text:
\section*{Acknowledgements}
This work is partially supported by U.S. Department of Energy under grants DE-SC0020270,
DE-SC0020418, and DE-SC0021077.

\bibliographystyle{ieeetr}

\section{Introduction}
Multifidelity Monte Carlo (MFMC) methods have shown great promise for the efficient and flexible estimation of statistical quantities.  As experimental data can take a variety of forms, the primary advantage of MFMC estimation is its ability to accommodate a diverse ensemble of information sources which may be unrelated apart from predicting the same quantity of interest.  However, despite their success in small proof-of-concept experiments where unlimited data are available, MFMC algorithms have not yet become standard in larger experiments of practical interest which are dominated by traditional Monte Carlo (MC) or Multilevel Monte Carlo (MLMC) techniques (see e.g. \cite{cubasch1994,gjerstad2003,leathers2004,hong2006,tomassini2007,mishra2012} and references therein).  While this is partly inertial, it is also true that MFMC algorithms are not always directly applicable to large scientific simulations, requiring nontrivial modification in order to function effectively.  A notable example of this is the popular MFMC estimation algorithm from \cite{peherstorfer2016} (hereafter known as MFMC), which divides computational resources through a global optimization with a unique closed-form solution. MFMC has been applied to great effectiveness on small-scale problems with functionally unlimited resources (e.g. \cite{patsialis2021,law2021,khodabakhshi2021}), but breaks down when the model costs are large relative to the available budget.  This is because the real-valued  optimization may call for some models to be evaluated less than once, producing an infeasible solution if integer parts are taken.

Despite this technical concern, it is highly desirable to have an estimation method such as MFMC which is applicable to problems of any size and can guarantee good accuracy without access to inter-model pointwise error metrics.  To that end, this work provides a simple extension of the MFMC algorithm \cite[Algorithm 2]{peherstorfer2016} to any computational budget (c.f. Algorithm~\ref{alg:mod}), which reduces to the original procedure when the budget is large enough.  It is proved that this extension preserves the optimality of MFMC in a precise sense (c.f. Theorem~\ref{thm:optim}), and numerical examples are presented which illustrate this fact.  Applying the modified MFMC presented here to estimate the expectations of quantities arising from benchmark problems provides empirical justification for the proposed method, demonstrating that MFMC has benefits over traditional MC estimation even when the amount of available resources is quite limited.

Before discussing MFMC and its extension in more detail, it is appropriate to mention other concurrent work on multifidelity methods for optimization and uncertainty quantification (see \cite{peherstorfer2018survey} for an introductory survey).  About 20 years ago, a multifidelity method for function approximation involving neural-based systems was formulated as a kriging problem in \cite{leary2003}, while a multifidelity optimization method based on co-kriging was developed in \cite{forrester2007} and applied to Gaussian processes.  Later \cite{narayan2014} developed a multifidelity approach for stochastic collocation, and \cite{ng2014} proposed a multifidelity method for optimization under uncertainty which grew into the MFMC method of present discussion.  Note that the optimization approach of \cite{ng2014,peherstorfer2016} was also explored by other authors including \cite{pauli2015}, who proposed a branch-and-bound algorithm for optimal parameter selection in MLMC with applications to fault tolerance in large-scale systems.  Besides direct estimation, multifidelity methods have also found applications in preconditioning \cite{peherstorfer2019transport} as well as plasma micro-instability analysis \cite{konrad2022}.  At the time of writing, many multifidelity methods also have guarantees on their accuracy, including the MFMC method of present discussion which is analyzed in \cite{peherstorfer2018} and proven (under some assumptions) to be efficient in the sense that it requires a cost bounded by $1/\epsilon$ to achieve a mean squared error (MSE) of $\epsilon$.  It will be shown in Section~\ref{sec:mfmcnew} that the proposed modifications which enable realistic computational budgets also preserve this optimality (see Corollary~\ref{cor:optim}).  In preparation for this, the next Section reviews the MFMC algorithm of \cite{peherstorfer2016} in greater detail.

\section{Overview of MFMC}\label{sec:mfmcalg}
Let $\Omega$ be a sample space, $\mathcal{D}\subset\mathbb{R}^n$ for some $n\in\mathbb{N}$, and $f^1,f^2,...,f^k:\mathcal{D}\to\mathbb{R}$ be a sequence of computational models depending on the random variable $Z: \Omega\to \mathcal{D}$ with decreasing computational complexity.  The goal of MFMC is to estimate the expectation $y = \EE{f^1}$ using a linear combination of standard MC estimators.  Recall that Monte Carlo sampling yields the unbiased estimators
\[ \bar{y}^i_{n} = \frac{1}{n}\sum_{j=1}^{n} f^i(\bb{z}_j), \]
where $\bb{z}_1,...,\mathbf{z}_{n}$ are i.i.d. realizations of $Z$ and unbiased implies that $\EE{\bar{y}^i_n} = \EE{f^i}$.  These can be used to define the MFMC estimator
\begin{equation}\label{eq:MFMC}
    \hat{y} = \bar{y}^1_{m_1} + \sum_{i=2}^k \alpha_i\left( \bar{y}^i_{m_i} - \bar{y}^i_{m_{i-1}}\right),
\end{equation}
which is unbiased by linearity and parameterized by weights $\alpha_i \in \mathbb{R}$ and an increasing sequence of integers $\{m_i\}$ defining the sampling.  If $w_i$ represents the cost of evaluating the $i^{th}$ model $f^i$, then the costs of computing the respective MC and MFMC estimators are given as
\begin{align*}
    c\left(\bar{y}^i_{n}\right) = nw_i, \qquad c(\hat{y}) = \sum w_im_i = \bb{w}\cdot\bb{m},
\end{align*}
where $\bb{w}=\{w_i\}_{i=1}^k$ and $\bb{m}=\{m_i\}_{i=1}^k$.  For a fixed computational budget $p = \bb{w}\cdot\bb{m}\in\mathbb{R}$, MFMC aims to construct an optimal sampling strategy $\bb{m}$ with weights $\bm{\alpha} = \{\alpha_i\}_{i=2}^k$ so that the mean squared error (MSE) of the estimator $\hat{y}$ is lower than that of the Monte Carlo estimator $\bar{y}^1_n$ for all $n\leq \lfloor p/w_1\rfloor$.  More precisely, denote the variance of the $i^{th}$ model and the correlation between models $i,j$ by
\[\sigma_i^2 = \VV{f^i(Z)}, \qquad \rho_{i,j} = \frac{\CC{f^i(Z)}{f^j(Z)}}{\sigma_i\sigma_j}.\]
Then, the MSE of the MC estimator $\bar{y}^i_n$ with respect to $y$ is simply
\[ e\left(\bar{y}^i_m\right) =  \EE{\left(\EE{f^i(Z)} - \bar{y}^i_n\right)^2} = \EE{f^i(Z)}^2 - \EE{f^i(Z)^2} = \frac{\sigma_i^2}{n}, \]
while the MSE of $\hat{y}$ can be expressed (see \cite{peherstorfer2016}) as
\begin{equation}\label{eq:MSE}
    e\left(\hat{y}\right) = \VV{\hat{y}} = \frac{\sigma_1^2}{m_1} + \sum_{i=2}^k \left( \frac{1}{m_{i-1}} - \frac{1}{m_{i}} \right)\left( \alpha_i^2\sigma_i^2 - 2\alpha_i\sigma_i\sigma_1\rho_{1,i} \right).
\end{equation}
With this, the optimal weights $\bm{\alpha}^*=\{\alpha_i^*\}_{i=2}^k$ and sampling numbers $\bb{m}^*=\{m_i^*\}_{i=1}^k$ of the MFMC sampling strategy are defined through a solution to the mixed integer minimization problem (MIP),
\begin{equation}\label{eq:MFMCmin}
\begin{aligned}
\argmin_{\bb{m}\in\mathbb{Z}^k,\bm{\alpha}\in\mathbb{R}^{k-1}} \,\,& e\left(\hat{y}\right) \\
\textrm{s.t.} \quad
    &m_1 \geq 1, \\
    &m_{i-1} \leq m_i, \qquad 2\leq i \leq k, \\
    &\bb{w}\cdot\bb{m} \leq p.
\end{aligned}
\end{equation}
While analytic solutions to the problem  \eqref{eq:MFMCmin} are not easy to find (if they even exist), it turns out that the continuous relaxation obtained by letting $m_i\in\mathbb{R}^+$ for all $1\leq i \leq k$ has a closed-form solution (c.f. \cite[Theorem 3.4]{peherstorfer2016}).  In particular, suppose the computational models are decreasingly ordered with respect to their correlations with the high-fidelity model $1 = |\rho_{1,1}| > |\rho_{1,2}| > ... > |\rho_{1,k}|$ and have costs satisfying the inequality conditions
\[ \frac{w_{i-1}}{w_i} > \frac{\rho_{1,i-1}^2 - \rho_{1,i}^2}{\rho_{1,i}^2 - \rho_{1,i+1}^2}, \qquad 2\leq i \leq k, \]
where $\rho_{1,k+1} := 0$; these conditions are always true for at least the subset $\{f^1\}$, in which case MFMC reduces to MC.  Introducing the notation
\[ r_i^* = \sqrt{ \frac{w_1\left( \rho_{1,i}^2 - \rho_{1,i+1}^2 \right)}{w_i\left( 1-\rho_{1,2}^2 \right)} }, \]
it follows from \cite[Theorem 3.4]{peherstorfer2016} that the optimal solution to the relaxation of \eqref{eq:MFMCmin} with $\bb{m}\in\mathbb{R}^{k}_+$ is given as
\begin{equation*}
    m_1^* = \frac{p}{\bb{w}\cdot\bb{r}}, \qquad
    \alpha_i^* = \frac{\rho_{1,i}\sigma_1}{\sigma_i}, \qquad m_i^* = m_1^* r_i^*, \qquad 2\leq i \leq k.
\end{equation*}
Temporarily neglecting the integrality of $\bb{m}^*$ necessary for evaluation, it follows that the MSE of the estimator $\hat{y}$ at this minimum becomes
\begin{equation}\label{eq:emfmc}
    e\left(\hat{y}\right) = \frac{\sigma_1^2\left(1-\rho_{1,2}^2 \right)p}{\left(m_1^*\right)^2 w_1},
\end{equation}
which is lower than the Monte Carlo equivalent with $c\left(\bar{y}^1_n\right) = p$,
\begin{equation}\label{eq:emc}
    e\left(\bar{y}^1_n\right)= \frac{\sigma_1^2}{n} = \frac{\sigma^2_1 w_1}{p},
\end{equation}
if and only if the ratio inequality
\begin{equation}\label{eq:compineq}
    \sqrt{\frac{e\left(\hat{y}\right)}{e\left(\bar{y}^1_m\right)}}=\sum_{i=1}^k \sqrt{ \frac{w_i}{w_1}\left(\rho_{1,i}^2 - \rho_{1,i+1}^2\right) } < 1,
\end{equation}
is satisfied (c.f. \cite[Corollary 3.5]{peherstorfer2016}). This provides a way to explicitly compute the potential reduction in error provided by MFMC as a function of the weights and model correlations, and displays a primary advantage of this method: the mean squared error $e\left(\hat{y}\right)$ can be rigorously estimated without assumptions on the pointwise errors $\nn{f^1(\bb{z}) - f^i(\bb{z})}$ for $\bb{z}\in\mathcal{D}$ which may be unattainable in practice.

On the other hand, the optimal sampling numbers $\bb{m}^*$ computed according to the above procedure are not guaranteed to be integers.  The issues with this have seemingly not arisen before now, as the standard application of MFMC formulated in \cite[Algorithm 2]{peherstorfer2016} and widely used thereafter simply chooses the integer parts $\bb{m}^*\gets \lfloor\bb{m}^*\rfloor$ of each optimal sampling number.  For large budgets, this is of little consequence\textemdash each model is evaluated hundreds of times or more, so the integer parts $\lfloor m^*_i\rfloor$ provide a simple and reasonable solution which is guaranteed to be feasible (if potentially sub-optimal) for the original problem \eqref{eq:MFMCmin}.  However, this approach is unsuitable for the smaller budgets found in large-scale problems and simulated in Section~\ref{sec:numerics}, where $p \ll \bb{w}\cdot\bb{r}$ and some $m_i^* \ll 1$.  Taking the integer part $\lfloor m_1^*\rfloor$ in this case not only violates the constraints of the problem but also biases the estimator $\hat{y}$, destroying the applicability of MFMC.  The remainder of this work discusses a relatively slight modification which addresses this concern and is guaranteed to preserve optimality in the sampling strategy.

\section{A Modified MFMC Algorithm}\label{sec:mfmcnew}

Before discussing the proposed modification to MFMC which will apply to both small and large budgets, it is prudent to recall the two stages of the original MFMC algorithm developed in \cite{peherstorfer2016}.  Given the models $\{f^i\}_{i=1}^k$, the first stage outlined in Algorithm~\ref{alg:select} performs an exhaustive search over $2^{k-1}$ subsets $\mathcal{M}\subset\{f^i\}$ containing the high-fidelity model $f^1$, choosing the models which yield the lowest predicted MSE $e\left(\hat{y}\right)$ relative to the MC estimator.  Note that this choice is independent of the computational budget $p$ by \eqref{eq:compineq}, so it suffices to choose e.g. $p=10w_1$ as a benchmark value during model selection.  The second stage uses the result  $\mathcal{M}^*\subset \mathcal{M}$ of Algorithm~\ref{alg:select} to compute the optimal MFMC estimator $\hat{y}$.  More precisely, optimal weights $\bm{\alpha}^*$ and model evaluation numbers $\bb{m}^*$ which parameterize the best possible MFMC estimator $\hat{y}$ are first computed, so that samples of the random variable $Z$ can be drawn based on $\bb{m}^* \gets \lfloor\bb{m}^*\rfloor$ and used to compute $\hat{y}$ according to \eqref{eq:MFMC}.  However, this procedure breaks down when $p<\bb{w}\cdot\bb{r}$ as discussed before, making it necessary to find an alternative which is applicable when the budget $p$ is too small to produce an unbiased sampling strategy.  The most obvious ``solution'' to this problem is to na\"{i}vely compute $m_i^*\gets \lceil m_i^*\rceil$ whenever $m_i^*<1$, so that any sampling numbers less than one are rounded up. While this simple change will produce a meaningful estimator $\hat{y}$, it is not an acceptable modification in general, since it is likely that $\bb{w}\cdot\bb{m}^*>p$ exceeds the allowed computational budget (see e.g. the examples in  Section~\ref{sec:numerics}).

\begin{algorithm}[!ht]
\caption{MFMC Model Selection (c.f. \cite[Algorithm 1]{peherstorfer2016})}\label{alg:select}
\begin{algorithmic}[1]

    \Ensure $\rho_{1,i}^2 > \rho_{1,i+1}^2$ for $1\leq i \leq k$ (reorder if necessary).
    \State Set computational budget $p=w_1$.
    \State Initialize $\mathcal{M}^* = \{f^1\}$.
    \State Initialize $v^* = \sigma_1^2w_1/p$.
    \For{$\mathcal{M} = \{f^1\} \cup \mathcal{N}$ where $\mathcal{N}\subseteq \{f^2,...,f^k\}$ }
        \State Set $k' = \nn{\mathcal{M}}$.
        \State Let $i_1,...,i_{k'}$ be the model indices of $\mathcal{M}$ such that $\rho_{1,i_j}^2 > \rho_{1,i_{j+1}}^2$ for $1\leq j \leq k'-1$.
        \State Set $\rho_{1,i_{k'+1}}=0$.
        \If{The following inequality holds:
        \[\frac{w_{i-1}}{w_i}\leq \frac{\rho_{1,i-1}^2-\rho_{1,i}^2}{\rho_{1,i}^2-\rho_{1,i+1}^2}, \quad\mathrm{for\,some\,\,} 2\leq i \leq k', \]\quad}
            \State Continue.
        \EndIf
        \State Compute the predicted variance
        \[v = \frac{\sigma_1^2}{p}\lr{\sum_{j=1}^{k'}\sqrt{\rho_{1,i_j}^2-\rho_{1,i_{j+1}}^2}}^2.\]
        \If{$v<v^*$}
            \State Set $\mathcal{M}^*=\mathcal{M}$.
            \State Set $v^*=v$.
        \EndIf
        \EndFor
    \State \Return $\mathcal{M}^*,v^*$.
\end{algorithmic}
\end{algorithm}

While simple rounding is not enough to repair MFMC estimation when $p<\bb{w}\cdot\bb{r}$, it will now be shown that an appropriately modified MFMC algorithm always yields a solution which is constraint-preserving, feasible for \eqref{eq:MFMCmin}, and optimal up to the rounding $m_i^*\gets \lfloor m_i^*\rfloor$ of some components $m_i^*$.  First, note that it can be shown by repeating the arguments in \cite{peherstorfer2016} that the Lagrangian,
\begin{equation}\label{eq:ifunc}
    L^i\lr{\bb{m},\bm{\alpha},\bm{\mu},\lambda,\xi} = e\left(\hat{y}\right) + \lambda \left(\sum_{j=i}^k w_jm_j - \left(p - \sum_{j=1}^{i-1} w_j\right)\right) + \sum_{j=i+1}^k\mu_j\left(m_j-m_{j-1}\right) - \xi m_i,
\end{equation}
has a unique global minimizer corresponding to a pair $(\bb{m}^*,\bm{\alpha}^*)\in\mathbb{R}^{k-i+1}\times\mathbb{R}^{k-i}$ for any $1\leq i \leq k$ (overloading the notation), where $\lambda\in\mathbb{R}, \mu_{i+1},...,\mu_k,\xi\in\mathbb{R}_+$ are dual variables encoding the constraints of \eqref{eq:MFMCmin}.  Observe that the minimization of $L^i$ for $i>1$ is almost the partial minimization of $L=L^1$ with the restriction $m_1=...=m_{i-1}=1$ on the model evaluation numbers, although they are not precisely equivalent.  Particularly, the $\lambda$-term enforces the budget constraint $\bb{w}\cdot\bb{m}=p$, and the $\mu_j$-terms enforce the increasing nature of the $m_j$ for $j\geq i$.  On the other hand, the constraint $m_i\geq m_{i-1}=1$ is not explicitly enforced, although the weaker constraint $m_i>0$ is still assured due to the $\xi$-term and the global minimizer of $L^i$ can be computed analytically for all $1\leq i \leq k$ using the following result.

\begin{algorithm}[!ht]
\caption{Modified MFMC Estimation}\label{alg:mod}
\begin{algorithmic}[1]

    \Ensure The models $f^1,...,f^k$ are the result of Algorithm~\ref{alg:select} and the budget satisfies $p \geq \sum_i w_i$.
    \State Set $\bm{\rho}_1 \gets \left[\bm{\rho}_1\,\, 0\right] \in \mathbb{R}^{k+1}$.
    \State Set the weights
    \[ \alpha_i = \frac{\rho_{1,i}\sigma_1}{\sigma_i},\qquad 2\leq i\leq k. \]
    \While{there is $1\leq i \leq k-1$ such that $m_i < 1$}
        \State Set $i$ equal to the first such index.
        \State $m_i \gets 1$ and $r_{i+1} \gets 1$.

        \State Set the coefficients $r_j$,
        \[r_j \gets \sqrt{\frac{w_{i+1}}{w_j}\left( \frac{\rho_{1,j}^2-\rho_{1,j+1}^2}{\rho_{1,i+1}^2 - \rho_{1,i+2}^2}\right)}, \qquad i+1<j\leq k.\]

        \State Set the $(i+1)$-st component of $\bb{m}$,
        \[m_{i+1} = \frac{p-\sum_{j=1}^i w_j}{\sum_{j=i+1}^k w_jr_j}.\]

        \State Set the remainder of $\bb{m}$,
        \[m_j = m_{i+1}r_j, \qquad i+1<j\leq k.\]

    \EndWhile
    \State Draw $m_k$ i.i.d. realizations $\bb{z}_1,...,\bb{z}_{m_k}$ of the random variable $Z$.
    \State Compute the MC estimators
        \[ \bar{y}_{m_i}^i = \frac{1}{m_i}\sum_{j=1}^{m_i}f^i\lr{\bb{z}_j}, \qquad 1\leq i \leq k. \]
    \State \Return the MFMC estimator
        \[\hat{y} = \bar{y}^1_{m_1} + \sum_{i=2}^k \alpha_i\left( \bar{y}^i_{m_i} - \bar{y}^i_{m_{i-1}}\right).\]
\end{algorithmic}
\end{algorithm}

\begin{remark}
Note that the budget restriction $p\geq \sum_i w_i$ necessary for Algorithm~\ref{alg:mod} is the minimum necessary for the MFMC sampling procedure to make sense.  Indeed, if $p<\sum_i w_i$, then it is not possible to both evaluate $m_1$ and ensure that $\{m_i^*\}_{i=1}^k$ forms an increasing sequence.  Therefore, the claim that Algorithm~\ref{alg:mod} applies to any computational budget should be read with the understanding that these degenerate cases are omitted.
\end{remark}

\begin{lemma}\label{thm:mod}
Let $\{f^i\}_{i=1}^k$ be computational models with correlation coefficients satisfying $\nn{\rho_{1,i-1}}>\nn{\rho_{1,i}}$ for all $2\leq i \leq k$ and computational costs $\{w_1,...,w_k\}$ satisfying
\[ \frac{w_{i-1}}{w_i} > \frac{\rho_{1,i-1}^2-\rho_{1,i}^2}{\rho_{1,i}^2-\rho_{1,i+1}^2}, \qquad 2\leq i \leq k. \]
Defining the ratios
\[ r^*_j = \frac{m_j^*}{m_i^*} = \sqrt{\frac{w_i}{w_j}\left( \frac{\rho_{1,j}^2-\rho_{1,j+1}^2}{\rho_{1,i}^2 - \rho_{1,i+1}^2}\right)},\qquad i\leq j \leq k, \]
for each $1\leq i \leq k$ the unique global minimum $\lr{\bb{m}^*,\bm{\alpha}^*}\in\mathbb{R}^{k-i+1}\times\mathbb{R}^{k-i}$ of the Lagrangian $L^i$ from \eqref{eq:ifunc} satisfies
\begin{equation*}
    m_i^* = \frac{p-\sum_{j=1}^{i-1}w_j}{\sum_{j=i}^k w_jr_j}, \qquad m_j^* = m_i^* r^*_j, \qquad \alpha_j^* = \frac{\rho_{1,j}\sigma_1}{\sigma_j}, \qquad i<j\leq k.
\end{equation*}
\end{lemma}

\begin{proof}
Let $1\leq i \leq k$ and $2\leq j \leq k$.  A straightforward calculation using the representation \eqref{eq:MSE} gives the partial derivatives of the MSE,
\begin{align*}
    e\left(\hat{y}\right)_{m_i} &= \frac{1}{m_i^2}\left(\alpha_i^2\sigma_i^2-  2\alpha_i\rho_{1,i}\sigma_1\sigma_i\right) - \frac{1}{m_i^2}\left(\alpha_{i+1}^2\sigma_{i+1}^2-  2\alpha_{i+1}\rho_{1,i+1}\sigma_1\sigma_{i+1}\right), \\
    e\left(\hat{y}\right)_{\alpha_j} &= 2\left( \frac{1}{m_{j-1}}-\frac{1}{m_j}\right)\left(\alpha_j\sigma_j^2-\rho_{i,j}\sigma_1\sigma_j\right),
\end{align*}
so that for any $i$ the first-order conditions for the optimality of \eqref{eq:ifunc} are
\begin{equation}\label{eq:optim}
\begin{alignedat}{2}
    &L^i_{m_j} = e\left(\hat{y}\right)_{m_j}+\lambda w_j+\mu_{j+1}-\mu_j - \xi\delta_{ij} = 0, \qquad && i\leq j\leq k,\\
    &L^i_{\alpha_l} = e\left(\hat{y}\right)_{\alpha_l} = 0, \qquad && i<l\leq k, \\
    &\sum_{j=i}^kw_jm_j = p - \sum_{j=1}^{i-1}w_j, \qquad && \\
    &-\xi m_i = 0, \qquad && \\
    &\mu_l\left(m_{l-1}-m_l\right) \leq 0, \qquad && i<l\leq k, \\
    &m_{l-1}-m_l \leq 0, \qquad && i<l\leq k, \\
    &\xi,\mu_{i+1},...,\mu_k \geq 0,
\end{alignedat}
\end{equation}
where $\delta_{ij}$ denotes the Kronecker delta function.  The reader can check by repeating the arguments in \cite{peherstorfer2016} that $L^i$ has a unique global minimum $\lr{\bb{m}^*,\bm{\alpha}^*,\bm{\mu}^*,\lambda^*,\xi^*}$ satisfying $0<m^*_{j-1}<m^*_j$ for all $j>i$.  To compute the minimizing vectors $\bb{m}^*,\bm{\alpha}^*$, it suffices to manipulate the system \eqref{eq:optim}. It follows immediately from the equations $L^i_{\alpha_l} = 0$ that $\alpha^*_l = \rho_{i,l}\sigma_1/\sigma_l$.  Moreover, $\xi^*, \bm{\mu}^* = 0$ at the minimum since the entries of $\bb{m}^*$ form a strictly increasing sequence, so using $\bm{\alpha}^*$ in the equation $L^i_{m_i} = 0$ gives
\[\lambda^* = \frac{\sigma_1^2\left(\rho_{1,i}^2-\rho_{1,i+1}^2\right)}{\lr{m_i^*}^2w_i},\]
which can be used in the remaining equations $L^i_{m_l}=0$ to derive the ratios,
\[ r_l^* = \frac{m_l^*}{m_i^*} = \sqrt{\frac{w_i}{w_l}\left( \frac{\rho_{1,l}^2-\rho_{1,l+1}^2}{\rho_{1,i}^2 - \rho_{1,i+1}^2}\right)}, \qquad i<l\leq k. \]
Extending this notation to include $r_i^*=1$ and using it alongside the fact
\[\sum_{j=i}^k w_jr_j^* = \frac{1}{m_i^*}\sum_{j=i}^k w_jm_j^*,\]
then yields the expression
\[ m_i^* = \frac{p-\sum_{j=1}^{i-1}w_j}{\sum_{j=i}^k w_jr^*_j}, \]
from which the remaining components of $\bb{m}^*$ are computed through $m^*_l = m^*_ir^*_l$.
\end{proof}

This suggests the provably budget-preserving MFMC algorithm which is outlined in Algorithm~\ref{alg:mod}.  At the first step, the optimal model set $\{f^1,...,f^k\}$ is selected as in \cite[Algorithm 1]{peherstorfer2016} and the continuous relaxation of \eqref{eq:MFMCmin} is solved using Lemma~\ref{thm:mod} with $i=1$.  This generates a solution pair $\lr{\bb{m}^*,\bm{\alpha}^*}\in\mathbb{R}^{k}\times\mathbb{R}^{k-1}$ which may or may not contain components $m_i^*<1$.  If no component of $\bb{m}^*$ is less than one, then no modification is needed and simple rounding produces the model evaluation numbers $\bb{m}^*\gets\lfloor\bb{m}^*\rfloor$ which are nearly optimal and preserve the computational budget.  Otherwise, at least the first component $m_1^*<1$ due to the increasing order of $\bb{m}^*$, so the first component is redefined to $m_1^*=1$ and the function $L^2$ is minimized for the remaining $k-1$ components of $\bb{m}^*$.  This gives a new solution vector $\lr{m_2^*,...,m^*_k}$ which may or may not contain entries less than 1.  If it does not, the model evaluation vector becomes $\bb{m}^* = \left(1,\lfloor m^*_2\rfloor,...,\lfloor m^*_k\rfloor\right)$.  Otherwise, this process is repeated and the functions $L^i$ are minimized until there are no more indices $i$ satisfying $m^*_i<1$.  The result is a pair $\lr{\bb{m}^*,\bm{\alpha}^*}\in\mathbb{R}^{k}\times\mathbb{R}^{k-1}$ which represents a small perturbation from continuous optimality at each step and is nearly optimal for the original problem \eqref{eq:MFMCmin}.  In fact, there is the following guarantee.

\begin{theorem}\label{thm:optim}
Suppose $\bb{m}^*$ is the solution to the relaxation of \eqref{eq:MFMCmin} with $m^*_1<1$.  Then, $m^*_1 \gets 1$ is optimal for \eqref{eq:MFMCmin}.  Similarly, $m^*_i\gets 1$ is optimal for \eqref{eq:MFMCmin} whenever $m^*_1 = ... = m^*_{i-1}=1$ are fixed and $m^*_i<1$ is defined by the global minimizer of $L^i$.
\end{theorem}

\begin{proof}
Choose  $1 = \bar{m}_1 = ... \bar{m}_{i-1}$ and consider varying $\bar{m}_i$.  Let $\lr{\bb{m},\bm{\alpha}} \in\mathbb{R}^{k-i}\times\mathbb{R}^{k-i-1}$ correspond to the unique global minimum of the Lagrangian $L^{i+1}$ from \eqref{eq:ifunc} for each $\bar{m}_i$,
\begin{equation*}
    m_{i+1} = \frac{p-\sum_{j=1}^{i}\bar{m}_jw_j}{\sum_{j=i+1}^k w_jr_j}, \qquad m_j = m_{i+1} r_j, \qquad \alpha_j = \frac{\rho_{1,j}\sigma_1}{\sigma_j}, \qquad i+1<j\leq k,
\end{equation*}
where $r_j$ is defined as in Lemma~\ref{thm:mod}.  Let $\tilde{\bb{m}}=\lr{\bar{m}_1,...,\bar{m}_i,m_{i+1},...,m_k}$ and consider the MSE at $\lr{\tilde{\bb{m}},\bm{\alpha}}$,
\begin{align*}
    \frac{e\left(\hat{y}\right)}{\sigma_1^2} &=  \frac{1}{\tilde{m}_1}-\sum_{j=2}^k\lr{\frac{1}{\tilde{m}_{j-1}}-\frac{1}{\tilde{m}_j}}\rho_{1,j}^2 \\
    &= \sum_{j=1}^{i-1}\frac{\rho_{1,j}^2-\rho_{1,j+1}^2}{\bar{m}_j}+\frac{\rho_{1,i}^2-\rho_{1,i+1}^2}{\bar{m}_{i}}+\frac{1}{m_{i+1}}\sum_{j=i+1}^k\frac{\rho_{1,j}^2-\rho_{1,j+1}^2}{r_j} =: f\lr{\bar{m}_i},
\end{align*}
where the first inequality uses the definition of $\bm{\alpha}$ while the second equality follows by re-indexing and the fact that $m_j = m_{i+1}r_j$ for all $j\geq i+1$.  We will show that $f$ is a convex function of $\bar{m}_i$.  Notice that for $j\geq i+1$,
\[\frac{1}{r_j} = \sqrt{\frac{w_j\lr{\rho_{1,i+1}^2-\rho_{1,i+2}^2}}{w_{i+1}\lr{\rho_{1,j}^2-\rho_{1,j+1}^2}}} = \frac{w_j\lr{\rho_{1,i+1}^2-\rho_{1,i+2}^2}}{w_{i+1}\lr{\rho_{1,j}^2-\rho_{1,j+1}^2}}\sqrt{\frac{w_{i+1}\lr{\rho_{1,j}^2-\rho_{1,j+1}^2}}{w_j\lr{\rho_{1,i+1}^2-\rho_{1,i+2}^2}}}, \]
where the term under the square root on the right-hand side is precisely $r_j$.  It follows that
\[ \sum_{j=i+1}^k\frac{\rho_{1,j}^2-\rho_{1,j+1}^2}{r_j} = \frac{\rho_{1,i+1}^2-\rho_{1,i+2}^2}{w_{i+1}}\sum_{j=i+1}^k w_jr_j,\]
and therefore $f$ has the form
\[ f\lr{\bar{m}_i} = C\lr{\bar{m}_1,...,\bar{m}_{i-1}} + \frac{\rho_{1,i}^2-\rho_{1,i+1}^2}{\bar{m}_{i}}+\frac{\rho_{1,i+1}^2-\rho_{1,i+2}^2}{w_{i+1}}\lr{\frac{\lr{\sum_{j=i+1}^k w_jr_j}^2}{p-\sum_{j=1}^i\bar{m}_jw_j}}, \]
where $C$ is constant in $\bar{m}_i$ and the definition of $m_{i+1}$ was used.  Now, differentiation in $\bar{m}_i$ yields
\begin{align*}
   f'\lr{\bar{m}_i} &= \frac{w_i\lr{\rho_{1,i+1}^2-\rho_{1,i+2}^2}}{w_{i+1}}\lr{\frac{\sum_{j=i+1}^k w_jr_j}{p-\sum_{j=1}^i\bar{m}_jw_j}}^2 - \frac{\rho_{1,i}^2-\rho_{1,i+1}^2}{\bar{m}_{i}^2}, \\
    f''\lr{\bar{m}_i} &= \frac{2w_i^2\lr{\rho_{1,i+1}^2-\rho_{1,i+2}^2}}{w_{i+1}}\frac{\lr{\sum_{j=i+1}^k w_jr_j}^2}{\lr{p-\sum_{j=1}^i\bar{m}_jw_j}^3} + \frac{2\lr{\rho_{1,i}^2-\rho_{1,i+1}^2}}{\bar{m}_i^3} \\
    &\coloneqq C_1\lr{p-\sum_{j=1}^i\bar{m}_jw_j}^{-3} + C_2\bar{m}_i^{-3},
\end{align*}
where $C_1,C_2$ are constants independent of $\tilde{\bb{m}}$.  Clearly $f'' > 0$ whenever $\sum_{j\leq i}\bar{m}_jw_j<p$ (i.e. whenever the choice of $\bar{m}_i$ is feasible for the budget), so that the MSE $e\left(\hat{y}\right)$ is convex in $\bar{m}_i$.  Therefore, $\bar{m}_i=1$ must be the optimal integer choice whenever $\bar{m}_i=m_i<1$ is defined by the global minimizer of $\bar{L}^i$.
\end{proof}

Theorem~\ref{thm:optim} provides a guarantee that the modification in Algorithm~\ref{alg:mod} is optimal for the original MIP \eqref{eq:MFMCmin}, and leads to the immediate conclusion that the modified MFMC Algorithm~\ref{alg:mod} is at least as optimal for \eqref{eq:MFMCmin} as the original algorithm from \cite{peherstorfer2016}.  Indeed, the above shows that $m^*_1 \gets 1$ is the integer value leading to the smallest $e\left(\hat{y}\right)$ whenever $m_1^*<1$ solves the relaxation of \eqref{eq:MFMCmin}.  Similarly, if $m_1^*=...=m_{i-1}^*=1$ and $m_i^*<1$ minimizes $L^i$, then $m_i^*\gets 1$ leads to the smallest $e\left(\hat{y}\right)$ and satisfies the constraints of the MIP \eqref{eq:MFMCmin}.  Repeating this sequentially until there are no more $m^*_i<1$ immediately yields the following Corollary.

\begin{corollary}\label{cor:optim}
The modification to MFMC in Algorithm~\ref{alg:mod} is optimal for \eqref{eq:MFMCmin} up to the rounding $\bb{m}^*\gets\lfloor\bb{m}^*\rfloor$ inherent in the original procedure.
\end{corollary}

Corollary~\ref{cor:optim} is analogous to the similar guarantees in \cite{peherstorfer2016,peherstorfer2018} which establish optimality of the model evaluation vector $\bb{m}^*$ for the relaxation of \eqref{eq:MFMCmin}.  On the other hand, the above result still cannot guarantee that the coefficients $m_i^*>1$ which solve the relaxed problem are optimal for the MIP \eqref{eq:MFMCmin}, although it is possible to find a small set of candidates which can be manually checked for optimality.  Indeed, the proof above shows that the MSE  $e\left(\hat{y}\right)$ is convex in the sub-leading entry $m^*_i$ of the global minimizer of $L^{i+1}$, so that given the minimizer for $L^i$ at each step the optimal choice for $m_i^*$ is either $\lfloor m_i^*\rfloor$ or $\lceil m_i^*\rceil$.  Therefore, a branch-and-bound algorithm based on this fact is guaranteed to produce a global minimum for \eqref{eq:MFMCmin} in at most $2^k$ steps.  As the simpler Algorithm~\ref{alg:mod} is already accurate enough for the present purposes, an approach based on this idea is left for future work.
\section{Numerical Experiments}\label{sec:numerics}

The results of the previous Section are now tested via two benchmark experiments which simulate low-budget use cases for MC and MFMC estimation.  More precisely, given computational models $\{f^i\}_{i=1}^k$ selected by Algorithm~\ref{alg:select} the modified MFMC Algorithm~\ref{alg:mod} is used to estimate the expectation of scalar quantities which arise from small-scale benchmark simulations in mechanics.  The performance of this estimation is evaluated for a range of computational budgets through comparison with simple MC estimation and a na\"{i}ve (generally infeasible) version of the original MFMC \cite[Algorithm 2]{peherstorfer2016}, whereby the optimal sampling $\bb{m}^*\in\mathbb{R}_+^k$ is rounded according to the rule
\[ m_i^* \gets \begin{cases}\lfloor m_i^*\rfloor & m_i^*\geq 1, \\
\lceil m_i^*\rceil & m_i^*<1\end{cases}. \]
Although this usually produces a sampling which exceeds the budget and is therefore impractical, its inclusion  provides a useful link between the modifications of Section~\ref{sec:mfmcnew} and the original MFMC algorithm.

The performance of each estimator $\tilde{y}$ produced by either MC or MFMC is measured using the MSE (or variance) $e\lr{\tilde{y}}$ and the relative MSE (or relative variance) $e\lr{\tilde{y}}/\lr{y_{\mathrm{ref}}^2}$, which provide a measure of spread around the desired expectation.  Since evaluating these MSEs requires an ``exact'' value $y_{\mathrm{ref}} \approx \EE{f^1}$ for comparison, for each example an MC approximation to $\EE{f^1}$ is constructed using $M>0$ i.i.d. samples of $Z$ for some large $M$,
\[y_{\mathrm{ref}} = \frac{1}{M}\sum_{m=1}^{M}f^1\lr{\bb{z}_m}.\]
Moreover, to further mitigate inherent dependence on stochasticity during these examples, the expectations and MSEs generated by MC and MFMC will be averaged over $N>0$ independent runs of each method, so that the estimators and MSEs used in practice are computed as
\[\bar{y} = \frac{1}{N}\sum_{j=1}^Ny_j, \qquad e\lr{\bar{y}} = \frac{1}{N}\sum_{j=1}^N \left(y_j - y_{\mathrm{ref}}\right)^2, \]
where $y_j \approx \EE{f^1}$ denotes the approximator produced by the relevant method (MC or MFMC) at run $j$.  Empirical results show that this facilitates a comparison which is more reproducible than if each method is carried out only once.  Finally, note that all experiments in this Section are carried out in Python 3.9 on a 2022 Macbook Pro with 32GB of RAM using a NumPy \cite{harris2020} implementation of Algorithms~\ref{alg:select} and \ref{alg:mod}.

\subsection{Analytical model of a short column}
The first example considered here is taken from \cite{peherstorfer2016} and involves an analytic model of a short column with rectangular cross-sectional area subject to bending and axial force.  The high-fidelity model is
\[ f^1(\bb{z}) = 1 - \frac{4z_4}{z_1z_2^2z_3} - \left(\frac{z_5}{z_1z_2z_3}\right)^2,\]
where $\bb{z} \in [5,15]\times[15,25]\times\mathbb{R}^+\times\mathbb{R}^2 \subset \mathbb{R}^5$.  The random variable $Z$ is chosen such that the width and depth $z_1,z_2$ are distributed uniformly, the yield stress $z_3$ is distributed log-normally with mean 5 and standard deviation 0.5, the bending moment $z_4$ is distributed normally with mean 2000 and standard deviation 400, and the axial force $z_5$ is distributed normally with mean 500 and standard deviation 100.  Since the computational cost of $f^1$ is arbitrarily, the low-fidelity surrogates and computational costs are chosen semi-arbitrarily. In particular, the surrogates are taken to be
\begin{align*}
    f^2(\bb{z}) &= 1 - \frac{3.8z_4}{z_1z_2^2z_3} - \left(\frac{z_5\left(1 + \frac{z_4-2000}{4000}\right)}{z_1z_2z_3}\right)^2, \\
    f^3(\bb{z}) &= 1 - \frac{z_4}{z_1z_2^2z_3} - \left(\frac{z_5(1+z_4)}{z_2z_3}\right)^2, \\
    f^4(\bb{z}) &= 1 - \frac{z_4}{z_1z_2^2z_3} - \left(\frac{z_5(1+z_4)}{z_1z_2z_3}\right)^2, \\
    f^5(\bb{z}) &= 1 - \frac{z_4}{z_1z_2^2z_3} - \left(\frac{z_5}{z_1z_2z_3}\right)^2,
\end{align*}
where we note that $f^3$ and $f^5$ are reversed from their definitions in \cite{peherstorfer2016}.  Denoting $\boldsymbol{\rho}_1 = \{\rho_{1,i}\}_{i=1}^k$, the weights and model correlation coefficients are
\begin{align*}
    \bb{w} &= \begin{pmatrix} 100 & 50 & 20 & 10 & 5 \end{pmatrix}^\intercal,\\

    \boldsymbol{\rho}_1 &= \begin{pmatrix} 1.0000000 & 0.99994645 & 0.6980721 & 0.92928154 & 0.99863737 \end{pmatrix}^\intercal,
\end{align*}
where 1000 i.i.d. samples of $Z$ are used to compute the displayed correlations.  With this data, the model selection Algorithm~\ref{alg:select} produces the subset $\mathcal{M}^*=\{f^1,f^2,f^5\}$ to be used in constructing the MFMC estimator $\hat{y}\approx\EE{f^1}$.  Here, the reference value $y_\mathrm{ref}$ is computed from $M=10^7$ i.i.d. samples of $Z$ and the estimators/MSEs are averaged over $N=1000$ independent runs.

\begin{table}[!ht]
\caption{Results of the short column experiment for various computational budgets $p$.  All errors are reported in units of $10^{-6}$.}\small
\smallskip
\centering
\begin{tabular}{ccccccccc}\toprule
\multicolumn{3}{c}{MC} & \multicolumn{6}{c}{Modified MFMC (Rounded MFMC)}
\\\cmidrule(lr){1-3}\cmidrule(lr){4-9}
           $p/w_1$ & $\hat{e}\left(\bar{y}^i_m\right)$ & $e\left(\bar{y}^i_m\right)$ & $p/w_1$ & \# $f^1$ & \# $f^2$ & \# $f^5$ & $\hat{e}\left(\hat{y}\right)$ & $e\left(\hat{y}\right)$ \\\midrule
2 & 25.94 & 25.29 & 2 (3.15) & 1 (1) & 1 (1) & 10 (33) & 5.020 (1.496) & 4.893 (1.458) \\
4 & 11.47 & 11.18 & 4 (4.8) & 1 (1) & 1 (1) & 50 (66) & 1.082 (0.8865) & 1.055 (0.8641) \\
8 & 5.860 & 5.712 & 7.5 (8.6) & 1 (1) & 1 (2) & 120 (132) & 0.4986 (0.4329) & 0.4860 (0.4219) \\
16 & 2.911 & 2.837 & 15.9 (16.2) & 1 (1) & 4 (4) & 258 (264) & 0.2434 (0.2159) & 0.2372 (0.2104) \\
32 & 1.463 & 1.426 & 31.45 (31.45) & 1 (1) & 8 (8) & 529 (529) & 0.1059 (0.1167) & 0.1032 (0.1137) \\
64 & 0.7411 & 0.7223 & 63.45 (63.45) & 2 (2) & 17 (17) & 1059 (1059) & 0.05698 (0.05485) & 0.05554 (0.05346) \\\bottomrule
\end{tabular}
\label{tab:toy}
\end{table}

Table~\ref{tab:toy} gives a numerical summary of the comparison between MC and both variants of MFMC when estimating the expectation $y=\EE{f^1}$, showing explicitly that the sampling Algorithms~\ref{alg:mod} and \cite[Algorithm 2]{peherstorfer2016} are generally inequivalent but reduce to the same procedure for large enough computational budgets.  Figure~\ref{fig:toy} plots the values resp. relative MSEs of the estimators obtained from MC and each MFMC method, along with shading indicating the standard deviation resp. variance in these quantities over the $N=1000$ runs.  As expected, each MFMC procedure is sufficient for a variance reduction of about an order of magnitude relative to MC regardless of the available budget.  However, the na\"{i}ve rounding procedure necessary for a direct application of \cite[Algorithm 2]{peherstorfer2016} clearly exceeds the budget in most cases, rendering it unreliable as an estimation method when computational resources are limited.  Despite this technical difficulty which has been addressed in Algorithm~\ref{alg:mod}, it is clear that MFMC estimation has benefits over traditional MC when a variety of information sources e.g. surrogate models are available.  Since MFMC is able to utilize more than just the high-fidelity model $f^1$, much of the computational budget can be loaded onto the cheaper surrogate models in order to decrease the MSE of the estimator $\hat{y}$.

\begin{figure}[!ht]
    \centering

    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/toy_Values.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/toy_Values_close.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \centering
        \includegraphics[width=\textwidth]{figs/toy_MSE.png}
    \end{minipage}
    \caption{Pictures corresponding to the short column experiment.  Plotted (left, middle) are the mean estimates and (right) MSEs over $N=1000$ independent runs of each method, along with shaded regions representing the standard deviations (left, middle) and variances (right, imperceptible) in these quantities over the $N$ runs.}
    \label{fig:toy}
\end{figure}

\subsection{Inviscid Burgers equation}

The next experiment involves a parameterized version of the Inviscid Burgers equation seen in  \cite{gruber2022comparison}, which is a common model for systems which experience shocks.  Let $x\in[0,100]$, $t\in[0,10]$, and $\bb{z} = \begin{pmatrix}z_1 & z_2\end{pmatrix}^\intercal \in [0.5,3.5]\times [2\times 10^{-4},2\times 10^{-3}]$ be a vector of parameters.  Consider the initial value problem,
\begin{align}\label{eq:burgers}
\begin{split}
    w_t + \frac{1}{2}\left(w^2\right)_x &= 0.02 e^{z_2 x}, \\
    w(0, t, \bb{z}) &= z_1, \\
    w(x,0,\bb{z}) &= 1,
\end{split}
\end{align}
where subscripts denote partial differentiation and $w=w(x,t, \bb{z})$ represents the conserved quantity of the fluid.  Discretizing the spatial interval $[0,100]$ with finite differences converts \eqref{eq:burgers} to a system of $n$ ordinary differential equations defined at the nodal points $\{x^\alpha\}_{\alpha=1}^{n}$, which can be further discretized in time and solved for any $\bb{z}$ using a simple upwind forward Euler scheme.  More precisely, let $\Delta x = 100/256, \Delta t = 1/10$ be stepsizes and let $w^\alpha_k$ denote the solution $w$ at time step $k$ and spatial node $\alpha$.  Then, forward differencing applied to \eqref{eq:burgers} gives the algebraic system
\begin{equation*}
    \frac{w_{k+1}^\alpha - w_k^\alpha}{\Delta t} + \frac{1}{2}\frac{\left(w^2\right)_k^\alpha - \left(w^2\right)_k^{\alpha-1}}{\Delta x} = 0.02e^{z_2 x}, \qquad 1\leq\alpha\leq n
\end{equation*}
which expresses the solution $w^\alpha(t,\bb{z})\coloneqq w(x^\alpha,t,\bb{z})$ at time $t = (k+1)\Delta t$ in terms of its value at time $t = k\Delta t$.  Collecting this for each $1\leq \alpha\leq n$ yields the vectorized solution $\bb{w}(t,\bb{z}) \in \mathbb{R}^n$ which is stored in practice and can be used to define a quantity of interest for statistical estimation.  In particular, let $Z$ be a random variable chosen so that its realizations $(z_1,z_2)$ are distributed uniformly in their ranges of definition.  Then, $\bb{w} = \bb{w}(Z)$ becomes a random variable and it is meaningful to consider the high-fidelity model
\[ f^1\lr{\bb{w}(Z)} = \frac{1}{n}\sum_{\alpha=1}^n w^\alpha_{T}, \]
where the notation $w^\alpha_{T}$ indicates the value of the solution at node $\alpha$ and final time $T=10$.  This gives an average measure of $w$ which is useful to track as a function of the random variable $Z$ and whose expectation $y = \EE{f^1}$ can be estimated using MC and MFMC.

To simulate a low-budget use case for MC/MFMC estimation in this context, access to the high-fidelity model $f^1$ is assumed along with an ensemble of reduced-order models (ROMs) based on proper orthogonal decomposition (POD) which will now be described.  Recall that POD uses snapshots of the high-fidelity solution at various time instances to generate a reduced basis of size $d \ll n$ whose span has minimal $\ell_2$ reconstruction error.  For this example, snapshot matrices $\bb{S} = \left(s^\alpha_j\right)$ with entries $s^\alpha_j = w^\alpha(t_j)$ are collected from simulations corresponding to 50 uniform i.i.d. samples of $Z$ and preprocessed by subtracting the relevant initial condition from their columns, generating matrices $\bar{\bb{S}} = \left(\bar{s}^i_j\right)$ with $\bar{s}^i_j = s^i_j - w^i(t_0)$.  These 50 matrices are then concatenated column-wise to form the full snapshot matrix $\bar{\bb{S}} = \bb{U}\bm{\Sigma}\bb{V}^\intercal$ (overloading notation), so that the optimal POD basis of dimension $d$ is contained in the matrix $\bb{U}_d$ containing the first $d$ columns of $\bb{U}$.  Denoting $\bb{w}_0 = \bb{w}(t_0)$ and $\hat{\bb{w}}\in\mathbb{R}^d$ a coefficient vector, this enables a POD approximation to the high-fidelity solution $\tilde{\bb{w}} = \bb{w}_0 + \bb{U}_d\hat{\bb{w}}, \approx \bb{w}$ which can be substituted into the discretization of \eqref{eq:burgers} to yield the reduced-order system
\begin{equation}\label{eq:burgersROM}
    \begin{split}
        \frac{\hat{\bb{w}}_{k+1}-\hat{\bb{w}}_k}{\Delta t} + \frac{1}{2}\left(\bb{a} + \bb{B}\hat{\bb{w}}_k + \bb{C}(\hat{\bb{w}}_k,\hat{\bb{w}}_k)\right) &=  0.02 \bb{U}_d^\intercal e^{z_2\bb{x}}, \\
        \hat{\bb{w}}_0 &= \bb{U}_d^\intercal(\bb{w}_0-\bb{w}_0) = \bm{0}.
    \end{split}
\end{equation}
where $0 \leq k \leq N_t-1$, $e^{z_2\bb{x}}$ acts component-wise, and $\bb{a},\bb{B},\bb{C}$ are vector-valued quantities which can be precomputed.  To describe this more explicitly, let $\bb{D}_x \in \mathbb{R}^{N\times N}$ denote the bidiagonal forward difference matrix with entries $\{0,1/\Delta x,...,1/\Delta x\}$ on its diagonal and $\{-1/\Delta x,..,-1/\Delta x\}$ on its subdiagonal. A straightforward computation then yields
\begin{align*}
\bb{a} &= \bb{U}_d^\intercal\bb{D}_x(\bb{w}_0)^2, \\
\bb{B} &= 2\bb{U}_d^\intercal\bb{D}_x\mathrm{Diag}(\bb{w}_0)\bb{U}_d, \\
\bb{C} &= \bb{U}_d^\intercal\bb{D}_x \mathrm{diag}_{1,3}(\bb{U}_d \otimes \bb{U}_d),
\end{align*}
where $(\bb{w}_0)^2$ is a component-wise operation taking place in $\mathbb{R}^N$, $\mathrm{Diag}(\bb{w}_0)$ indicates the diagonal matrix built from $\bb{w}_0$, and $\mathrm{diag}_{1,3}(\bb{U}_d\otimes\bb{U}_d)$ indicates the diagonal of $\bb{U}_d\otimes\bb{U}_d$ that occurs when components 1 and 3 are the same, i.e. the rank 3 tensor with components $u^\alpha_j u^\alpha_k$.  Solving the $d$-dimensional system \eqref{eq:burgersROM} for $\hat{\bb{w}}$ gives an approximate solution $\tilde{\bb{w}}$ which is computationally less expensive and can be made arbitrarily accurate by increasing the  dimension $d$.

\begin{figure}[!ht]
    \centering
    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/burgers_FOM_20.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/burgers_FOM_60.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \centering
        \includegraphics[width=\textwidth]{figs/burgers_FOM_100.png}
    \end{minipage}
    \caption{Snapshots of the high-fidelity PDE model and its POD-ROM surrogates corresponding to the inviscid Burgers example at times $t = 2,6,10$ and parameter value $\bb{z} = \begin{pmatrix}2.75 & 0.0275\end{pmatrix}^\intercal$.}
    \label{fig:burgersFOM}
\end{figure}

In view of this, the ROM \eqref{eq:burgersROM} can be used to obtain multifidelity surrogates for $f^1(Z)$ by varying the approximation quality of $\tilde{\bb{w}}$.  The present experiment considers a sequence of ROMs defined by $d = \{3, 5, 10, 15\}$ and their associated surrogate models $\{f^2,f^3,f^4,f^5\}$ defined analogously to $f^1$.  It is evident from Figure~\ref{fig:burgersFOM} that approximation quality increases with $d$, although this benefit comes with an increased computational cost.  Using 100 i.i.d. samples of $Z$ to compute the computational costs and inter-model correlations for the model set $\{f^i\}_{i=1}^5$ yields the data
\begin{align*}
    \bb{w} &= 10^{-4}\begin{pmatrix} 30.5625 & 5.5174 & 5.8633 & 6.3854 & 7.4522 \end{pmatrix}^\intercal,\\
    \boldsymbol{\rho}_1 &= \begin{pmatrix} 1.0000000 & 0.99766585 & 0.98343683 & 0.99999507 & 0.99999882 \end{pmatrix}^\intercal,
\end{align*}
where the cost vector $\bb{w}$ is computed as the average time (in seconds) necessary to evaluate each model.  From this, the model selection Algorithm~\ref{alg:select} chooses the subset $\{f^1, f^4, f^2\}$ for use in constructing the MFMC estimator $\hat{y}$ which will approximate $y=\EE{f^1}$.  To evaluate the quality of MC and MFMC estimation, a reference MC approximation $y_\mathrm{ref}\approx \EE{f^1}$ is generated from $10^5$ i.i.d. samples of $Z$, and each estimator/MSE computed is averaged over $N=100$ independent runs of its relevant method.

\begin{table}[!ht]
\caption{Results of the Inviscid Burgers experiment for various computational budgets $p$.  All errors are reported in units of $10^{-3}$.}\small
\smallskip
\centering
\begin{tabular}{ccccccccc}\toprule
\multicolumn{3}{c}{MC} & \multicolumn{6}{c}{Modified MFMC (Rounded MFMC)}
\\\cmidrule(lr){1-3}\cmidrule(lr){4-9}
           $p/w_1$ & $\hat{e}\left(\bar{y}^i_m\right)$ & $e\left(\bar{y}^i_m\right)$ &   $p/w_1$ & \# $f^1$ & \# $f^4$ & \# $f^2$ & $\hat{e}\left(\hat{y}\right)$ & $e\left(\hat{y}\right)$ \\\midrule
2 & 17.35 & 36.96 & 1.93 (3.01) & 1 (1) & 1 (1) & 4 (10) & 8.077 (2.404) &  17.20 (5.120) \\
4 & 8.256 & 17.20 & 3.92 (4.82) & 1 (1) & 1 (1) & 15 (20) & 1.745 (1.493) & 3.716 (3.181) \\
8 & 3.428 & 7.302 & 7.92 (8.64) & 1 (1) & 2 (2) & 36 (40) & 1.145 (0.9163) & 2.440 (1.952) \\
16 & 1.852 & 3.945 & 15.7 (16.7) & 1 (1) & 4 (5) & 77 (81) & 0.3487 (0.2590) & 0.7427 (0.5517) \\
32 & 0.9291 & 1.979 & 31.8 (32.5) & 1 (1) & 10 (10) & 159 (163) & 0.1713 (0.1588) & 0.3649 (0.3383) \\
64 & 0.4641 & 0.9885 & 63.9 (64.2) & 1 (1) & 20 (20) & 325 (327) & 0.07745 (0.1039) & 0.1651 (0.2213) \\\bottomrule
\end{tabular}
\label{tab:burgers}
\end{table}

The results of this experiment are displayed in Table~\ref{tab:burgers} and Figure~\ref{fig:burgersROM}.  Again it is clear that simply rounding the solution of \cite[Algorithm 2]{peherstorfer2016} is not sufficient for producing a feasible MFMC estimator in the presence of limited computational resources, as the target budget is exceeded (sometimes greatly so) in every case.  This confirms that the modified MFMC Algorithm~\ref{alg:mod} is necessary for applying MFMC estimation in these cases, and gives confidence that Algorithm~\ref{alg:mod} will also be effective on large-scale problems where such resource restrictions are commonplace.  Moreover, it is also evident from the results presented here that Algorithm~\ref{alg:mod} preserves the accuracy benefits of MFMC estimation over simple MC, producing a notably lower MSE regardless of the size of computational budget.

\begin{figure}[!ht]
    \centering
    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/burgers_Values.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \includegraphics[width=\textwidth]{figs/burgers_Values_close.png}
    \end{minipage}%
    \begin{minipage}{0.33\textwidth}
        \centering
        \includegraphics[width=\textwidth]{figs/burgers_MSE.png}
    \end{minipage}
    \caption{Pictures corresponding to the inviscid Burgers experiment.  Plotted (left, middle) are the mean estimates and (right) MSEs over $N=100$ independent runs of each method, along with shaded regions representing the standard deviations (left, middle) and variances (right, almost imperceptible) in these quantities over the $N$ runs.}
    \label{fig:burgersROM}
\end{figure}

\section{Conclusion}
A method for the multifidelity Monte Carlo estimation of statistical quantities has been presented which is applicable to computational budgets of any size and reduces variance when compared to simple Monte Carlo estimation.  To accomplish this, existing MFMC technology from \cite{peherstorfer2016} has been adapted and modified, leading to the MFMC estimation Algorithm~\ref{alg:mod} which is fast to compute and simple to implement.  It has been shown through Theorem~\ref{thm:mod} and Corollary~\ref{cor:optim} that the proposed algorithm solves the sampling problem \eqref{eq:MFMCmin} at least as optimally as its namesake, and numerical experiments have been conducted which validate this fact. Forthcoming work in  \cite{gruber2022climate} will investigate applications of the present MFMC method to complex systems governed by partial differential equations, particularly in the context of climate modeling.

redpajama_set_name: RedPajamaArXiv


text:
Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan: Time to say good-by to the old remedies
Osuke Iwata, Toshiki Takenouchi
Brain and Development
https://doi.org/10.1016/j.braindev.2011.06.008
Induced Hypothermia
Iwata, O., & Takenouchi, T. (2012). Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan: Time to say good-by to the old remedies. Brain and Development, 34(2), 163-164. https://doi.org/10.1016/j.braindev.2011.06.008
Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan : Time to say good-by to the old remedies. / Iwata, Osuke; Takenouchi, Toshiki.
In: Brain and Development, Vol. 34, No. 2, 02.2012, p. 163-164.
Iwata, O & Takenouchi, T 2012, 'Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan: Time to say good-by to the old remedies', Brain and Development, vol. 34, no. 2, pp. 163-164. https://doi.org/10.1016/j.braindev.2011.06.008
Iwata O, Takenouchi T. Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan: Time to say good-by to the old remedies. Brain and Development. 2012 Feb;34(2):163-164. https://doi.org/10.1016/j.braindev.2011.06.008
Iwata, Osuke ; Takenouchi, Toshiki. / Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan : Time to say good-by to the old remedies. In: Brain and Development. 2012 ; Vol. 34, No. 2. pp. 163-164.
@article{a0b4fe9c9f364161b1161c6e966f7627,
title = "Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan: Time to say good-by to the old remedies",
author = "Osuke Iwata and Toshiki Takenouchi",
doi = "10.1016/j.braindev.2011.06.008",
journal = "Brain and Development",
T1 - Past, present and future of hypothermic neuroprotection for neonatal encephalopathy in Japan
T2 - Time to say good-by to the old remedies
AU - Iwata, Osuke
AU - Takenouchi, Toshiki
U2 - 10.1016/j.braindev.2011.06.008
DO - 10.1016/j.braindev.2011.06.008
JO - Brain and Development
JF - Brain and Development
10.1016/j.braindev.2011.06.008

redpajama_set_name: RedPajamaCommonCrawl


text:
est une série d'animation américaine produit par Teamto sous la supervision de l'auteur-producteur Eric Rogers, inspirée de la série de jeux vidéo Skylanders créée par Activision, produite par Activision Blizzard Studios et diffusée depuis le  sur Netflix.

Synopsis
Dans la lutte contre les forces des ténèbres sur la Terre des Skylands, l'Académie des Skylanders dirigée par Maître Eon représente le meilleur espoir de paix. Alors que Stealth Elf, Spyro, Eruptor, Pop Fizz et Jet-Vac forment l'équipe la plus efficace pour repousser le Mal, Kaos, l'ennemi le plus acharné des Skylanders, élabore sans cesse de nouveaux complots afin que les Ténèbres règnent en maître.

Production
La série a été annulée après 3 saisons le .

Réalisateurs
Arthur Qwak (saison 1,2,3)
Solène Azernour (saison 1,2)
Alban Rodriguez (saison 3)

Distribution
La voix exprimée pour la plupart des personnages est différente afin de donner à cette série une identité distincte des jeux vidéo. Les versions anglaise et française sont touchées par cela. Aussi dans cette série, Alexandre Gillet qui avait déjà doublé Spyro dans la série de jeux video The Legend of Spyro et non dans celle des Skylanders, reprend le rôle du dragon violet pour le doublage français de la série télévisée. Nathalie Homs ne reprend pas le rôle de Cynder, bien qu'elle l'avait doublé dans la trilogie de The Legend of Spyro.

Voix originales

Voix françaises

 Alexandre Gillet : Spyro
 Patrice Baudrier : Eruptor
 Marc Saez puis Xavier Fagnon : Pop Fizz
 Céline Melloul : Stealth Elf
 Geneviève Doang : Cynder
 Gilbert Lévy : Kaos
 Martial Le Minoux : Jet-Vac
 Frédéric Cerdal : Maître Eon
 Jérôme Pauwels : Glumshanks
 Sophie Riffont : Golden Queen
 Cédric Dumond : Broccoli Guy
 Boris Rehlinger : Bad Breath
 Xavier Fagnon : Chompy Mage
 Antoine Tomé : Pepper Jack
 Bernard Métraux : Strykore, Snapshot
 Frédéric Popovic : Wolfgang
 Isabelle Leprince : Kaossandra
 Stéphane Roux : Crash Bandicoot
 Peggy Martineau : Coco Bandicoot
 Pauline Moulène : Sprocket
 Oscar Douieb : Fire Viper
 Emmanuel Rausenberger : Hugo
 Emmanuel Curtil : Flynn
 Michel Vigné : Malefor
 François Jerosme : King Pen
 Stéphane Ronchewski, Sylvain Lemarié, Magali Rosenzweig, Pascal Casanova, Cécile Gatto, Raphaël Cohen, François Delaive, Sacha Petronijevic, Marie Diot, Christian Diaz, Benjamin Bollen, Gilles Bellomi, Brigitte Guedj, Alain Dumas, Nathalie Kanoui, Élisa Bourreau : voix additionnelles

 Version française
 Studio de doublage : Deluxe Media Paris (saison 1), BTI Studios (saisons 2-3)
 Direction artistique : Jérôme Pauwels
 Adaptation : Ilana Castro, Corinne Hyafil

Épisodes

Le premier épisode a été présenté lors de la Paris Games Week 2016.

Première saison (2016)
 La réunion des Skylanders ()
 Méthode Skylanders ()
 Tous pour un ()
 Manipulations ()
 L'évasion ()
 Les envahisseurs de l'esprit ()
 La colère d'Eruptor ()
 Une célébrité parmi nous ()
 La science de la barbe ()
 Skylands en danger ()
 Loin de chez soi ()
 L'invasion de l'Académie Skylanders ()

Deuxième saison (2017)
Cette saison de treize épisodes a été mise en ligne le .
 Spyromanie ()
 Rêve de Ninjini ()
 À la recherche de Cynder ()
 Un Thankstaking mémorable ()
 Élémentaire, mon cher Eruptor ()
 Avis partagé ()
 Le procès de Pop Fizz ()
 Le virus des Skylands ()
 Le ventre du monstre ()
 Qui est ton papa ? ()
 Rêves de mouton(-ball) ()
 À deux, c'est mieux ()
  Une touche maléfique ()

Troisième saison (2018)
Le , la série est renouvelée pour une troisième saison prévue pour le .
 La lutte pour le pouvoir
 La vérité est ici
 Une rencontre explosive
 Une traitresse parmi nous
 Le fameux capitaine
 Un week-end chez Eon
 Enragée
 Trois versions pour une histoire
 Retour vers le futur
 La course des courses
 Trouble de la personnalité
 Les aventuriers de l'île perdue,  partie
 Les aventuriers de l'île perdue,  partie

Références

Liens externes

Série télévisée créée en 2016
Série télévisée américaine des années 2010
Série d'animation de Netflix
Série d'animation américaine

redpajama_set_name: RedPajamaWikipedia


text:
Q: read Json contents on Restful webserver I have a JSON content as below and able to parse the data with following portion of code.However I have stored the content in a flat file and parsed it. Need to do the same when I get the request in Restful webserver.
Code to read from file:(Working)
data = new Gson().fromJson(new FileReader("/home/raj/apache-tomcat-8.0.3/webapps/file.json"), Data.class);

JSON data:
{
        "DeviceCommon": {
            "ASIdentifier": "123",
            "DatadeliveyMechanism": "notify",
            "MobileOriginatorCallbackReference": {
                "url": "http://application.example.com/inbound/notifications/modatanotification/"
            },
            "AccessiblityCallbackReference": {
                "url": "http://application.example.com/inbound/notifications/accessibilitystatusnotification"
            }
        },
        "DeviceList": [{
            "ExternalIdentifer": "123456@mydomain.com",
            "msisdn": "123456",
            "senderName": "Device1",
            "MobileOriginatorCallbackReference": {
                "notifyURL": "http://application.example.com/inbound/notifications/modatanotification/"
            },
            "ConfigurationResultCallbackReference": {
                "notifyURL": "http://application.example.com/inbound/notifications/configurationResult"
            },
            "ASreferenceID": "AS000001",
            "NIDDduration": "1d"
        }]
    }

POJO class(Data)
import com.google.gson.annotations.SerializedName;

class Data {

    @SerializedName("DeviceCommon")
    DeviceCommon deviceCommon;

    @SerializedName("DeviceList")
    DeviceListEntry[] deviceList;

    @Override
    public String toString() {
        return "Data{" +
                "\n  deviceCommon=" + deviceCommon +
                "\n  deviceList=" + Arrays.toString(deviceList) +
                "\n}";
    }
}

REST server code(not working)
@POST
    @Path("/adddevice")
    @Produces(MediaType.APPLICATION_JSON)
    @Consumes(MediaType.APPLICATION_JSON)
    public Response  createConfigRequest(Data configRequest) {
             Gson gson = new Gson();
            String json = gson.toJson(Data.class);
            log.info("json string is"+json);

}
error:
Unrecognized field "DeviceCommon" (class com.AS.service.Data), not marked as ignorable (0 known properties: ])
 at [Source: org.glassfish.jersey.message.internal.ReaderInterceptorExecutor$UnCloseableInputStream@724206cc; line: 2, column: 17] (through reference chain: com.affirmed.AS.service.Data["DeviceCommon"])

redpajama_set_name: RedPajamaStackExchange


text:
President Coolidge, Taken on the White House Ground is een Amerikaanse korte film uit 1924. De film duurt 4 minuten en bestaat volledig uit een opname van president Calvin Coolidge, die een speech geeft. Het is de eerste filmopname van een Amerikaanse president met geluidsopname.

Verwijzingen

 President Coolidge, Taken on the White House Ground op Internet Archive

Amerikaanse film
Film in het publiek domein
Korte film
Film uit 1924


text:
Q: What is the probability that in a sample of $5$ people that all of them are blonde? at least two of them are blonde? I am dealing with this probability problem and I am stuck. I would really appreciate if you could please help me. Thanks in advance.

In a population of $80$ people, $30\%$ are blonde and $70\%$ are black-haired. If we take a sample of $5$ guys,

*

*what is the probability that all of them are blonde?

*what is the probability that at least two of them are blonde?

I have thought for the first one that it should be , by Laplace's rule, $\frac{24\choose 5}{80\choose 5}$. But, I also would say that it is true that it is a binomial distribution and thus it should be $(0.3)^5$. Nevertheless, these answers do not coincide.

A: If it is a selection of $5$ distinct persons out of $80$ then hypergeometric distribution must be applied (as you did). Not binomial. That must be applied if the persons are replaced after selection so that a person can be selected more than once. So your answer of 1) is okay.
Also for 2) we use hypergeometric distribution and if $W$ denotes the number of blondes we find probability:$$P(W\geq2)=\sum_{w=2}^5P(W=w)=\sum_{w=2}^5\frac{\binom{24}w\binom{56}{5-w}}{\binom{80}5}$$
A summation of $4$ terms. We can make it a bit less complicated like this:$$P(W\geq2)=1-P(W\leq1)=1-P(W=0)-P(W=1)=1-\frac{\binom{24}0\binom{56}{5}}{\binom{80}5}-\frac{\binom{24}1\binom{56}{4}}{\binom{80}5}$$


text:
Union's Derrick Jones ready to make next step with Bedoya away
By Dave Zeitlin @DaveZeitlin
CHESTER, Pa. — Before departing for US national team camp, Alejandro Bedoya left Derrick Jones with a message: Don't get another red card.
Jones, who indeed drew a questionable red in his last start on June 18, recounted that story with a laugh from the Philadelphia Union's training facility Friday. But the 20-year-old midfielder knows the opportunity in front of him is a serious one as he prepares to take over Bedoya's starting role, beginning with Sunday's nationally televised matchup vs. the New England Revolution (5 pm ET | FS1, FOX Deportes, MLS LIVE in Canada).
"I look up to him and he's a very good player," Jones told MLSsoccer.com. "Now that he's not here I have to just fill in his spot and work as hard as he does."
Bedoya — who could miss the next five Union games while at the Gold Cup — told reporters after Philly's 1-0 win over D.C. United this past weekend that he has "so much faith" in Jones filling in for him while he's gone.
Union head coach Jim Curtin also believes the time is right for Jones to take the next step in his career.
"Through his fearlessness, through how brave he is, through how competitive he is, he gets better and better each and every day he goes out on the field," Curtin said Friday during his weekly press conference. "Are there still gonna be mistakes? Absolutely. But this is a big time for him now with Ale being out for these next two games. He'll have his hands full. But his fearlessness is what [sporting director Earnie Stewart] and I love in him. It's what we see every day in training, and we know he'll be up to the challenge."
Jones, who was signed to a Homegrown deal out of the Union's youth academy a little less than a year ago, has been thrown into the fire before. He started the Union's first five games of the season, including a daunting opener at Vancouver, before being shifted to a reserve role when Bedoya was dropped into a deeper role. And the Ghana-born midfielder burst onto the international scene as a key starter during the U-20 World Cup just after getting clearance to play for the United States.
He and Curtin know there's still plenty of room for improvement, with the Union coach pointing to his "short-range and long-range passing" and Jones to his finishing.
Now seems like like a good opportunity for him to work on those things and continue developing into the star the Union believe he can be.
"I think maybe people know me a little bit," said Jones, who admits he's hoping to get noticed by the senior national team after his U-20 run. "But I have to keep playing good and working hard for a lot of people to know me more."
Derrick Jones Philadelphia Union
What each Eastern Conference team should be worried about in 2022
MLS Eastern Conference roster build status for 2022 season
Leeds United, AC Milan, RB Leipzig chasing USMNT, ex-Union midfielder Brenden Aaronson
Toronto FC's GoalMachine takes home eMLS League Series One title
Transfer Tracker
Inter Miami CF sign Jamaica international defender Damion Lowe
Reports: Brian Rodriguez, Ezequiel Barco linked to Brazilian Serie A loans
Charlotte FC sign free-agent goalkeeper Adrián Zendejas
Chile's Unión Española announce arrival of Nashville attacker Rodrigo Piñeiro
Ex-Orlando City SC star Nani makes big debut for Venezia
RECAP: GoalMachine (TOR) vs Joksan (NSH) | 2022 League Series 1 Final
GoalMachine (TOR) match winner vs PauloNeto (ATL) | 2022 League Series 1 Final
GoalMachine (TOR) goal #3 vs PauloNeto (ATL) | 2022 League Series 1 Final


Get bored with trousers? Summer comes and just wear shorts! Welcome to Dresslily and you can enjoy free shipping and up to 60% off for all shorts. Just come and buy!
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Unleashed Multi-Site Manager is an intuitive, easy to use Linux based software for building and managing small business Wi-Fi networks across multiple sites. It enables owners of Unleashed networks to easily and securely monitor, manage and configure the network from anywhere in the world with a single sign-on.
Unleashed Multi-Site Manager provides a single pane of glass visibility and management of Unleashed networks.
Unleashed Multi-Site Manager enables customers to perform remote monitoring, centralized management and detailed reporting of all Unleashed networks. Unleashed Multi-Site manager can also be leveraged to manage ZoneDirector (ZD) based networks from a single pane of glass.


Fran O'Brien '65 was never particularly political. Slight and a little shy, O'Brien wanted to be a teacher, spending her days with children. She wasn't one to demonstrate on a street corner or shout at a rally.
But in the upheaval of the early 1960s, O'Brien found an unexpected passion — not political so much as religious.
A deeply involved member of Pacific University's Student Religious Council, O'Brien had a fundamental belief in social justice. It was that belief that led her to become the only Pacific University student to participate in the Freedom Summer of 1964.
The idea was to send white college students to Mississippi — then the heart of America's racial inequity — to help black citizens navigate the murky, and often dangerous, process of voter registration, to supplement the education of students whose schools were still separate and far from equal, and to bring white America's attention to the burgeoning Civil Rights Movement.
It was an idealistic — and controversial — idea, and not one that immediately appealed to O'Brien.
She didn't have the knowledge to teach Mississippi's constitution to adults or to give black high-schoolers the physics or algebra lessons they needed, she told herself.
Then, she heard they needed teachers to take care of younger children too, to run recreation programs and art and crafts.
Still, she hesitated — an application completed, but waiting to be submitted.
It was the film Judgment of Nuremburg that changed her mind.
"It would be one thing if I didn't know, but I had in my hand the opportunity to do something. If a door is locked and you don't go through, that's one thing. But if it's wide open and someone says, 'Come in,' that's different.
It was never supposed to be an easy job.
Volunteers for Freedom Summer were carefully screened to keep out anyone who might not be able to follow the principles of complete nonviolence. And, volunteers were required to come with money for transportation and living costs — plus a fund for bail, if and when they were arrested for their efforts.
The costs became apparent quickly. O'Brien was in the second wave of volunteers to go to Oxford College in Ohio for training before riding buses south to Mississippi.
"We arrived on Sunday afternoon. On Monday, we heard that three of the men who had gone down the week before were missing.
At no time did I doubt I should have been there. Sometimes I wondered why; often I wondered what I would do and how on earth I could possibly be useful; but never did I use the question word "IF." It did not occur to me that I might have made a mistake in coming. I KNEW I was meant to be exactly where I was. That is faith. In my case, it led to a unique type of activism.
O'Brien spent the rest of her week-long training gathering ideas for simple art projects for her students and learning how to drop into the fetal position to protect her vital organs if attacked.
O'Brien was assigned to Vicksburg, a relatively safe — by Mississippi standards — community, where she found many people welcoming, if cautious.
The retired schoolteacher with whom she lived, Mrs. Garrett, was different.
O'Brien taught arts and crafts, led games, played the piano, and started a children's chorus. But when the children started asking about the American history lessons their older siblings were learning, she paused. There were no children's history books.
"Mrs. Garrett said, 'Fran, I didn't have books. If you know the material, you can teach it.' The problem was, I didn't know the material. I was minoring in history at Pacific, but I'd never heard of Fredrick Douglass or Harriet Tubman.
And so O'Brien did, helping her young students discover their role in American history, smiling as they added verses to songs like America the Beautiful, and helping inspire them to put on their own American history pageant.
Her work even drew the praise of Martin Luther King Jr., who visited the Vicksburg site in July 1964.
Then, getting serious as only Martin Luther King could get serious, he said, 'Young lady, don't you ever say you 'just work with the kids.' Our children are the future and you are forming it." After dinner, King spoke to a boisterous crowd, but Fran O'Brien would not remember a thing he said that was more important than what he said to her.
If King's words would leave an indelible mark in her heart in the years to come, another experience would leave a shadowy reminder of the Freedom Summer, one that would trigger nightmares and denial for years to come.
It was a rainy night, and she was waiting for a ride at the bottom of a hill in front of the Vicksburg Freedom House. When the car arrived, there wasn't room for everyone, so she offered to stay behind for the next car.
The men took O'Brien to a vacant lot nearby — or so she believes. At the time, fear distorted her senses. She had no idea where she was or how long she was gone.
When she woke up, she was back at the Freedom House. She thought she'd been gone for hours, but in reality it was only about 30 minutes. She hadn't even been missed. In shock, she blamed herself, thought herself foolish.
She carried the secret for the next 25 years.
When O'Brien returned to Pacific University for her senior year, she struggled to express what she'd seen and experienced.
"It suddenly seemed like everybody had changed. People who had just seemed to have a different opinion from mine were suddenly horribly racist and bigoted," she said.
"The most difficult were the people who I thought asked very stupid questions: 'Can't it just change gradually?' I'd forgotten that three months earlier I had thought the same thing.
Nightmares would follow her for years.
But so, too, would her renewed passion for teaching.
O'Brien had suffered polio when she was 12 and was drawn to teaching children with disabilities after volunteering at a summer camp shortly thereafter.
After graduating from Pacific, she went on to earn a master's in special education from San Francisco State University and spent some 36 years teaching.
Her Freedom Summer experiences have been written up in two books and a variety of articles, but it's likely the three and a half decades working with children that have made the most difference in the world.
"A lot of the issues (in special education) were very similar to the Civil Rights Movement," she said.
She remembers one young boy who had suffered a hand injury when he was 4.
"His first-grade teacher thought his hand was too unsightly to be in school and had put him in a special education class," she said. "He had gradually been doing less and less academically, because at the time, special ed had very low expectations.
That's what she had seen in Mississippi.
O'Brien retired 10 years ago, unable to keep up with the physical demands of working with children with disabilities. But she continues to tutor and teach Sunday school — and neither the lessons of Mississippi or the faith that took her there have faded.
Excellent, excellent article!! Fran truly is an inspirational and amazing lady! What lessons we can learn from this article and from her experiences.
Fran was a freshman with me in the dorm at Pacific University. I really liked her, and am thrilled to have read her story. Many of us in that freshman dorm did our best to work toward racial harmony. I graduated from Stanford, and eventually taught at Merritt College in Oakland for 30 years, in the Landscape Horticulture Department – working with people from all races and religions. Old values seem to "die hard," but her contribution is, indeed, wonderful! Thank you so much Fran!
We knew Fran from church and always found her faith and kindness an inspiration. We moved from California to Illinois and miss our old church family very much.


From internal communications, to annual reports, outdoor and experiential, I'm passionate about concept and design, and aligning them with strategy.
From helping people know where they stand financially, to understanding careers in the foreign service, the tools and sites I've worked on educate audiences in a fun and interactive way.
Whether it's an interactive tool, a website or an online campaign, I ensure that digital creative is a compelling experience that rewards users with knowledge.
From illustrating creative ways to afford college, to chronicling the transformation of New Jersey's largest city, tracing the 140-year mission of an iconic American brand, and developing a video for UBS's new Wealth Management unit, I have been the lead on projects from concept to storyboard, casting and production.
New Jersey's largest city is also a big success story. The award-winning #20StoriesofStrength brings Newark's people, places and promise to life through images and :15 video vignettes on Instagram.
As creative lead, I was involved in directing the photos shoots, videos, as well as editing the videos to a :15s story.
In the first two weeks, this social campaign quickly spread across Instagram, generating 32,536 total audience engagements, as well as thousands of video views and hundreds of positive conversations, resulting in over 2.3 million impressions across social media.
In Masterpiece of Love, four artists were commissioned to create one-of-a-kind works to tell the stories of people who had recently lost loved ones. My creative partner and I managed the artists' progress, designed the exhibit, oversaw its construction and installation, produced the debut event at the Newark Museum, and organized employee engagement efforts.
Disco Picnic was a one-of-a-kind picnic basket for the Picnic by Design event to raise funds to support nonprofits combating HIV/AIDS. I concepted and built the mirror-ball basket, designed the record plates, and staged the experience.
Grief is one of life's most isolating experiences. But what if we could make things just a little bit better for those who are grieving? This is Masterpiece of Love, a series of short films celebrating life, love, art and the power of the human spirit. From concept to art direction and working with composer on music, these films are creating a positive conversation about death and life insurance on Prudential's social media.


text:
Every school has its own 'dark social network' – WhatsApp or Facebook Messenger groups used by parents of children in the same class or year group and set to private viewing by members only. Parents must be told to use these responsibly – they might use advanced privacy settings, but that won't reduce the risk of offence being caused or prevent pictures from being widely shared. Parents should be reminded regularly to contact the school for facts, rather than spreading rumours and gossip – and to contact the school themselves if they feel a particular online exchange or message thread has gone too far and started to damage the school's reputation.
Yet while social media and messaging might have become parents' chief means of communicating, you also need to keep talking to them. Organising regular parent forums or a weekly 'open door' morning will provide opportunities for concerns to be shared face to face.
Defuse the situation
Taking the steps above should help you become aware of potential crises at an earlier stage. Having been made aware, here's what to do next:
1. Arrange a meeting and open it up to all those who have concerns. You should always have another member of staff or a governor present, with one of you taking notes.
2. Once you've heard the problem, begin your response by clearly setting out the facts and the purpose of any change or decision you've made, as there can often be a lot of miscommunication. In one recent example, parents heard that a school's new food supplier would not be supplying truly halal meat, but were reassured once they had a chance to question the school further.
3. While you won't want to change or roll back a policy (and in 95% of cases you'd be in the right), you should always listen to any concerns regarding its implementation. Every year, schools introduce new uniforms or hairstyle policies at short notice that can carry harsh penalties; allowing parents and students more time to adjust will often solve the issue.
4. Communicate what happened at the meeting and what was agreed back to your community, and again let them respond. That might seem to take up a lot of time, but it's nothing compared to what a full-blown crisis will involve.
5. Try to involve parents in solving the problem. In one school, a parent who had been upset by a lack of awareness raising around LGBT parenting was delighted when they were asked to research and select suitable books on the subject for the school library.
6. Consider making exceptions for individuals. An example of this might be allowing a child to carry a mobile phone so they can contact a surviving parent after a bereavement.
Seek help from within, and potentially outside your organisation. Tell your chair of governors, LA or trust about your problem and what you propose to do. Consult your crisis or reputation management policy, and consider bringing in additional support to help you monitor your social media or manage your media relations.
Create a short, clear statement that sets out the issues and actions you've taken. The statement should be easy to share by email, on your website and via your social media channels, so make sure you have access to these at all times, all year round.
If you have a good relationship with the local press, talk to them first. This can works well, for example, with issues revealed in a poor Ofsted report. There will still be a negative story, but at least the context for the issue can be communicated. You can also arrange to give a statement on camera – this will often be the best way of dealing with serious legal issues, such as convictions for abuse.
Negative stories on social media will inevitably reach the local papers. Share your statement widely with parents, staff and governors, and ask them to pass on any media enquiries to the school. This will reduce the pressure on them if they are 'doorstepped', and hopefully avoid them having to say 'no comment'!
Finally, it's rare for parents to progress to the stage of threatening members of staff, but it does happen. In such situations, you need to focus on safeguarding your team and your pupils – report any incidents to the police and LA and ask for their support.
Simon Hepburn is Director of Marketing Advice for Schools; Linda Tanner is a journalist, chair of governors and schools communications consultant with Local Voice Media.
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How Should You Respond to Parent Protests?
Parental Communications October 29, 2018, 14:14 GMT+1
Read in 7 minutes
One of your decisions has prompted some parents to set up a Facebook group and organise a protest outside the school gates. What should be your next move?
For many heads we've worked with, a parent protest is their worst nightmare – something seemingly out of their control, with the potential to seriously damage their own and their school's reputation.
However, there are steps school leaders can take to reduce the likelihood of such events happening in the first instance, and things they can do to minimise the impact of any that do occur. Here, we'll go through the options available to you for protecting your school from the start, defusing situations as they arise and getting your story across when issues escalate.
Protect your school
When setting up a communication channel with the wider community, the school must retain control and safeguard everyone involved, while making it easy to share facts.
With social media and online messaging dominating the way parents communicate, schools must own their social media platforms and have clear policies in place for those staff and parents who use them – the days of allowing a PTA group to manage your Facebook presence are long gone! Switching on the moderation function of a school Facebook page will enable negative messages to be intercepted, though they will, of course, still need to be acknowledged and addressed.
You also need to monitor wider social media, using applications such as Hootsuite or Social Mention. This will also help you find positive mentions of your school and could have other benefits. We know of one school PA who saw some comments one evening about a possible fire in her school building and immediately passed them on to the head, thus alerting others to the situation early and limiting the subsequent damage.
Parental Communications
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There's nothing more exciting, or more magical, than performing on a Disney stage in front of an international audience. Disney Performing Arts OnStage sets the standard for which all other performance opportunities are measured. Expose your group to the highest level of performance excellence, as well as a taste of show biz — Disney style. Reinforce lifelong lessons in teamwork, discipline and artistic growth at one of the most high-profile venues in the world.
For some, a Disney performance is a crowning achievement. For others, it's a launch pad to greatness. And everyone will experience the excitement of Disney's world-famous attractions and discover why this is the place where dreams come true.
After your performance, experience the excitement of Disney's world-famous theme parks and discover why Walt Disney World Resort® is the place where dreams come true.
Expose your instrumental group to the highest level of performance and get a taste of showbiz, Disney style.
Partake in the magic of our parks with specially priced tickets, and experience Disney excitement at every turn.
We just received word that we will be marching in Disney on Saturday April 13th at 9:45am! Come see us perform!!
We will be staying at Port Orleans Riverside on Disney Property!


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Tuesday's World Events - April 10, 2007
1. Match the following people and organizations mentioned in the article with their descriptions:
a) _______________ Robert Mugabe
b) _______________ Morgan Tsvangirai
c) _______________ Eddie Cross
d) _______________ Kerry Kay
e) _______________ Thabo Mbeki
f) _______________ ZANU-PF
g) _______________ MDC - Movement for Democratic Change
h) _______________ Central Intelligence Organization
--the ruling party of Zimbabwe (Robert Mugabe's party)
--general policy coordinator for MDC
--Zimbabwean secret police
--Deputy Health Secretary for MDC
--dictator of Zimbabwe
--the opposition party in Zimbabwe
--president of South Africa
--head of MDC
2. What is known about Robert Mugabe's personal life?
3. How did Mugabe defend the beating by his police force of opposition party chief Morgan Tsvangirai and 50 others?
4. How does Mugabe control Zimbabweans?
5. Why is it significant that Zimbabwe's armed forces are double the size of South Africa's?
6. What has happened to the people of Zimbabwe since Mugabe seized white-owned Zimbabwean commercial farms 7 years ago?
7. Why does Eddie Cross say that despite the dire conditions in Zimbabwe, the people will not rebel?
8. a) List the countries that border Zimbabwe.
b) Why do you think these countries do not strongly condemn Mr. Mugabe's actions?
9. What help can Zimbabweans hope for from the governments of surrounding African countries?


text:
Obituary: Allen Lee Bowman Jan. 19, 1931 – Oct. 14, 2022
Submitted by Carol A. Clark
ALLEN LEE BOWMAN Jan. 19, 1931 – Oct. 14, 2022
Allen Lee Bowman "Al" passed away peacefully on October 14, 2022 in Green Valley, Arizona.
He was born on January 19, 1931 in Arlington, VA to John Dee Perry Bowman & Loula Elizabeth Flanagan.
Al graduated from high school at the age of 16 because he was able to skip part of 4th and all of 5th grade. Al often attributed this fortuitous event to his obsession with reading, which he did every chance he had.
After high school, Al attended the College of William & Mary in Williamsburg, VA where he met the love of his life, Margaret Maxwell Braddock. As the story goes, Al and Margaret were both active in the drama club at William & Mary, and one day, after a rehearsal of "Guys and Dolls" it was Al's turn to put props away. Margaret waited for him for quite a while. After finding the courage, Al finally returned and asked Margaret to be his betrothed. Apparently, things worked out well as they were married 70 years ago on June 14, 1952, in Williamsburg at Bruton Parish Church.
After graduating from William & Mary, Al attended Officer Candidate School and gained the rank of Commander in the United States Navy. He served in the U.S. Naval Reserves and was also a Naval Recruiter in Annapolis, Maryland. The irony is that he served at the Reserves in Santa Fe, New Mexico, far from any sizeable bodies of water. It was a well-known inside family joke that although Al served in the Navy, he couldn't actually swim. Al received his Doctorate in Physical Chemistry in 1958 from Iowa State College, now known as Iowa State University.
Al moved to Los Alamos, NM in 1958 with Margaret and their young daughter, Brenda, where they had three more children, Catherine, Kenny, and Murray.
Al worked at Los Alamos Scientific Laboratory/Los Alamos National Laboratory for 33 years, as a chemist per his doctorate, but also primarily as a physicist, spending time in the weapons division as well as other areas.
While in Los Alamos, Al spent considerable time playing bridge and reading. He volunteered as a coach for Little League Baseball and Soccer for both of his sons.
In 1991, Al retired, and he and Margaret moved to Green Valley, AZ where he was very active in the community.
Al was an amazing story teller, whether it was about his days at Los Alamos National Lab, his bridge playing, or his children.
His love of reading led to a collection of over 5,000 books which are stashed away in his tiny den in Green Valley. Amongst these are technical writings, some of which were authored by Al, several copies from the New Mexico Geology Society in which he spent numerous weekends traveling the Southwest and Mexico to learn about "Odd Rock Formations". Let's not forget his Erle Stanley Gardner Collection, of which it is believed he has every book and article written by Gardner. Al also had a for love and interest in railroads and trains.
To say Al was an avid bridge player is like saying "The Grand Canyon is a big hole in the ground." There are some interesting tidbits about Al's bridge playing as eloquently stated by his bridge partner of 50 years, Bill Barkley. Al learned the game of bridge while a student at the College of William & Mary. It began a 70-year love affair with the game as he learned the challenging intricacies of bidding, declarer play, and defense.
Al's bridge activities expanded into ACBL duplicate bridge with both local games and sectional, regional, and national tournaments. He played regularly as a partner with John Hancock, a nationally known player of great skill. In the 1970's he won his first regional tournament contest – the first of many to follow. When he retired and moved to Green Valley, he became active in a leadership role with the GV Recreation duplicate bridge club, which has regularly scheduled games three times a week and more than 300 members. He undertook the challenging role of club manager for more than 20 years. In his leadership role he helped to maintain a large, vibrant, and challenging bridge club. Al will be missed both as a player and leader by his bridge friends. He ended his life as an ACBL Emerald Life Master.
Al was preceded in death by his parents and his younger brother, Bruce.
Al is survived by his wife Margaret, his daughters Brenda Giese of Tahoe, CA, Catherine Bowman of Los Alamos, NM, and sons Kenny Bowman and his wife Carol of Prosper, TX, and Murray Bowman of Green Valley, AZ, six grandchildren, Dov Shlachter, Orli Shlachter, Jennifer Bowman, Kailey Bowman, Cathy Blake, and Travis Weatheread, as well as four great-grandchildren.
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A native of Augusta, Dr. Trotter attended the University of Georgia and graduated from the Medical College of Georgia (MCG) School of Dentistry in 1973. Upon graduation he was elected to membership in Omicron Kappa Upsilon Dental Honor Society. After practicing general dentistry for six years, he returned to MCG for a two-year residency in orthodontics and has been in the private practice of orthodontics since 1981.
Dr. Trotter participates in 60-70 hours of continuing education each year in order to provide the best and most efficient treatment techniques for his patients. He is also a part-time assistant professor of orthodontics at Georgia Health Science University.
Dr. Trotter is one of 30% of the nation's orthodontists who have achieved certification by the American Board of Orthodontics. He is past president of the Augusta Dental Society and the School of Dentistry Alumni Association. He is a recipient of the MCG School of Dentistry Distinguished Alumnus Award and the Georgia Dental Association Honorable Fellow Award.
Dr. Trotter and his wife, Anne, have three sons— Jim, Dennis, and Paul— and eight grandchildren. They also have a long-haired dachshund named Callie.
Dr. Trotter is a member of the Augusta Area Chamber of Commerce, Columbia County Chamber of Commerce, and St. Mary on the Hill Catholic Church, and enjoys playing golf-- especially with his sons and grandchildren.


My preferred transportation is my Vespa, by a wide margin.
Today the destination was Washington DC. The Vespa wasn't an option.
Yet there was quite a rare treat in store. Transportation was courtesy of the aeronautical equivalent of a Vespa, in this case a Beechcraft 9 passenger King Air.
The cross-winds at Dulles were ferocious. Being able to see out of the cockpit windows and the side windows simultaneously during the landing was a treat, and made counter steering the Vespa at full throttle on a curving expressway ramp seem quite tame by comparison.
The plane seemed headed straight for the runway, but was gyrating on its axis by what seemed to be 15 to 20 degrees to the left then to the same extent to the right. Turbulence was simultaneously tossing us up and down like a toy.
The amazing thing was that the plane touched down light as a feather, with no wrenching at all. Seeing all the left-right rotation in the split-seconds before the plane touched down made that smooth landing all the more improbable and remarkable. I guess that gets chalked up to the pilot's skill.
All things considered, I didn't miss the scoot commute today.
Everybody seems to be going places these days.
This transportation is a worthy alternative to Vespa scootering.
I love the small planes where, as you say, you get to look out the pilot's window as well. Quite a different perspective than with the large jets.
Why do I have a feeling their is a new hobby for you in the future?
did you bring your GoProHD Hero2 ? It is an easy way to bring back "stealth" video. I use mine more OFF bike, than ON. Don't forget it also takes "stills"
Did you thank the pilot as you left the plane?
I've flown on the little Air BC planes before that you have to walk out to the tarmac and up the stairs and don't seat very many people. They are an experience in winter.
Those sideways approaches must be somewhat scary! It is cool the way the pilot turns right at the last minute to land straight.
Canajun, it's odd that I felt more safe in that small aircraft with all that rocking and rolling than if I had been experiencing the same thing in a large commercial plane.
Trobairitz, you're darn right I thanked the pilot.
Robert, that's a hobby I couldn't afford. I have enough trouble keeping up with my modest little Vespa love affair.
Gary, that's got to be exactly how it was handled. Just an unbelievably smooth poised landing with all that drama just before.
Sonja, we need pics from down under.... I'll scoot over to your blog to take a peak.


text:
Q: Python Zipfile module is throwing errors I'm trying to get a python Zip module to compress data.
But all it's doing is throwing an error:
with ZipFile(O_file7,mode='w',compression=ZipFile.ZIP_DEFLATED,compressionlevel=9) as file:
AttributeError: type object 'ZipFile' has no attribute 'ZIP_DEFLATED'

I've tried using
ZipFile.ZIP_DEFLATED, and
zipfile.ZipFile.ZIP_DEFLATED, but get the exact same result.
If I remove the compression and compression level parameters, the code runs just fine, but the whole point of using it is to compress the data.
Any help would be greatly appreciated.
Here is the code.
    import zlib
    from zipfile import ZipFile
    HOME2 = "h:\\\\passport\\tran-14d - ollw mass print\\"
    ARCHIVE = HOME2+"\\Archive\\"
    I_file1 = "masterfile.txt"
    O_file7 = "New.zip"
    def ZIPPER():
        os.chdir(HOME2)
        with ZipFile(O_file7,mode='w',compression=ZipFile.ZIP_DEFLATED,compressionlevel=9) as file:
            os.chdir(HOME2)
            file.write(I_file1)

A: Fairly sure it's zipfile.ZIP_DEFLATED. Note the the docs mention the need for zlib dependency for ZIP_DEFLATED. You might find these examples here useful: https://pymotw.com/2/zipfile/#creating-new-archives.


Web Hosting Services – ALISO VIEJO, CA – Alentus Corporation (PINKSHEETS: ALNS), a leading provider of web hosting solutions including managed services, sharepoint hosting, dedicated servers, Windows and Linux web hosting, and Microsoft exchange email services, today announced that the Company and its senior lender have entered into a Letter of Intent (LOI) which will provide the required flexibility for Alentus to substantially restructure its balance sheet including debt obligations. The Company anticipates completing the restructuring process in the first quarter of 2010.
Alentus has previously reported $6,664,219 in revenue and $451,385 in EBITDA for the trailing twelve months ending September 30, 2009. Interest expense in the same period was $903,607. The Company expects the interest expense to decline significantly after the restructuring.
Alentus Corporation is a next-generation web hosting and infrastructure company. We are a premier provider for both individuals and businesses looking for website hosting, dedicated servers, virtual private servers (VPS), Microsoft Exchange and Sharepoint hosting, IP transit, and collocation.


Planning a family vacation is full of tough choices. On one hand you have the kids that may be hoping for the adrenaline rush of rides, roller coasters and other theme park attractions! A great time to be sure, but you return home needing a vacation! Mom and dad may be longing for some R&R at sea on a well-appointed cruise ship that includes fine dining, great entertainment and maybe some pampering at the onboard day spa. What's a family to do?
Why not both? A land and sea combo vacation! There was no way this could not be successful…the partnership between Royal Caribbean and Universal Studios, that is. Their special land and sea package combines a 3 or 4-night stay at Universal Orlando Resort with a 3 or 4-night Bahamas cruise aboard Royal Caribbean's Monarch of the Seas sailing out of Port Canaveral. For those that want a longer vacation, that same land package could be added to a 7-night Eastern or Western Caribbean cruise aboard Freedom of the Seas, also out of Port Canaveral.
How do you make this extraordinary vacation package even better? Extend the deal to those cruising out of Tampa! Now guests can combine a 3 or 4-night Universal Orlando vacation with a 4 or 5-night Western Caribbean cruise on the lovely Jewel of the Seas. This ship has a glamorous feel with wide-open spaces and acres of towering glass that fills the interior with natural light. Her pools, restaurants, bars, lounges, kid's clubs, casino, spa and much, much more will keep the whole family happy!
All land and sea packages include hotel, shipboard accommodations, multi-day passes to Universal Studios Florida and Islands of Adventure, and ground transfers between airport, hotel, theme parks and the port. Let a cruise consultant at Direct Line Cruises help your family plan this amazing vacation by calling 1-800-352-8088."