 \mathbf{A} \right)_{i,j} = \langle \psi_i, \zeta_j \rangle$
 for $i=1,\ldots,m$ and $j=1,\ldots,n$.
 
The singular value problem is practically solved using the Python routine \texttt{svd} of \texttt{numpy}: the approximation error is expected to be negligible. The lack of effectivity that is observed in the numerical results is due to the fact that $\beta(V_n,W_m)$ is associated with the least well-represented mode of the reduced space $V_n$; therefore, the error bound provides an upper bound for the actual prediction error.

\section{Reviewer 2: question 2}

We thank the reviewer for raising this issue.  
In the data assimilation literature, Maday et al. proposed to augment the background space $V_n$ based on the results of previous data assimilation tasks on related inputs (cf. \cite[section 5.8]{maday2015pbdw}): we envision that the same approach can be applied in our setting.
We remark that in the MOR literature several authors have proposed to combine snapshots of variable fidelity to enhance predictive capabilities of ROMs (see, e.g., \cite{bertram2018towards,chakir2019non}); however, we are not aware of MOR data compression techniques that explicitly leverage two sets of snapshots of variable fidelity.



\bibliographystyle{abbrv}	


\section*{Introduction }

In the field of nuclear engineering, numerical methods play a crucial role at several stages: they are involved in important assessments and decisions related to design, safety, energy efficiency, and reactor loading plans. In this paper, we focus on the task of providing real time information about the spatial distribution of the nuclear power generated by a nuclear reactor from a limited number of measurement observations. We combine this data with physical models in order to provide a complete spatial reconstruction of the power field. This task is a state estimation problem, and we work with the Parametrized Background Data Weak (PBDW), originally introduced in \cite{MPPY2015}. The method has the appealing feature of providing very fast reconstructions by leveraging techniques from model order reduction of parametric Partial Differential Equations (PDEs). We refer to \cite{binev2017, DPW2017, CDDFMN2020, CDMN2022} for theoretical analysis of the method, optimal recovery results and nonlinear extensions. A recent overview may be found in~\cite{Mula2022}.


The main ideas of the above state estimation methodology have been applied to the field of nuclear physics for applications connected to neutronics (see \cite{ABGMM2017, ABGMM2017-proc, ABCGMM2018}). We could also cite other works such as \cite{SCLR2016,GCZC2020,ChenEtAll2021,GERMAN2019144,LORENZI2018245} which study the forward reduced modeling problem for neutronics (compared to these works, note that there is a salient difference in the nature of the task that we consider, which is inverse state estimation). In this paper, we again consider neutronics but our goal is to study the impact of inaccuracies in the physical model that is involved in the reconstruction algorithm, and which is often assumed to perfectly describe reality. This assumption goes beyond the present application on neutronics but studying it for this particular topic has the advantage that we have two very well identified models with different levels of accuracy, thereby allowing to examine synthetically what one can expect when working in a real application scenario.

In neutronics, the most accurate physical model is the so-called neutron  transport equation which describes the evolution of the neutronic population in a reactor core by expressing it in the form of a balance between produced and lost neutrons~\cite{Hebert2009}. This model is often approximated at the reactor core scale by a neutron diffusion model to save computing time. This is why in this work, we explore the impact of model inaccuracies by applying a reconstruction based on a diffusion model for the neutron flux, and then  assuming that the true physical system is  governed by a neutron transport model.

The paper is organized as follows. Section~\ref{sec:inverse_state_estimation} is devoted to presenting inverse state estimation problems and the PBDW method. Section~\ref{sec:application_reconstruction_power} details the application of the methodology to the reconstruction of nuclear power. Section~\ref{sec:numerics} provides some numerical results.



\section{Inverse State Estimation with PBDW}
\label{sec:inverse_state_estimation}
In this section, we introduce the problem of state estimation, and the Parametrized Background Data Weak method which combines measurement observations and reduced models from parametric PDEs. We refer the reader to \cite{Mula2022} for an overview of inverse problem algorithms using these elements.

Let $\ensuremath{\mathcal{R}}$ be a fixed given domain of $\ensuremath{\mathbb{R}}^d$ with dimension $d\geq 1$, and let $V$ be a Hilbert space defined over $\ensuremath{\mathcal{R}}$. In our application, $\ensuremath{\mathcal{R}}$ will be defined as the nuclear reactor domain. The space is endowed with an inner product $\<\cdot, \cdot\>$ and induced norm $\Vert \cdot \Vert$. The choice of $V$ must be relevant for the problem under consideration:
 typical options are $L^2$, $H^1$; for pointwise measurements, a Reproducing Kernel Hilbert Space should be considered.

Our goal is to recover an unknown function $u\in V$ from $m$ measurement observations
\begin{equation}
\label{eq:meas}
y_i = \ell_i(u),\quad i=1,\dots,m,
\end{equation}
where the $\ell_i$ are linearly independent linear forms from the dual $V'$. Note that  
we have assumed that experimental observations are perfect; however, the methodology could be extended to deal with noisy measurements (see, e.g., \cite{Taddei2017, GMMT2019, EF2021}). In practical applications, each $\ell_i$ models a sensor device which is used to collect the measurement data $\ell_i(u)$. In the applications that we present in our numerical tests, the observations come from sensors for the neutron flux which are placed in the reactor.

We denote by $\omega_i \in V$ the Riesz representers of the $\ell_i$. They are defined via the variational equation
$$
\left< \omega_i, v \right> = \ell_i(v),\quad \forall v \in V.
$$
Since the $\ell_i$ are linearly independent in $V'$, so are the $\omega_i$ in $V$ and they span an $m$-dimensional space
$$
\ensuremath{{W_m}}={\rm span}\{\omega_1,\dots,\omega_m\} \subset V.
$$

When there is no measurement noise, knowing the observations $y_i=\ell_i(u)$ is equivalent to knowing the orthogonal projection
\begin{equation}
\omega =P_{\ensuremath{{W_m}}} u.
\end{equation}
In this setting, the task of recovering $u$ from the measurement observation $\omega$ can be viewed as building a recovery algorithm
$$
A:\ensuremath{{W_m}}\mapsto V
$$
such that $A(P_{\ensuremath{{W_m}}}u)$ is a good approximation of $u$ in the sense that $\Vert u - A(P_{\ensuremath{{W_m}}}u) \Vert$ is small.

Recovering $u$ from the measurements $P_{\ensuremath{{W_m}}} u$ is a very ill-posed problem since $V$ is generally a space of very high or infinite dimension so, in general, there are infinitely many $v\in V$ such that $P_{\ensuremath{{W_m}}} v = \omega$. It is thus necessary to add some a priori information on $u$ in order to recover the state up to a guaranteed accuracy. In the following, we work in the setting where $u$ is a solution to some parameter-dependent PDE of the general form 
\begin{equation*}
\ensuremath{\mathcal{P}}(u, \mu) = 0,
\end{equation*}
where $\ensuremath{\mathcal{P}}$ is a differential operator and $\mu$ is a vector of parameters that describe some physical property and  belong to a  given set $\ensuremath{\mathcal{D}}\subset \ensuremath{\mathbb{R}}^p$. For every $\mu \in \ensuremath{\mathcal{D}}$, we assume that the PDE has a unique solution $u=u(\mu)\in V$. Therefore, our prior on $u$ is that it belongs to the so-called {\em solution manifold}
\begin{equation}
\label{eq:manifold}
\ensuremath{\mathcal{M}} \coloneqq \{ u(\mu) \in V \; :\; \mu\in\ensuremath{\mathcal{D}} \}.
\end{equation}
In practical applications, the PDE model $\mathcal{P}$ might not be known exactly or might be too expensive to evaluate:  we should thus rely on a surrogate approximate model to perform state estimation.

\medskip
\noindent
\textbf{Performance Benchmarks:} The quality of a recovery mapping $