\section{Uncertainty quantification}
Uncertainty quantification \cite{Avramova2010} is the science that aims to characterize an experiment or computational model estimating the associated uncertainties. Its purpose is to determine the likelihood of a certain output if some of the aspect of the system are not exactly known. The methodology is widely spread in the nuclear field in general and in neutronic calculations in particular to provide confidence intervals of the relevant quantities evaluated. Uncertainty quantification is needed as it complements the information given by the best estimate coupling it with a reliability analysis; moreover it is relevant from a more technological point of view  to verify and properly establish the many safety margins foreseen for the nuclear installations. In a neutronic simulation the uncertainty associated to the output is a combination of different sources that must be investigated separately to assess the global variability of the parameter of interest.

\subsection{Sources of uncertainty}
The sources of uncertainty to take into account in the case study, transport and depletion calculation in MYRRHA, are the possible variabilities of the model with respect to the real reactor design and operation.\\
The first source of uncertainty is related to the input parameters, since the model is fed by the best-available information on the design but there is still a certain flexibility. For example the compositions of MOX fuel and LBE coolant are not defined accurately yet and the deviation from reference compositions have to be taken into account. Besides, slight variation in operational history can influence the power and the burnup bringing deviations to the nuclide vector of the fuel for example.\\
Neutronic and depletion calculations strongly rely on nuclear data evaluated through experiments and mathematical models. Specific data for each nuclide are provided as best-estimate evaluation and associated uncertainties in the form of covariance matrices. The impact of nuclear data on criticality parameters and \acrshort{SNF} concentrations is well assessed and several methodologies are adopted in the field \cite{Rochman2009UncertaintiesDistributions}, \cite{Rochman2014NuclearBurnup}.

\subsection{Uncertainty propagation and statistical sampling}\label{stSMPS}
In this section the methodologies to perform uncertainty propagation are described \cite{Rochman2011} with particular focus to nuclear data uncertainties and statistical sampling procedure.
The first method, based on perturbation theory, makes use of the covariance matrices of nuclear data and sensitivity profiles. Sensitivity profile is defined as the relative change in a response parameter caused by a small variation in a cross section data at a certain energy as shown in \ref{sens_p}:  
\begin{equation}
    S_R = \frac{\delta R / R}{\delta \sigma_g / \sigma_g}
    \label{sens_p}
\end{equation}
The sensitivity profiles obtained are combined with nuclear data covariance matrix $\Sigma$ to propagate nuclear data uncertainties using the so-called first-order uncertainty propagation formula (or sandwich rule) \cite{cacuci2005sensitivity}, as shown in eq. \ref{SandRule}.
\begin{equation}
    Var_R = S_R \ \Sigma \ S_R^T 
    \label{SandRule}
\end{equation}

The upscaling of computational power in the last few years gave the possibility to investigate new methods to propagate uncertainty. One of the most prominent is based on stochastic sampling of nuclear data \cite{Fiorito2017NuclearSANDY}. Perturbed nuclear data files are generated to be fed in the model. The uncertainty of the output response under investigation is determined as the standard deviation of the output population using the standard definitions reminded in eq. \eqref{mean_std}.\\
\begin{equation}
\begin{gathered}
    \mu = \frac{\sum_i^N (x_i - \Bar{x})}{N} \\
    Var = \frac{1}{N-1} \sum_i^N (x_i - \Bar{x})^2
    \label{mean_std}
\end{gathered}
\end{equation}
The main advantages of statistical sampling procedure are the independence from the model and the possibility to investigate higher order effects thanks to the inherent characteristics of the method. The limit is found in the computational cost since a large sample is required to get statistically significant results.\\
Samples of nuclear data are generated from a multivariate normal distribution $\mathcal{N}(\mu,\Sigma)$. The procedure to obtain $n$ samples consists of applying $\Sigma$ to a matrix $X$ $m x n$ of identically and independent variables distributed according $N (0,1)$. The perturbation coefficients are produced according to eq. \eqref{applyCM}:
\begin{equation}
    P = L X + 1
    \label{applyCM}
\end{equation}
where $$\Sigma = L L^T.$$
The matrix $L$ that fulfills the relation is obtained decomposing the covariance matrix. The use of this specific factorization is possible thanks to the properties of the covariance matrix itself, being symmetric and positive semi-definite. P, the matrix of perturbation coefficients is applied to nuclear data such as cross sections, neutron fission product yields, angular and energy distributions to estimate the uncertainty coming from these parameters. In this analysis the contribution of neutron cross sections and fission yields is assessed.\\
The procedure is validated for a set of 200 sample (the number used for the analysis), assessing the convergence of the perturbation coefficients standard deviation to the standard deviation given in input.
\begin{figure}[h]
  \centering
    \begin{minipage}{0.99\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/uncertainty/input_Pu239.png} 
    \end{minipage}
    \hfill
      \centering
    \begin{minipage}{0.99\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/uncertainty/input_Cm245.png} 
    \end{minipage}
    \hfill
\caption{Convergence analysis on the perturbation coefficients of $^{239}$Pu fission cross section and $^{245}$Cm neutron capture cross section @Energy = 0.1 MeV in top and bottom figure respectively. The perturbation coefficients sample distribution is shown on the right part of the plot for both cross section data.}
\label{inputConvergence}
\end{figure}
In addition the output distribution of the perturbation coefficients is is shown, to qualitatively assess the correspondence to the selected sampling distribution. They are presented in Fig. \ref{inputConvergence}.
The perturbation coefficients chosen come from the fission cross section of $^{239}$Pu in the energy range around 0.1 MeV, that is the most relevant for this application. The fission \acrshort{XS} has low uncertainty in this range and it is fast to converge. Then the perturbation coefficients of $^{245}$Cm capture \acrshort{XS} are shown. The reaction is relevant since it produces $^{246}$Cm that is the nuclide with largest uncertainty among those analysed in this work, as presented in Sect. \ref{unc_act_xs_sect}. The convergence of this cross section is much slower compared to the previous one, as shown in the bottom plot of Fig. \ref{inputConvergence}.


