_{c}}$, $\rho_{{\bm{\lambda}}^{1}}$ is the random normalized residue of
the constrained learned set without taking into account the constraints and we
have, for $i=1$,
$E\\{\,\rho_{{\bm{\lambda}}^{1}}\\}={E\\{\,\hat{\rho}_{{\bm{\lambda}}^{1}}\\}}/{\hat{\underline{\rho}}_{\,0}}=1$.

###  5.8 Defining function ${\bm{h}}^{c}$ related to the constraints and
defining the targets represented by ${\bm{b}}^{c}$

We define the function
${\bm{\eta}}\mapsto{\bm{h}}^{c}({\bm{\eta}}):{\mathbb{R}}^{\nu}\rightarrow{\mathbb{R}}^{n_{c}}$
related to the constraints (see Eq. (2.2)) and we define the target
represented by vector ${\bm{b}}^{c}$ given in ${\mathbb{R}}^{n_{c}}$. Three
constraints are introduced, the second-order moment of the random normalized
residue (see Section 5.7) and two statistical moments: the normalized mean
value of the random effective elasticity matrix
$[{\mathbb{C}}^{\hbox{{\ppppcarac eff}}}]$ and its coefficient of dispersion
(see Section 5.5). Below we consider iteration $i$, and then
${\bm{\lambda}}^{1},\ldots,{\bm{\lambda}}^{\,i-1},{\bm{\lambda}}^{\,i}$ are
known.

(i) The random normalized residue $\rho_{{\bm{\lambda}}^{\,i}}$ whose
realizations are defined by Eq. (5.30), is an implicit function of
${\bm{H}}_{{\bm{\lambda}}^{\,i}}$, that we can rewrite as
$\rho_{{\bm{\lambda}}^{\,i}}({\bm{H}}_{{\bm{\lambda}}^{\,i}})$. The second-
order moment of the normalized random residue is then written as
$E\\{(\rho_{{\bm{\lambda}}^{\,i}}({\bm{H}}_{{\bm{\lambda}}^{\,i}}))^{2}\\}$
yielding
$h_{\rho}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})=\left(\rho_{{\bm{\lambda}}^{\,i}}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\right)^{2}$.
Therefore, using the notation of Eq. (3.55), we have
$E\\{\,h_{\rho}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\\}=b^{c}_{\rho}$, in
which we choose $b_{\rho}^{c}=1$ (this value is close to the value of the
second-order moment of the random normalized residue without constraint) and
where ${\bm{\eta}}\mapsto
h_{\rho}^{c}({\bm{\eta}}):{\mathbb{R}}^{\nu}\rightarrow{\mathbb{R}}$ is a
positive-valued implicit function.

(ii) The second constraint is given by Eq. (5.13). Transforming the upper
triangular matrix of $[{\mathbb{C}}_{n}^{\hbox{{\ppppcarac
eff}}}]\in{\mathbb{M}}_{6}^{+}$ in a ${\mathbb{R}}^{21}$-vector, Eq. (5.13) is
rewritten, similarly to Eq. (3.55), as
$E\\{\,{\bm{h}}_{\mathbb{C}}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\\}={\bm{b}}_{\mathbb{C}}^{c}$,
in which
${\bm{\eta}}\mapsto{\bm{h}}_{\mathbb{C}}^{c}({\bm{\eta}}):{\mathbb{R}}^{\nu}\rightarrow{\mathbb{R}}^{21}$
is an implicit function and where
${\bm{b}}_{\mathbb{C}}^{c}\in{\mathbb{R}}^{21}$ is the reshaping of the upper
triangular matrix of $[\,{\underline{{\mathbb{C}}}}_{\,n}^{{\hbox{{\ppppcarac
exp}}}}]\in{\mathbb{M}}_{6}^{+}$ defined by Eq. (5.12).

(iii) The last constraint is given by Eq. (5.17) that is rewritten, using Eq.
(3.55), as
$E\\{\,h_{\delta}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\\}=b^{c}_{\delta}$, in
which $b_{\delta}^{c}=(\mu_{\hbox{{\ppppcarac
eff}}}\,\delta^{\,{\hbox{{\ppppcarac exp}}}}/\mu_{\hbox{{\ppppcarac
exp}}})^{2}$ and where ${\bm{\eta}}\mapsto
h_{\delta}^{c}({\bm{\eta}}):{\mathbb{R}}^{\nu}\rightarrow{\mathbb{R}}$ is a
positive-valued implicit function.

(iv) Finally, for given ${\bm{\lambda}}$, and in particular for
${\bm{\lambda}}={\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}$ yielding
${\bm{H}}^{c}={\bm{H}}_{\bm{\lambda}}$, the constraint is defined by Eq.
(3.55) with

${\bm{\eta}}\mapsto{\bm{h}}^{c}({\bm{\eta}})=(h_{\rho}^{c}({\bm{\eta}}),{\bm{h}}_{\mathbb{C}}^{c}({\bm{\eta}}),h_{\delta}^{c}({\bm{\eta}})):{\mathbb{R}}^{\nu}\rightarrow{\mathbb{R}}^{n_{c}}={\mathbb{R}}\times{\mathbb{R}}^{21}\times{\mathbb{R}}\,,$
(5.31)
$b^{c}=(b_{\rho}^{c},{\bm{b}}_{\mathbb{C}}^{c},b_{\delta}^{c})\in{\mathbb{R}}^{n_{c}}={\mathbb{R}}\times{\mathbb{R}}^{21}\times{\mathbb{R}}\,,$
(5.32)

in which $n_{c}=23$. Using the mathematical developments presented in [88], it
can be verified that ${\bm{h}}^{c}$ satisfies Hypothesis 1. We will also
assume that Eq. (2.6) holds. Indeed, verifying a priori that the components of
${\bm{h}}^{c}$ are algebraically independent is very difficult for the
considered problem. This hypothesis will indirectly be verified by checking
that the solution ${\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}$ is well
identified (see Proposition 1-(c) and Section 5.9).

###  5.9 Error function and convergence analysis of the sequence of MCMC
generator

The error function is defined by Eqs. (3.58) and (3.59). The constraint on the
random normalized residue (see Section 5.8-(i)) is introduced with
$b_{\rho}^{c}=1$ in order to avoid the increasing of
$E\\{\,h_{\rho}^{c}({\bm{H}}^{c})\\}$ with respect to
$E\\{\,h_{\rho}^{c}({\bm{H}}_{{\bm{\lambda}}^{1}})\\}$ (first iteration
without constraints effects). This constraint, which is taken into account in
Algorithm 1 for computing
$\\{{\bm{\lambda}}^{i},i=1,\ldots,i_{\hbox{{\ppppcarac max}}}\\}$, is not
taken into account in the error function (see Eq. (3.58)) to identify the
index $i_{\hbox{{\ppppcarac sol}}}$ (see Eq. (3.57)) of the optimal value
${\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}={\bm{\lambda}}^{i_{\hbox{{\ppppcarac
sol}}}}$ of ${\bm{\lambda}}^{i}$. Consequently, Eq. (3.58) is written, for
$i\in\\{1,\ldots,i_{\hbox{{\ppppcarac max}}}\\}$,

${\hbox{{err}}}(i)=\sqrt{\left({{\hbox{{err}}}_{\mathbb{C}}(i)}\,/\,{{\hbox{{err}}}_{\mathbb{C}}(1)}\right)^{2}+\left({{\hbox{{err}}}_{\delta}(i)}\,/\,{{\hbox{{err}}}_{\delta}(1)}\right)^{2}}\,,$
(5.33)
${\hbox{{err}}}_{\mathbb{C}}(i)=\|\,{\bm{b}}_{\mathbb{C}}^{c}-E\\{\,{\bm{h}}_{\mathbb{C}}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\\}\,\|\,/\,\|\,{\bm{b}}_{\mathbb{C}}^{c}\|\quad,\quad{\hbox{{err}}}_{\delta}(i)=|\,b_{\delta}^{c}-E\\{\,h_{\delta}^{c}({\bm{H}}_{{\bm{\lambda}}^{\,i}})\\}\,|\,/\,b_{\delta}^{c}\,.$
(5.34)

For the three cases, SC1, SC2,and SC3, Fig. 2 displays the error function
$i\mapsto{\hbox{{err}}}(i)$ defined by Eq. (5.33), computed with the
constrained learned set ${\mathcal{D}}_{{\bm{H}}_{{\bm{\lambda}}^{\,i}}}$ for
$N=1000$, $2000$, $6000$, and $10\,000$. It can be seen that convergence is
reached for $N=10\,000$ and that, at convergence, function
$i\mapsto{\hbox{{err}}}(i)$ is relatively smooth (that is not the case for
$N=1000$). These graphs show a good illustration of the convergence of the
sequence in $N$ of the MCMC generator using the statistical surrogate model
$\hat{\bm{h}}^{N}$ of ${\bm{h}}^{c}$ (see Proposition 4). When the convergence
is reached for $N=10\,000$, Table 3 gives the value of $i_{\hbox{{\ppppcarac
sol}}}$ such that ${\bm{\lambda}}^{\hbox{{\ppppcarac
sol}}}={\bm{\lambda}}^{\,i_{\hbox{{\ppppcarac sol}}}}$ (see Eq. (3.57)) and
the corresponding value ${\hbox{{err}}}(i_{\hbox{{\ppppcarac sol}}})$ of the
error.

Figure 2: For scale separation, SC1 (left figure), SC2 (central figure), and SC3 (right figure), graph of error function $i\mapsto{\hbox{{err}}}(i)$ for $N=1000$, $2000$, $6000$, and $10\,000$. Table 3: For cases SC1, SC2, and SC3 of scale separation, value ${\hbox{{err}}}(i_{\hbox{{\ppppcarac sol}}})$ of the error function for the solution ${\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}={\bm{\lambda}}_{i_{\hbox{{\ppppcarac sol}}}}$ computed with the constrained learned set for $N=10\,000$. Case | SC1 | SC2 | SC3
---|---|---|---
$i_{\hbox{{\ppppcarac max}}}$ | 30 | 30 | 20
$i_{\hbox{{\ppppcarac sol}}}$ | 27 | 30 | 17
${\hbox{{err}}}(i_{\hbox{{\ppppcarac sol}}})$ | 0.5924 | 0.5495 | 0.5364

###  5.10 Residue and posterior second-order moments of the random effective
elasticity matrix estimated with the constrained learned set

The posterior statistics of the random normalized residue and the random
effective elasticity matrix are estimated with the constrained learned set
${\mathcal{D}}_{{\bm{H}}^{c}}={\mathcal{D}}_{{\bm{H}}_{{\bm{\lambda}}^{{\hbox{{\ppppcarac
sol}}}}}}$ as a function of $N$ for the three cases, SC1, SC2, and SC3.

(i) The posterior second-order moment of the random normalized residue is
$E\\{\,\rho_{c}^{2}\\}$ and compared to $1$, see Section 5.8-(i).

(ii) The posterior mean value is
$[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac
eff}}}]=E\\{[{\mathbb{C}}_{\,c}^{\hbox{{\ppppcarac eff}}}]\\}$ and is compared
to $[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]$ (see Eq. (5.13)
for the normalized version). We also consider the Frobenius norm
$\|\,[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]\,\|_{F}$
that is compared to $\|\,[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac
exp}}}]\,\|_{F}$.

(iii) For the posterior dispersion coefficient of
$[{\mathbb{C}}_{\,c}^{\hbox{{\ppppcarac eff}}}]$, instead of comparing
$\delta_{c}^{\,{\hbox{{\ppppcarac eff}}}}$ to $\delta^{\,{\hbox{{\ppppcarac
exp}}}}$ (see Eq. (5.16)), it is more efficient to use the maximum likelihood
consisting in comparing $\delta_{c,{\hbox{{\pppppcarac
ML}}}}^{\,{\hbox{{\ppppcarac eff}}}}$ to $\delta^{\,{\hbox{{\ppppcarac
exp}}}}$ in which

$\delta_{c,{\hbox{{\pppppcarac ML}}}}^{\,{\hbox{{\ppppcarac
eff}}}}=\sqrt{\delta_{2,{\hbox{{\pppppcarac ML}}}}^{\,{\hbox{{\ppppcarac
eff}}}}}\quad,\quad\delta_{2,{\hbox{{\pppppcarac ML}}}}^{\,{\hbox{{\ppppcarac
eff}}}}=\max_{\delta_{2}}\,p_{\Delta_{2,C}^{\hbox{{\ppppcarac
eff}}}}(\delta_{2})\,,$ (5.35)

where $p_{\Delta_{2,C}^{\hbox{{\ppppcarac eff}}}}$ is the pdf of the random
variable $\Delta_{2,c}^{\hbox{{\ppppcarac eff}}}$ defined by Eq. (5.18), at
convergence ${\bm{\lambda}}={\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}$
(introduction of subscript ”$c$”). For the three cases, SC1, SC2, and SC3,
Fig. 3 displays the graph of the posterior pdf $\delta_{2}\mapsto
p_{\Delta_{2,C}^{\hbox{{\ppppcarac eff}}}}(\delta 2)$ of random variable
$\Delta_{2,c}^{\hbox{{\ppppcarac eff}}}$, estimated with the constrained
learned set for $N=10\,000$ and its prior counterpart estimated with the
training set (constructed using the prior probability model). Fig. 4 (left
figure) displays the graph of the Frobenius norm
$N\mapsto\|\,[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac
eff}}}(N)]\,\|_{F}$ of the posterior mean value
$[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac
eff}}}(N)]=E\\{\,[{\mathbb{C}}_{\,c}^{\hbox{{\ppppcarac eff}}}(N)]\\}$ of the
random effective elasticity matrix $[{\mathbb{C}}_{\,c}^{\hbox{{\ppppcarac
eff}}}(N)]$ as a function of $N$ and estimated using the constrained learned
set, while Fig. 4 (right figure) shows the graph of the maximum likelihood
$N\mapsto\delta_{c,{\hbox{{\pppppcarac ML}}}}^{\,{\hbox{{\ppppcarac
eff}}}}(N)$ of the coefficient of dispersion of
$[{\mathbb{C}}_{\,c}^{\hbox{{\ppppcarac eff}}}(N)]$ defined by Eq. (5.35).
Fig. 5 shows the graph of the posterior pdf $r\mapsto p_{\rho_{c}}(r)$ of the
random normalized residue $\rho_{c}$, estimated with the constrained learned
set for $N=10\,000$ and its counterpart for the estimation performed with the
training set (constructed using the prior probability model).

(iv) For cases SC1, SC2, and SC3, Table 4 gives the posterior statistics
computed with the constrained learned set for $N=10\,000$ (subscript ”c”), the
prior statistics computed with the training set (subscript ”d”), and the
targets (superscript ”exp”), for the second-order moment of the random
normalized residue, for the Frobenius norm of the mean value of the random
effective elasticity matrix, and for the coefficient of dispersion of this
random matrix. For the same three cases, Table 5 gives the values of the
entries of the mean matrices
$[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]$ computed with
the training set, $[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac
eff}}}]$ computed with the constrained learned set for $N=10\,000$, and
$[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]$ for the targets. Note
that entries $(4,5)$, $(4,6)$, and $(5,6)$, which are small with respect to
the other entries, are not given.

Figure 3: For scale separation, SC1 (left figure), SC2 (central figure), and
SC3 (right figure), graph of the posterior pdf of random variable
$\Delta_{2,c}^{\hbox{{\ppppcarac eff}}}$ (thick blue line) estimated with the
constrained learned set for $N=10\,000$ and its prior counterpart (thin black
line) corresponding to an estimation with the training set (constructed using
the prior probability model).

Figure 4: For scale separation, SC1, SC2, and SC3, graph of
$N\mapsto\|\,[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac
eff}}}(N)]\,\|_{F}$ (left figure) and $N\mapsto\delta_{c,{\hbox{{\pppppcarac
ML}}}}^{\,{\hbox{{\ppppcarac eff}}}}(N)$ (right figure), computed with the
constrained learned set.

Figure 5: For scale separation, SC1 (left figure), SC2 (central figure), and SC3 (right figure), graph of the posterior pdf of random normalized residue $\rho_{c}$ (solid line) estimated with the constrained learned set for $N=10\,000$ and its prior counterpart (dotted line) corresponding to an estimation with the training set (constructed using the prior probability model). Table 4: For cases SC1, SC2, and SC3, posterior statistics computed with the constrained learned set for $N=10\,000$ (subscript ”c”), prior statistics (subscript ”d”) computed with the training set, and targets. | SC1 | SC2 | SC3
---|---|---|---
$E\\{\rho_{c}^{2}\\}$ | 1.2938 | 1.2687 | 1.2413
$b_{\rho}^{c}$ | 1 | 1 | 1
$\|\,[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]\,\|_{F}\times 10^{11}$ | 4.2106 | 4.1925 | 4.1943
$\|\,[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]\,\|_{F}\times 10^{11}$ | 4.6294 | 4.6923 | 4.6816
$\|\,[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]\,\|_{F}\times 10^{11}$ | 4.6317 | 4.6549 | 4.6706
$\delta^{\,{\hbox{{\ppppcarac eff}}}}_{d,{\hbox{{\pppppcarac ML}}}}$ | 0.2257 | 0.2469 | 0.2701
$\delta^{\,{\hbox{{\ppppcarac eff}}}}_{c,{\hbox{{\pppppcarac ML}}}}$ | 0.1329 | 0.1476 | 0.1671
$\delta^{\,{\hbox{{\ppppcarac exp}}}}$ | 0.0946 | 0.1374 | 0.1825
Table 5: For cases SC1, SC2, and SC3, values of $[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]_{ij}$ computed with the training set, $[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]_{ij}$ computed with the constrained learned set for $N=10\,000$, and $[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]_{ij}$ for the targets. | SC1 | SC2 | SC3
---|---|---|---
Entries of |  |  |  |  |  |  |  |  |
$(6\times 6)$ | $[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]$ | $[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]$ | $[\,\underline{{\mathbb{C}}}_{\,d}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}_{\,c}^{\hbox{{\ppppcarac eff}}}]$ | $[\,\underline{{\mathbb{C}}}^{\hbox{{\ppppcarac exp}}}]$
matrix |  |  |  |  |  |  |  |  |
$(1,1)$ | 2.0904 | 2.2600 | 2.2652 | 2.0792 | 2.2914 | 2.2810 | 2.0465 | 2.2751 | 2.2946
$(1,2)$ | 0.7427 | 0.8809 | 0.8753 | 0.7269 | 0.8982 | 0.8809 | 0.7140 | 0.8874 | 0.8824
$(1,3)$ | 0.7458 | 0.8804 | 0.8745 | 0.7324 | 0.8983 | 0.8800 | 0.7381 | 0.8999 | 0.8826
$(2,2)$ | 2.0832 | 2.2603 | 2.2668 | 2.0786 | 2.2946 | 2.2846 | 2.0917 | 2.2830 | 2.2822
$(2,3)$ | 0.7451 | 0.8802 | 0.8734 | 0.7486 | 0.8950 | 0.8754 | 0.7471 | 0.9045 | 0.8808
$(3,3)$ | 2.0839 | 2.2647 | 2.2680 | 2.0777 | 2.2841 | 2.2697 | 2.1038 | 2.2928 | 2.2812
$(4,4)$ | 0.6702 | 0.6909 | 0.6958 | 0.6785 | 0.6985 | 0.7003 | 0.6835 | 0.6976 | 0.7027
$(5,5)$ | 0.6714 | 0.6903 | 0.6949 | 0.6732 | 0.6950 | 0.6960 | 0.6726 | 0.6872 | 0.6980
$(6,6)$ | 0.6713 | 0.6924 | 0.6960 | 0.6727 | 0.6958 | 0.6970 | 0.6749 | 0.6933 | 0.6991

###  5.11 Posterior probability model of parameters

The prior probability model concerns the ${\mathbb{R}}^{n_{g}}$-valued random
variable ${\bm{\mathcal{G}}}$ that corresponds to the spatial discretization
of the ${\mathbb{R}}^{21}$-valued random field ${\bm{G}}$, and the
${\mathbb{R}}^{3}$-valued random variable ${\bm{W}}$ that is related (see Eq.
(5.6)) to the random bulk modulus $C_{\hbox{{\ppppcarac bulk}}}$ and the
random shear modulus $C_{\hbox{{\ppppcarac shear}}}$, which control the
elasticity tensor of the mean isotropic model at mesoscale, and to the
dispersion coefficient $\delta_{\mathbb{C}}$ that controls the level of
anisotropic statistical fluctuations of the random apparent elasticity field
at mesoscale (see Section 5.2). For cases SC1, SC2, and SC3, Fig. 6 displays
the posterior pdf $c\mapsto p_{C_{\hbox{{\ppppcarac bulk}}}}(c)$ of
$C_{\hbox{{\ppppcarac bulk}}}$ (left figure), $c\mapsto
p_{C_{\hbox{{\ppppcarac shear}}}}(c)$ of $C_{\hbox{{\ppppcarac shear}}}$
(central figure), and $c\mapsto p_{\delta_{\mathbb{C}}}(c)$ of
$\delta_{\mathbb{C}}$ (right figure), estimated with the constrained learned
set for $N=10\,000$, and their prior counterparts estimated with the training
set constructed using the prior probability model. It should be noted that for
each one of the random variables $C_{\hbox{{\ppppcarac bulk}}}$,
$C_{\hbox{{\ppppcarac shear}}}$, and $\delta_{\mathbb{C}}$, its prior
probability model is the same for the three cases and consequently, does not
depend on the case contrary to its posterior probability model that depends on
it.

Figure 6: Graphs of the posterior pdf $c\mapsto p_{C_{\rm{bulk}}}(c)$ (left
figure), $c\mapsto p_{C_{\rm{shear}}}(c)$ (central figure), and $\delta\mapsto
p_{\delta_{\mathbb{C}}}(\delta)$ (right figure), estimated with the
constrained learned set for $N=10\,000$, for cases, SC1 (thin solid line), SC2
(med solid line), and SC3 (thick solid line), and the corresponding prior pdf
estimated with the training set constructed using the prior probability model
(dotted line).

###  5.12 Discussion about the presented results

(i) The results obtained with the posterior model (see Tables 4 and 5) show
that the constrained learned set significantly improves the prior probability
model used for generating the training set. The comparison of the posterior
statistics with the targets are good.

(ii) As explained in Section 5.9 the residue constraint is taken into account
for the generation of the constrained learned set, but does not intervene in
the estimation of the optimal value ${\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}$
of ${\bm{\lambda}}$ using the error function. Nevertheless and as expected,
Fig. 5 and Table 4 show that the residue is controlled and stayed small with
respect to the reference (the training) for the optimal solution.

(iii) The convergence of the sequence of MCMC generators with respect to the
number of points generated in the constrained learned set is good as shown by
Figs. 2 and 4 in accordance to Proposition 4.

(iv) The dispersion of the target, measured by the value of
$\delta^{\,{\hbox{{\ppppcarac exp}}}}$, is smaller than the one of the prior
probability model (with which the training set has been constructed). So, it
was expected that the posterior random effective elasticity matrix be less
dispersed than the one exhibited by the prior probability model. The results
confirm this point as it can be seen in Tables 4 and 5 and also in Figs. 3 and
6.

(v) Comment (ii) is extended as follows. It can be seen that the target is
”well” reached for the mean value of the random effective elasticity matrix
(see Table 5), while it is ”less well” reached for its coefficient of
dispersion computed using the maximum likelihood. This can be explained by the
fact that the second-order moment $E\\{\rho_{c}^{2}\\}$ of the random
normalized residue of the equation is kept small during the probabilistic
learning of the posterior probability measure, what prevents the dispersion
coefficient from reaching its target. We have carried out numerical tests
without imposing the constraint related to $E\\{\rho_{c}^{2}\\}$, which should
stay close to 1. We have observed that the dispersion coefficient ”reasonably”
reached its target but that $E\\{\rho_{c}^{2}\\}$ did not stay close to $1$
but took on significant values greater than $1$. There is indeed a choice of
objective between (1) correctly satisfying the constraint on the dispersion
coefficient while degrading the value of $E\\{\rho_{c}^{2}\\}$ or (2)
preserving a small value of $E\\{\rho_{c}^{2}\\}$ to the detriment of
perfectly reaching the target for the dispersion coefficient. We have chosen
to present the compromise consisting in taking into account the constraint on
the residue during the probabilistic learning process, but the error function
that we have chosen to identify the optimal value
${\bm{\lambda}}^{\hbox{{\ppppcarac sol}}}$ of ${\bm{\lambda}}$ does not take
into account the constraint on the residue.

(vi) A last comment concerns the effects of no scale separation. As expected,
for the three cases SC1, SC2, and SC3, Table 4 shows that the coefficient of
dispersion is significant and increases with the spatial correlation lengths
of the random apparent elasticity field at mesoscale, inducing statistical
fluctuations of the effective elasticity tensor at macroscale. It should be
noted that, even for the case SC1, for which homogenization in the plane of
the plate (domain $\Omega$) is guaranteed (the correlation lengths
$\underline{L}_{C1}$ and $\underline{L}_{C2}$ being much lower than $1$), this
is not the case for the correlation length $\underline{L}_{C3}$ that is equal
to the thickness of the plate. Consequently, there is no homogenization at the
macroscopic scale and the effective elasticity tensor remains random and is
not deterministic.

##  6 Conclusions

In this paper, we have presented a general methodology to estimate a posterior
probability model for a stochastic boundary value problem from a prior
probability model. The given targets are statistical moments for which the
underlying realizations are not available. Under these conditions, it has been
proposed to use the Kullback-Leibler divergence minimum principle for
estimating the posterior probability measure, given the prior probability
measure and the constraints related to the targets of the statistical moments.
We have proposed the construction of a statistical surrogate model of the
implicit mapping that represents the constraints. The constrained learned set,
which defines the posterior model, is constructed using only a training set
constituted of a small number of points. A mathematical analysis of the
proposed methodology has been presented. We have defined the required
mathematical hypotheses, which have allowed us to prove the convergence of
introduced approximations. We have also given all the necessary numerical
elements, which facilitate the implementation of the methodology in a parallel
computing framework.

The application presented to illustrate the proposed theory is also, as such,
a contribution to the three-dimensional stochastic homogenization of
heterogeneous linear elastic media in the case of a non-separation of the
microscale and macroscale, that is to say, when there are significant
statistical fluctuations in the effective elasticity tensor at macroscale. The
prior stochastic model of the elasticity tensor field at the mesoscopic scale
is an advanced model, recently proposed, which takes into account
uncertainties on its spectral measure. In addition to the statistical moments
of the random effective elasticity tensor, for the construction of the
posterior probability measure by probabilistic learning inference, the second-
order moment of the random normalized residue of the stochastic partial
differential equation has been added as a constraint. This constraint
guarantees that the algorithm seeks to bring the statistical moments closer to
their targets while preserving a small residue. The results obtained are those
which were expected and give a very good illustration of the theory developed
for a non-trivial application.

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