text:
\section{Introduction}

	Many practical applications such as nuclear waste storage and
	reservoir simulations require multiphase transport simulations
	in porous media.
	Detailed reservoir models with a wide range of scales are available for more accurate simulations. In this paper, we consider a two-phase flow model, in which a pressure equation is coupled with a specific transport equation. As a main component of the model, we consider the permeability that exhibits high heterogeneities with large contrast, which creates a significant difficulty to resolve by traditional approaches. In particular, due to the multiscale nature of the problem, solving on a fine-scale that is sufficient to capture the heterogeneity results considerable computation, which motivates upscaled or multiscale methods and some other model reduction methods. The idea in upscaled method involves using one effective media parameter in each coarse block so that the concerned  equation can be solved in a reduced model \cite{wu2002analysis,DDDAS_upscale_2004,Arbogast_Boyd_06,durlofsky1991numerical,efendiev2002numerical,efendiev2000modeling}. However, this approach may encounter inaccuracy for missing fine-scale information that has strong effect on the overall flow. Another way is to solve the equation
	in coarse grid with multiscale methods \cite{egw10,efendiev2009multiscale,Arbogast_two_scale_04,chung2015mixed,chen2003mixed,jennylt03,Wheeler_mortar_MS_12, Arbogast_PWY_07, mortar_elliptic,mortar_online,yang2019multiscale,yang2018online}.
	Underlying idea in this method is to incorporate small-scale information in the basis functions and then capture this effect on a larger scale. Parallel computations can be used since the construction of multiscale basis in different local neighborhoods is independent, and this enhances the efficiency.

	An important consideration for this coupled pressure-convection-diffusion system is that local conservation of mass is required, which can be hardly guaranteed within the framework of continuous Galerkin method. This requirement has motivated a variety of methods, for example, multiscale finite volume methods \cite{cortinovis2014iterative,jenny2003multi,lunati2004multi}, mixed multiscale finite element methods \cite{aarnes2004use,aarnes2008mixed,chen2003mixed,chung2015mixed}, mortar multiscale methods \cite{arbogast2007multiscale, peszynska2005mortar,peszynska2002mortar}, discontinuous Galerkin (DG) methods \cite{du2018adaptive,kim2013staggered,cockburn2002local}, and postprocessing methods \cite{odsaeter2017postprocessing,bush2013application}.  In this paper, we utilize the mixed multiscale finite element method. In a mixed finite element formulation, one uses a first-order system for pressure and velocity, while the pressure space is spanned by piecewise constant pressure basis functions. The support for each pressure basis function is a single coarse block while for velocity basis function, it vanishes outside a coarse neighborhood, which is composed of two coarse blocks sharing a common coarse edge. Contributed by this construction, local conservation in coarse scale can be attained, which is sufficient in most circumstances. On the other hand, when oscillations appear in the source term, conservation property is not satisfied in the relatively small scale, which motivates the need for some postprocessing procedures. In this paper, some relevant methods are reviewed in Section \ref{post}. Since we only need to postprocess the velocity field when the source term is not a constant, the additional computation is limited.
	Furthermore, one advantage of some postprocessing procedures is that the implementation can be conducted in parallel if the corresponding regions are non-overlapping, which improves the efficiency significantly.

	A typical strategy to simulate the multi-phase flow is the implicit pressure explicit saturation (IMPES) method.  That is, the pressure equation  is solved at
	the current time step with mobility coefficients defined with saturation values from
	the previous time step. The saturation equation is then computed with an explicit Euler scheme using velocity field  from the previous time step.
	The dominant computational cost exists in solving the pressure equation.
	Most multiscale methods for multi-phase simulation are trying to reduce the
	computational cost of solving the pressure equation.
	We note that in \cite{kippe2008comparison}, the authors compared several popular
	multiscale methods for the two-phase flow simulation.
	However, no large-scale three-dimensional test was provided.

	Our approach is based on  multiscale finite element method (MsFEM), the pioneering work of which can be found in \cite{babuvska1983generalized,hou1997multiscale}, where it was initially proposed to
	solve the elliptic equation in second order formulation. Furthermore, it was then extended to mixed
	multiscale finite element method (MMsFEM). The key ingredient of the method is the construction of multiscale space which contains the fine-grid information of permeability field. Besides, due to the independent nature of the construction of each local multiscale space, we can implement in parallel to save computational time. Based on traditional MMsFEM, a new MMsFEM using limited global information \cite{aarnes2008mixed,jiang2012some} is introduced. Without ambiguity, we use MMsFEM to denote the new method for simplicity. A main attraction of the method is that one may make use of global information obtained by single-phase flow in the construction of multiscale basis functions, while only local information is used in traditional MMsFEM. In particular, we make some modifications in the boundary conditions in the local problems which are aimed at constructing the local multiscale space. This novel modification is powerful when fine-scale feature of two-phase flow depends on single-phase flow. Moreover, since only one velocity multiscale basis function is utilized in each local neighborhood, the simulation time is relatively shortened compared to the case where multiple basis functions are used, which will discussed in the numerical results. A limitation of this method is that one can not flexibly adjust the dimension of approximation space by changing the number of multiscale basis functions, which motivates the other method mixed Generalized Multiscale method (MGMsFEM).

	As is mentioned in previous papers \cite{efendiev2013generalized,Efendiev2011ADD,chung2014adaptive},
	one basis function on each edge is not sufficient especially when there are long channels and non-separable scales in the permeability field. MGMsFEM is designed to construct multiple basis functions on some coarse edges, which provides accuracy in this case and guarantee conservation of mass simultaneously. This method is divided into two steps, constructing snapshot space and multiscale space. First, one may construct a snapshot space as approximate space, where there are basis functions carry important properties of solution. Furthermore, a reduced space called offline space is obtained by some well-designed spectral problems. As mentioned in the paper \cite{CHUNG201669}, a good approximation of the reduced model is only attained when the offline information can effectively represents the real case.  At the same time, when the number of offline basis functions exceeds some level, the error decreases will slow down. In particular, the convergence rate is proportional to $1/\Lambda$, where $\Lambda$ is the smallest eigenvalue we abandon during the construction of multiscale space. Consequently, some online enrichment \cite{chan2016adaptive,chung2015residual,online_mixed,online_cg} is necessary for the sake of more accurate velocity fields, which will be inherited in saturation by the following simulation.

	The main purpose of the paper is to compare the MGMsFEM with MMsFEM. As is mentioned above, the MMsFEM is efficient as only one multiscale basis function is utilized in each local region, which results in smaller coarse-grid system and hence shorter simulation time. However, it lacks flexibility compared with MGMsFEM as one can adapt the dimension of the approximation space in the latter one method.
	We will show that MGMsFEM serves as an efficient and flexible way to reduce the relevant errors. Different combinations of offline and online basis functions can be selected to provide most satisfying outcomes. Due to the power of the online basis functions, one may use relatively small number of basis functions to get comparable accuracy with MGMsFEM. Moreover, in context of two-phase flow model, there is no need to solve single-phase problem in advance, which is compulsory in MMsFEM. On the other hand, the construction of snapshot space and solving subsequent local spectral problems in MGMsFEM are not necessary in MMsFEM. As for the similarities of these two methods, once multiscale space is constructed, it can be applied to different cases, e.g. source terms, boundary conditions, mobility on coarse grid, etc., and no further update
	will be conducted. It is computationally cheap since flow equations need to be solved multiple times during in multi-phase transport simulation.

	The paper is organized as follows. In Section 2, we introduce the two-phase model. Main ingredients of two concerned methods are discussed respectively in the following two sections. In Section 5, the postprocessing procedures are briefly reviewed in Section 5. Finally, some extensive numerical results are exhibited.

	\section{Model formulation}
	In this paper, we consider the two-phase flow model.
	We can write the flow equations as follows:
	\begin{eqnarray}\label{eq:pressure}
	\left\{\begin{aligned}
	(\lambda(S) k(x))^{-1} \mathbf{v}-\nabla p &=0 \quad \text { in } \quad D \\
	\operatorname{div}( \mathbf{v}) &=f \quad \text { in } \quad D \\
	\mathbf{v}\cdot \mathbf{n} &=g(\mathbf{x}) \quad \text { on } \quad \partial D \label{model}
	\end{aligned}\right.
	\end{eqnarray}
	First equation is obtained applying the Darcy's law and the second states the conservation of mass. $\lambda(S)$ is the total mobility. In particular, $\lambda(S)=\lambda_o(S)+\lambda_w(S)$. And we have $\lambda_i(S)=\frac{k_{ri}(S)}{\mu_{i}}$ , for $i=o,w$. The $k_{rw}(S)$ and $k_{ro}(S)$ are permeabiliy for water and oil while $\mu_{w}(S)$ and $\mu_{o}(S)$ are viscosities for water and oil phases, correspondingly. The dynamics of saturation $S$ affects the flow equation. One can write the dynamics of saturation in the following transport equation.
	\begin{eqnarray}
	\dfrac{\partial S}{\partial t}+ \operatorname{div} (\mathbf{v}\cdot f(S))=q\label{transport}
	\end{eqnarray}
	where $f(S)=\frac{\lambda_w(S)}{\lambda_w(S)+\lambda_o(S)}$ and $q$ is some external force term. \\
	To solve (\ref{transport}), we need to solve (\ref{model}) in advance. Solution algorithm for the two-phase problem is presented in table \ref{two-phase algorithm}. \\
	\begin{table}[!htbp]
		\centering
		\begin{tabular}{c l}
			\hline
			&Two-phase algorithm  \\
			\hline
			\textbf{Input} & $S_{n-1}$ obtained in previous time step\\
			\textbf{Output}& $S_{n}$ \\
			&1. Solving (\ref{model}) to get $p_n$ and $\mathbf{v}_n$\\
			&2. Using $\mathbf{v}_n$ and $S_{n-1}$  in (\ref{transport}) to get $S_{n}$ \\
			\hline
		\end{tabular}
		\label{two-phase algorithm}
	\end{table}

	To solve (\ref{transport}), we integrate it with $[t_{n-1},t_n]$ and some volumn  $V_i\subset D$
	\begin{eqnarray}
	\text{meas}(V_i)(S_{z,n}-S_{z,n-1})+\Delta t\int_{\partial_{V_i}}\mathbf{v}\cdot n f(S_{z,n-1})d l=\Delta t\int_{V_i} q_w d x
	\end{eqnarray}
	where we have neglected the error terms and
	\begin{eqnarray}
	S_{z,n}\approx\dfrac{1}{meas(V_i)}\int_{V_i}S(x,t_n)d x
	\end{eqnarray}
	We use $meas(A)=\int_{D} 1_{A}d x$ with $1_A=1$ when $x\in A$ while 0 elsewhere.\\
	To evaluate the term $\int_{\partial_{V_i}}\mathbf{v}\cdot n f(S_{z,n-1})d l$, we use an upwinding scheme. A review of upwinding on a rectangular mesh can be found for example in \cite{thomas2013numerical}. It is imperative that the numerical approximation of $\mathbf{v}$ satisfies the following local conservation property. In particular, it is desirable to have
	\begin{eqnarray}
	\int_{\partial{V_i}}\mathbf{v}\cdot \mathbf{n}d l=\int_{V_i} q d x.
	\end{eqnarray}
	We will consider more details about the mass conservation property in Section \ref{post}.
	\section{Mixed GMsFEM}
	In this section, we briefly review the Generalized multiscale finite element method for (\ref{eq:pressure}). The construction consists of two steps.
	\begin{enumerate}
		\item Constructing a snapshot space.
		\item Solve a spectral problem.
	\end{enumerate}
	Let $T^{h}$ be a partition of the domain $D$ into fine finite elements. Here $h>0$ is the fine grid mesh size.
	The coarse partition, $T^{H}$ of the domain $D$, is formed such that each element in $T^{H}$ is a connected union of fine-grid blocks. More precisely, $\forall K_{j} \in T^{H}$, $ K_{j}=\bigcup_{F\in I_{j} }F$ for some $I_{j}\subset T^{h}$. The quantity $H>0$ is the coarse-mesh size. We will consider the rectangular coarse elements and the methodology can be used with general coarse elements. An illustration of the mesh notations is shown in the Figure \ref{mixed neighbor}. We denote the interior coarse edges of $T^{H}$ by $E_i,i=1,\cdots,N_{\text{in}}$,
	where $N_\text{in}$ is the number of interior coarse edges. The coarse elements
	of $T^{H}$ are denoted by $K_j,j=1,2,\cdots,N_e$, where $N_e$ is the number of coarse elements. We define the coarse neighborhood of the edge $E_i$ by $\omega_i:=\cup\{K_j\in T_{H}:E_i\subset \overline{K_j}\}$.
	The snapshot space is linearly spanned by a set of extensive basis functions with all possible boundary conditions. We utilize a well-designed local spectral problem to select dominant modes, providing the solution with rapidly decreasing residual, which is illustrated in detail in error analysis section in \cite{efendiev2013generalized}.
	\begin{figure}[htbp!]
		\centering
		\includegraphics[scale=0.3]{Figures/mixed_neighbor.png}
		\caption{Illustration of coarse neighborhood $\omega_i=K_{i,1}\cup K_{i,1}$ and oversampled neighborhood $\omega_i^+$}
		\label{mixed neighbor}
	\end{figure}
	\subsection{Snapshot space}
	We let $V_h$ be the lowest Raviart Thomas vector field ($RT_0$) and $Q_h$ be space of piesewise constant functions on the fine mesh. We set $V_h=\text{span}\{\psi_{\text{fine},1},\cdots,\psi_{\text{fine},N_{\text{in,fine}}}\}$ and
	$Q_h=\text{span}\{p_{\text{fine},1},\cdots,p_{\text{fine},N_{\text{e,fine}}}\}$, where $N_{\text{in,fine}}$ and $N_{\text{e,fine}}$ are the number of inner edges and blocks in fine grid. To construct the multiscale basis functions, we need to construct a snapshot space. In particular, we need to solve the following local problem in each $\omega_i$:
	\begin{eqnarray}
	\left\{
	\begin{aligned}
	(\lambda(S) k(x))^{-1} \mathbf{\psi}^{(i)}_j+\nabla p_j^{(i)}&=0 \quad\text{in}\quad \omega_i\\
	\text{div}(\mathbf{\psi}_j^{(i)})&=\alpha_j^{(i)}\quad \text{in}\quad \omega_i \label{mixed gms}\\
	\mathbf{\psi}_j^{(i)}\cdot n_i&=0\quad \text{on } \partial\omega_i\\
	\mathbf{\psi}_j^{(i)}\cdot m_i&=\delta_j^{(i)}\quad \text{on } E_i \label{snapshot}
	\end{aligned}\right.
	\end{eqnarray}
	$E_i$ is the common edge of the two elements $K_p,K_q\subset\omega_i$. And $E_i=\bigcup_{l_j\subset E_i} l_j$, as is shown in figure \ref{mixed neighbor}.
	\begin{eqnarray}
	\delta_j^{(i)}=
	\left\{
	\begin{aligned}
	&1,\quad\text{on }l_j\\
	&0,\quad\text{elsewhere on }E_i
	\end{aligned}
	\right.
	\end{eqnarray}
	We also set $\alpha_j^{(i)}$ such that the compatibility condition holds:$\int_{K_l} \alpha_j^{(i)}=\int_{E_i}\delta_j^{(i)}$ is satisfied for all $K_l\subset\omega_i$.\\
	In matrix formulation, we solve
	\begin{equation*}
	\left[\begin{array}{cc}
	A_{h}^i& {B_{h}^i}^T \\
	B_{h}^i  & 0
	\end{array}\right]\left[\begin{array}{l}
	\phi_{j,c}^{(i)} \\
	p_{j,c}^{(i)}
	\end{array}\right]=\left[\begin{array}{c}
	0 \\
	\tilde{\alpha}_j^{(i)}
	\end{array}\right],
	\end{equation*}
	where $\phi_{j,c}^{(i)}$ and $p_{j,c}^{(i)})$ are coefficient vectors of $\phi_{j}^{(i)}$ and $p_{j}^{(i)}$ in expansion of space $V_h\times Q_h$. $A_h^i$ and $B_h^i$ are corresponding matrix in $\omega_i$. $\tilde{\alpha}_{i}$ is a vector and in particular,  $\tilde{\alpha}_{i}[j]=\int_{\omega_i} \alpha_i p_i^j$ for all j, where $p_i^j$ is the $j-th$ pressure basis function in space $Q_h$ constrained in $\omega_i$.\\
	We remark that (\ref{mixed gms}) is solved separately in the coarse elements forming $\omega_i$.
	Besides $n_i$ is a unit outer normal vector to $\partial \omega_i$ and $m_i$ is a fixed unit normal vector for $E_i$.\\
	From (\ref{snapshot}), we can solve snapshot basis $\mathbf{\psi}_j^{(i)}$ for $1\leq j\leq J_i$.
	We can further define $V_{\text{snap}}^{i}=\text{span}
	\{\mathbf{\psi}_1^{(i)},\cdots,\mathbf{\psi}_{J(i)}^{(i)}\}$ for $1\leq i\leq N_{\text{in}}$, where $N_{\text{in}}$ is the number of inner edges for coarse elements. Finally, $V_{\text{snap}}=\bigoplus_{i=1}^{N_{\text{in}}}V_{\text{snap}}^{i}$.

	\subsection{Offline space}
	We proceed to the space reduction in this subsection. We aim at solving the eigenpair $(\lambda,\mathbf{v})$ from the following spectral problem,
	\begin{eqnarray}
	a(\mathbf{v},\mathbf{w})=\lambda s(\mathbf{v},\mathbf{w}),\label{spectral}
	\end{eqnarray}
	where $a(\mathbf{v},\mathbf{w})$ and $s(\mathbf{v},\mathbf{w})$ are symmetric positive definite bilinear forms defined on $V_{\text{snap}}\times V_{\text{snap}}$. We can view the spectral problem as a way to select the dominating components of a specific residual operator. In particular, we can define $\mathcal{A}:V_{\text{snap}}\rightarrow V_{\text{snap}}$ as follows.
	\begin{eqnarray}
	s(\mathcal{A}\mathbf{v},\mathbf{w})=a(\mathbf{v},\mathbf{w})
	\end{eqnarray}
	where $s(\mathbf{v},\mathbf{w})$ is an inner product on $V_{\text{snap}}$. We assume $\mathcal{A}$ has a rapidly decaying eigenvalues. In practice, we need to solve a local spectral problem, which is more sufficient than the global one. In particular, in $\omega_i$ corresponding to $E_i$ we find a real number $\lambda$ via solving the following problem:
	\begin{eqnarray}
	a_i(\mathbf{v},\mathbf{w})=\lambda s_i(\mathbf{v},\mathbf{w})\label{spectral_i}
	\end{eqnarray}
	where $a_i(\mathbf{v},\mathbf{w})=\int_{E_i}(\lambda(S) k(x))^{-1}(\mathbf{v}\cdot m_i)(\mathbf{w}\cdot m_i)$ and $s_i(\mathbf{v},\mathbf{w})=\int_{\omega_i}(\lambda(S) k(x))^{-1}\mathbf{v}\cdot \mathbf{w}+\int_{\omega_i}\text{div}(\mathbf{v})\cdot\text{div}(\mathbf{w})$. \\
	Suppose we have solved $\lambda_{1}^{(i)}\cdots\lambda_{J_i}^{(i)}$ corresponding to eigenvectors $\mathbf{v}_1^{(i)}\cdots \mathbf{v}_{J(i)}^{(i)}$ in $\omega_i$, we further select first $l_i$ eigenvectors and consequently the offline basis function
	$\mathbf{\phi}_j^{(i)}=\sum_{k=1}^{J_i} v_{jk}^{(i)}\psi_k^{i}$, where $v_{jk}^{(i)}$
	is the k-th component of $\mathbf{v}_{j}^{(i)}$.\\
	And we can construct local offline space in $\omega_i$.
	$V_{\text{off}}^{(i)}=\text{span}\{\mathbf{\phi}_1^{(i)},\cdots,\mathbf{\phi}_{l_i}^{(i)}\}$.
	We set the global offline basis function to be $V_{\text{off}}=\bigoplus_{E_{l} \in \mathcal{E}_{n}} V_{\text{off}}^{i}$, where $\mathcal{E}_{n}$ is the collection of all inner coarse faces.
	\subsection{Online enrichment}
	To improve the accuracy of velocity solution in (\ref{model}), we consider enriching the offline space based on some residual functions. Our enrichment is performed iteratively. Based on the space $V_{H}^l$ from previous iteration, we can obtain $V_{H}^{l+1}$ through the following 5 steps. In particular, we have $V_{H}^0=V_{\text{off}}$.
	First, we define a residual operator on this iteration $$R_{D}^l(\mathbf{v})=\int_{D} (\lambda(S) k(x))^{-1}\mathbf{v}_H^l \cdot \mathbf{v}-\int_{D} \text{div}(\mathbf{v})p_H^l\quad \forall
	v\in V_{\text{snap}},$$

	Let $\hat{V}_{\text{snap}}$ be the divergence free subspace of ${V}_{\text{snap}}$, and we have
	$$R_{D}(\mathbf{v})=\int_{D} (\lambda(S) k(x))^{-1}\mathbf{v}_H \cdot \mathbf{v}\quad \forall
	\mathbf{v}\in\hat{V}_{\text{snap}}.$$\\
	Step 1: Find the multiscale solution  $\mathbf{v}_{H}^{l} \in V_{H}^{l}$ and $p_{H}^{l} \in Q_{H}$ satisfying
	\[
	\begin{aligned}
	\int_{D} (\lambda(S) k(x))^{-1} \mathbf{v}_{H}^{l} \cdot \mathbf{w}-\int_{D} \operatorname{div}(\mathbf{w}) p_{H}^{l} &=0 \quad \forall w \in V_{H}^{l}, \\
	\int_{D} \operatorname{div}\left(\mathbf{v}_{H}^{l}\right) q &=\int_{D} f q \quad \forall q \in Q_{H}.
	\end{aligned}
	\]
	In matrix formulation, we have
	\begin{equation}
	\left[\begin{array}{cc}
	\left(V^{l}\right)^{T} A_{h} V^{l} & \left(V^{l}\right)^{T} B_{h}^{T} P_{H} \\
	P_{H}^{T} B_{h} V^{l} & 0
	\end{array}\right]\left[\begin{array}{l}
	\mathbf{v}_{H, c}^{l} \\
	\mathbf{p}_{H, c}^{l}
	\end{array}\right]=\left[\begin{array}{c}
	0 \\
	P_{H}^{T} F_{h}
	\end{array}\right].\label{matrix gms}
	\end{equation}
	Step 2:  Let $\{\omega_i\}$ be a non-overlapping partition of $D$, which corresponds to common coarse faces $\{E_i\}$. We let $\{\omega_i^+\}$ be oversampled neighborhoods from $\{\omega_i\}$. We compute $R_{\omega_{i}^{+}}(\mathbf{v})$ for each $\mathbf{v}\in V_{\text {snap }}^{i ,+}$. We note that we need to project the $\mathbf{v}_{H}^{l}$ and $p_{H}^{l}$ into the fine-grid spaces $V_{h}$ and $Q_{h}$, respectively, in order to compute $R_{\omega_{i}^{+}}^l(v)$.  \\
	Step 3: For each $\omega_{i}^{+},$ we solve for $\mathbf{\phi}_{i}^{+} \in V_{\text {snap }}^{i ,+}$ such that
	\[
	\int_{\omega_{i}^{+}} (\lambda(S) k(x))^{-1} \mathbf{\phi}_{i}^{+} \cdot \mathbf{v}=R_{\omega_{i}^{l,+}}(\mathbf{v}) \quad \forall \mathbf{v} \in V_{\text {snap }}^{i ,+}
	\]
	\\
	Step 4: We take the restriction of $\mathbf{\phi}_{i}^{+} \cdot n_{i}^{+}$ on the coarse face $E_{i}$ and normalize it, and we denote it by $\lambda_{i}$.\\
	Step 5: The online basis $\mathbf{\chi}_{i}$ for the coarse neighborhood $\omega_{i}$ can be constructed by solving
	\begin{eqnarray}
	\begin{aligned}
	(\lambda(S) k(x))^{-1} \mathbf{\chi}_{i}+\nabla p_{i} &=0 \quad \text { in } \omega_{i} \\
	\operatorname{div}\left(\mathbf{\chi}_{i}\right) &=\alpha_{i} \quad \text { in } \omega_{i} \\
	\mathbf{\chi}_{i} \cdot n_{i} &=\lambda_{i} \quad \text { on } E_{i} \\
	\mathbf{\chi}_{i} \cdot n_{i} &=0 \quad \text { on } \partial \omega_{i} \label{online glo}
	\end{aligned}
	\end{eqnarray}
	where $\alpha_{i}$ is chosen to satisfy the condition $\int_{K} \alpha_{i}=\int_{E_i} \lambda_i$ for every $K \subseteq \omega_{i}, n_{i}$ is a fixed unit-normal vector for the coarse face $E_{i} .$\\
	In matrix formulation, we have
	\begin{equation*}
	\left[\begin{array}{cc}
	A_{h}^i& {B_{h}^i}^T \\
	B_{h}^i  & 0
	\end{array}\right]\left[\begin{array}{l}
	\chi_{i,c} \\
	p_{i,c}
	\end{array}\right]=\left[\begin{array}{c}
	0 \\
	\tilde{\alpha}_{i}
	\end{array}\right],
	\end{equation*}
	where $\chi_{i,c}$ and $p_{i,c}$ are coefficient vectors of $\chi_{i}$ and $p_{i}$ in expansion of $V_h\times Q_h$ constrained in $\omega_i$.\\
	Those $\{\mathbf{\chi}_i\}_{i=1}^{N_{\text{in}}}$ are the new online basis functions. We update the velocity space by letting $V_{H}^{l+1}=V_{H}^l \cup \operatorname{span}\{\mathbf{\chi}_{1},\mathbf{\chi}_{2}, \ldots, \mathbf{\chi}_{N_{\text{in}}}\}$.\\

	\section{Multiscale Finite element Method with limited global information}
	Different from standard Multiscale Finite Element Method, we use construct the multiscale basis function using the single-phase flow information. In particular, we can split this procedure into the following two steps.
	\begin{enumerate}
		\item Step1: Using standard finite element method to solve the fine-scale solution $\mathbf{v}_{sp}$ for the single-phase flow system.
		\item Step2: Construct the Multiscale basis using the global limited information $\mathbf{v}_{sp}$.
	\end{enumerate}

	For convenience, we first rewrite the two-phase flow equation as:
	\begin{eqnarray}
	\left\{\begin{aligned}
	(\lambda(S) k(x))^{-1} \mathbf{v}-\nabla p &=0 \quad \text { in } \quad D, \\
	\operatorname{div}( \mathbf{v}) &=f \quad \text { in } \quad D, \\
	\mathbf{v}\cdot \mathbf{n} &=g(x) \quad \text { on } \quad \partial D.
	\end{aligned}\right.
	\end{eqnarray}
	More specifically, the system is reduced to single-phase when $\lambda(S)=1$.

	The variational formulation associated with (\ref{model}) is to seek $(\mathbf{v},p)\in H(\text{div},\Omega)\times L^2(\Omega)$ such that $\mathbf{v}\cdot\mathbf{n}=g$ on $\partial \Omega$ and
	\begin{eqnarray}
	\begin{aligned}
	\int_{D} (\lambda(S) k(x))^{-1} \mathbf{v}\cdot \mathbf{w}-\int_{D} \text{div}(\mathbf{w})p_h&=0 \quad\forall \mathbf{w}\in \mathbf{V}_{0},\\
	\int_D \text{div}(\mathbf{v})q&=\int_D f q \quad\forall q\in Q.\label{varia}
	\end{aligned}
	\end{eqnarray}
	where
	$$\mathbf{V}_{0}=\{\mathbf{v}\in H(\text{div},\Omega)|\mathbf{v}\cdot\mathbf{n}=0\}\text{ and } Q= L^2(\Omega).$$

	We denote
	$$a(\mathbf{u},\mathbf{v})=((\lambda k)^{-1}\mathbf{u},\mathbf{v}),\quad b(\mathbf{v},p)=(\text{div}(\mathbf{v}),p).$$

	Then we can rewrite the above system as below:
	\begin{eqnarray}
	\begin{aligned}
	a(\mathbf{v},\mathbf{w})+b(\mathbf{w},p)&=0 \quad\forall \mathbf{w}\in \mathbf{V}_{0},\\
	b(\mathbf{v},q)&=(f,q)\quad\forall q\in Q.\label{varia matrix}
	\end{aligned}
	\end{eqnarray}

	We denote the $(\mathbf{v}_{h},p_h)$ to be the reference solution in $(V_h,Q_h)$ by solving
	\begin{eqnarray}
	\begin{aligned}
	a(\mathbf{v}_{h},\mathbf{w})+b(\mathbf{w},p_{h})&=0 \quad\forall \mathbf{w}\in \mathbf{V}_h^0,\\
	b(\mathbf{v}_{h},q)&=(f,q)\quad\forall q\in Q_h.\label{fine matrix}
	\end{aligned}
	\end{eqnarray}
	where $\mathbf{v}_{h}\cdot n=g_h$ on $\partial D$ and $\mathbf{V}_h^0=V_h\cap\{\mathbf{v}\in V_h:\mathbf{v}\cdot n=0 \text{ on } \partial D\}$.

	We can also write (\ref{fine matrix}) in the following matrix form:
	\begin{equation*}
	\left[\begin{array}{cc}
	A_{h}& B_{h}^T \\
	B_{h}  & 0
	\end{array}\right]\left[\begin{array}{l}
	\mathbf{v}_{h,c} \\
	\mathbf{p}_{h,c}
	\end{array}\right]=\left[\begin{array}{c}
	0 \\
	F_{h}
	\end{array}\right]
	\end{equation*}

	where we let $v_{h,c}$ and $p_{h,c}$ to be the corresponding coefficient vector for reference solution $v_h$ and $p_h$. $A_h[i,j]=a(\psi_{\text{fine},i},\psi_{\text{fine},j})$ and $B_h[i,j]=b(\psi_{\text{fine},i},p_{\text{fine},j})$.

	We first seek $\mathbf{v}_{sp}$ for the single-phase system in  $\mathbf{V}_h$ by solving (\ref{fine matrix}) with $\lambda(S)=1$.

	Then we consider the constructing multiscale basis functions by using the $v_h$ obtained above. We compute $\phi_{i}^K$ by solving the following local problem on coarse neighborhood $\omega_i$.
	\begin{eqnarray}
	\left\{\begin{aligned}
	(\lambda(S) k(x))^{-1} \mathbf{\phi}_{i}^K+\nabla(p_{i}^K)&=0 \text{ in } \omega_i\\
	\text{div}(\mathbf{\phi}_{i}^K)&=\alpha_i \text{ in } \omega_i\\
	\mathbf{\phi}_{i}^K\cdot \mathbf{n}_i&= \mathbf{v}_{sp}\cdot n_i\text{ on } E_i\\
	\mathbf{\phi}_{i}^K\cdot \mathbf{n}_i&=0 \text{ on } \partial{\omega_i}
	\end{aligned}\right.
	\end{eqnarray}
	Remark that $\alpha_{i}$ is chosen to satisfy the condition $\int_{K} \alpha_{i}=\int_{E_i} \lambda_i$ for every $K \subseteq \omega_{i}.$ And the multiscale finite element space $\mathbf{V}_{ms}$ for velocity will be defined by
	\begin{eqnarray}
	\begin{aligned}
	\mathbf{V}_{ms} &=\bigoplus_{K}\left\{\mathbf{\phi}_{i}^K\right\} \subset H(d i v, \Omega), \\
	\mathbf{V}_{ms}^{0} &=\mathbf{V}_{ms} \cap H_{0}(d i v, \Omega).
	\end{aligned}
	\end{eqnarray}
	We then seek the multiscale velocity solution in $(\mathbf{v}_{\text{ms}},p_{\text{ms}})\in (\mathbf{V}_{\text{ms}},Q_{\text{ms}})$ by solving the following
	\begin{eqnarray}
	\begin{aligned}
	a(\mathbf{v}_{\text{ms}},\mathbf{w})+b(\mathbf{w},p_{\text{ms}})\quad\forall \mathbf{w}\in \mathbf{V}_{\text{ms}},\\
	b(\mathbf{v}_{\text{ms}},q)=(f,q)\quad\forall q\in Q_{\text{ms}},\label{coar matrix}
	\end{aligned}
	\end{eqnarray}
	where $Q_{\text{ms}}$ is spanned by piecewise constant functions.

	Again, we can use matrix representation in the following.
	\begin{equation}
	\left[\begin{array}{cc}
	\left(V_{Hc}\right)^{T} A_{h} V_{Hc} & \left(V_{Hc}\right)^{T} B_{h}^{T} P_{Hc} \\
	P_{Hc}^{T} B_{h} V_{Hc} & 0
	\end{array}\right]\left[\begin{array}{l}
	\mathbf{v}_{H,c}^{l} \\
	p_{H,c}^{l}
	\end{array}\right]=\left[\begin{array}{c}
	0 \\
	P_{H}^{T} F_{h}
	\end{array}\right].\label{matrix lim}
	\end{equation}
	where $V_{H,c}$ and $P_{H,c}$ are the matrix form of the velocity multiscale space and pressure multiscale space. In particular, we have $V_{H,c}=\text{span}\{\phi_1,\cdots,\phi_n\}$ and $P_{H,c}=\text{span}\{p_1,\cdots,p_m\}$, where $n$ and $m$ are the dimension for velocity multiscale space and pressure multiscale space.
	\subsection{Some remarks about MsFEM with limited global information}
	As in \cite{jiang2012some}, the $(\mathbf{v}_{\text{ms}},p_{\text{ms}})$ is exact for the single-phase system. For the error analysis, we first need some assumptions as below:
	\begin{eqnarray}
	a(\mathbf{u}_{\text{ms}},\mathbf{u}_{\text{ms}})\text{ is  ker} B_{\text{ms}} -coercive \label{assump1}
	\end{eqnarray}
	\begin{eqnarray}
	\inf _{\mathrm{q} \in Q_{\text{ms}}} \sup _{\mathbf{v}\in \mathrm{V}_{\text{ms}}} \frac{b\left(\mathbf{v}, q\right)}{\left\|\mathbf{v}\right\|_{H(d i v, \Omega)}\left\|q\right\|_{L^{2}(\Omega)}} \geq C.
	\label{assump2}
	\end{eqnarray}
	Then we will have the following approximation property:
	\begin{lemma}
		If $(\mathbf{v}_h,p)$ and $(\mathbf{v}_{\text{ms}},p_{\text{ms}})$ are solutions for (\ref{fine matrix}) and (\ref{coar matrix}) respectively and the conditions (\ref{assump1}), (\ref{assump2}) hold, then we have
		\begin{eqnarray}
		\left\|\mathbf{v}_h-\mathbf{v}_{\text{ms}}\right\|_{H(\text {div}, \Omega)}+\left\|p_h-p_{\text{ms}}\right\|_{0, \Omega} \leq \displaystyle\inf_{
			\begin{array}{cc}
			&\mathbf{u} \in \mathbf{V}_{\text{ms}}\\
			&\mathbf{u}-g_{\text{ms}} \in \mathbf{V}^0_{\text{ms}}
			\end{array}
		}\left\|\mathbf{v}_h-\mathbf{u}\right\|_{H(\text {div}, \Omega)}+\inf _{q\in Q_{\text{ms}}}\left\|p_h-q\right\|_{0, \Omega},
		\end{eqnarray}
		where $g_{\text{ms}}$ is the multiscale interpolation in the space $\mathbf{V}_{\text{ms}}$.
	\end{lemma}
	Contributed by this property, we aim at seeking a good enough velocity solution in approximation space to have satisfying approximation effect.
	$(\mathbf{v}_h,p_h)$ and $(\mathbf{v}_{\text{ms}},p_{\text{ms}})$ are solutions for (\ref{fine matrix}) and (\ref{coar matrix}) respectively, we have
	\begin{theorem}\label{the}
		\begin{eqnarray}
		\left\|\mathbf{v}_h-\mathbf{v}_{\text{ms}}\right\|_{H(\text {div}, \Omega)}+\left\|p_h-p_{\text{ms}}\right\|_{0, \Omega}\leq C\delta+CH,
		\label{thm}
		\end{eqnarray}
		where $\delta$ is a very small number and $H$ is the size of the coarse grid.
	\end{theorem}
	Consequently, we can base our multiscale space for two-phase model on the single-phase reference solution to get relatively satisfying estimation.
	\section{Postprocessing procedures}\label{post}
	In mixed formulation, contributed by fact that $Q_H$ is spanned by piecewise constant basis functions, we can have mass conservation in coarse scale as shown in (\ref{coarse conser}), which is not sufficient under some specific circumstances, for example, when the source term has some oscillations in fine scale. In order to obtain the conservation in fine scale, we can make use of postprocessing procedures introduced in \cite{chung2015mixed,guiraldello2020velocity}. \\
	\begin{eqnarray}
	\int_{K} \operatorname{div}\left(\mathbf{v}_{H}\right)  &=\int_{K} f  \quad \forall K\subset D.\label{coarse conser}
	\end{eqnarray}
	In \cite{guiraldello2020velocity}, three types of approaches to postprocess the flux are mentioned, the Mean method, the Patch method and the Stitch method. The latter two are developed from the first one, which we will give a brief review as follows. The basic idea is that after we obtained the multiscale solution $\mathbf{v}_H$, we further solve a new velocity field $\mathbf{v}_{h,p}$ by solving a downscaled local problem, where previous solution information is contained in boundary condition as shown in (\ref{fine conser}).
	\begin{eqnarray}
	\begin{aligned}
	\int_{K} (\lambda(S) k(x))^{-1}\mathbf{v}_{h,p}\cdot w_h-\int_{D} \text{div}(\mathbf{w}_h)p_h&=0 \quad\forall w_h\in V_h(K),\\
	\int_K \text{div}(\mathbf{v}_{h,p})q_h&=\int_D f q_h \quad\forall q_h\in Q_h(K).\label{fine conser}\\
	\mathbf{v}_{h,p}\cdot \mathbf{n} &=\mathbf{v}_{H}\cdot \mathbf{n} \quad \partial K.
	\end{aligned}
	\end{eqnarray}
	Since different coarse blocks are disjoint,implementations can be conducted in parallel to enhance efficiency. In the Patch method, there is a modification in the boundary condition, where we need to solve an auxiliary field in some local problems with small size, but in return higher accuracy is obtained with similar complexity. In terms of Stitch method, on the other hand, computational cost is reduced by restricting the local problems to smaller size.

	In some flow problems such as single-phase flow and multi-phase flow model, we will utility the postprocessing techniques to guarantee the mass conservation in fine scale. Despite the fact that computation in fine scale is demanding, this approach is necessary when the source term is non-constant, which means the computations of $v_{h,p}$ are very limited and efficient.
	\section{Numerical results}
	In this section, we present numerical results for two-phase flow problem, where permeability field comes from SPE10. As one may refer Figure \ref{fig:model}, the most significant features of this permeability are the channelized structure and high contrast, which give rise to some inaccuracy using local approaches. In our experiments, we take the last 80 layers, which contain most heterogeneous part of information. One can observe that it performs better to utilize global information in constructing multiscale basis. In our experiments, We compare the results of Mixed finite element method (MFEM), Mixed multiscale finite element method with limited global information (MMsFEM), Mixed generalized multiscale finite element method (MGMsFEM).
	We performed all the computation on a workstation with Intel(R) Xeon(R) CPU E5-2650 CPU and Matlab, where 40 cores were used in the basis computation stage in parallel.

	The permeability field $\kappa(x)$ is given on fine mesh with size $220\times60\times80$ while different scales of coarse mesh are used. In the simulation, we consider $n=20,10,5$, where $n$ is the number of fine elements in each coarse element in each axis. In particular, there are $n^3$ fine elements in each coarse element. We compare two circumstances, where the difference lies in location of
	injectors and producers. In case 1 the water is injected from four edges and produced in the center of the three-dimensional domain while in case 2 we exchange the injectors ans producers. We denote four producers by producer 1, producer 2, producer 3 and producer 4.

	We use $L^2$ error for the average relative saturation error $e_s$, in particular, we have
	\begin{eqnarray*}
		e_{s,i}&=\sqrt{\dfrac{\int_{D}(s_{ref,i}-s_{ms,i})^2}{\int_{D}{s_{ref,i}}^2}},\quad     e_s=\dfrac{\sum_{i}^{T_{\text{final}}}e_{s,i}}{n},
	\end{eqnarray*}
	where $s_{ref,i}$ and $s_{ms,i}$ are reference and multiscale saturation at $i-th$ time instant respectively while $T_{\text{final}}$ is the final time instant.

	In Table \ref{time1} and Table \ref{time2} (for case 1 and 2 respectively), time in seconds for constructing multiscale basis ($T_{\text{setup}}$ in the table), simulation time and average saturation error $e_s$ among iterations are exhibited. The shown $T_{\text{setup}}$ in two tables can be shortened if more cores are used.

	"Dof" corresponds to the dimension of (\ref{matrix gms}) or (\ref{matrix lim}). Specifically, in each coarse element, we use one pressure basis function which is a piecewise constant function support in this coarse element. As for velocity basis functions, we construct them in each coarse neighborhood. For MMsFEM, there is only one for each coarse neighborhood while multiple basis functions are constructed in MGMsFEM. We use $a+b$ to denote there are $a$ offline and $b$ online basis functions in each local neighborhood. We compare six different combinations of basis functions, $1+0,6+0,8+0,2+1,2+2,4+2$. With same $n=20$, there is a significant improvement in accuracy when we enrich the multiscale space with online basis functions. The average errors shown in the lase column of Table \ref{time1} are approximately diminished by half from $8\%$ in the case $8+0$ to $4\%$ in case $4+2$. In the second example, there is a degradation in accuracy after injectors and producers are exchanged. However, the error decay is still apparent especially when online basis functions are incorporated. The case $2+2$ has smaller average error $7.7\%$ compared with $8.6\%$ in the case $8+0$, where the corresponding dimension is nearly doubled and simulation time also increased by almost a half, nearly 1500 seconds. The case $4+2$ with $n=20$ performed practically as well as the case $2+2$ with $n=10$, while in the latter case the dimension is much larger, which gives rise to almost twice the length of simulation time, i.e. more than 8000 seconds. Consequently, it is evident that the enrichment, in both offline and online levels, can improve the approximation performance. However, online enrichment is more efficient than offline enrichment, where only local information is utilized in the latter case.

	As for error convergence, one may easily verify from Table \ref{time1} and Table \ref{time2} that error decay is apparent with finer coarse elements for MMsFEM. We consider three sizes of coarse elements, $n=20,10,5$. Notably, when $n$ is twice larger, the simulation time is much shorter without losing much accuracy. For instance, the average error is increased by $2\%$ when we resize the coarse elements from $n=5$ to $n=10$ while the dimension is largely diminished. In addition, the simulation time is reduced by nearly $80\%$, which contributes to higher efficiency. In Theorem \ref{the}, there is error bound $\delta$, which represents how well the two-phase velocity field can be approximated by the single-phase velocity field in each local patch. As we decrease the coarse-mesh size, the error will converge to the bound. From Figure \ref{fig:e_1} and \ref{fig:e_2}, it is remarkable that the largest saturation error is nearly reduced by half in $4+2$ compared to $8+0$, which is consistent with the aforementioned average error. In addition, one can observe that it converges faster in iterations with bigger dimension. In particular, before 300 time instants, the accuracy in case $n=5$ is almost steady while for case $n=20$, the error keeps increasing in the first 500 instants.

	When we compare the MMsFEM and MGMsFEM, we can see that both of these two methods can provides relatively accurate solutions. Compared to the case when only offline basis functions are used in MGMsFEM, MMsFEM is more efficient in reducing error while the dimension of the system is much smaller than MGMsFEM, which costs shorter computation time. It is apparent that limited global information efficiently improves the accuracy without increasing much computation time by comparing the case between MMsFEM and MGMsFEM with $1+0$ basis. For example when $n=20$, MMsFEM gives $9.1\%$ compared with $39.3\%$ MGMsFEM with $1+0$ basis while the time spent in multiscale space construction and simulation is similar in these two cases. In addition, when we consider the case when $n=20$, the MMsFEM has average error $9\%$ compared to $8\%$ in MGMsFEM(8+0). However, the simulation time in MMsFEM is shortened by half (about 3000 seconds). Once the online enrichment is implemented in MGMsFEM, error decay is remarkable while computation time is comparable with MMsFEM. In Table \ref{time1}, when $n=20$, we can see that MMsFEM has error $9.1\%$ with near 3363 seconds while MGMsFEM$(2+1)$ has error $11.2\%$ with about 3700 seconds. However, MGMsFEM$(2+2)$ achieves smaller error $6.4\%$ with not much longer simulation time about 4000 seconds. Consequently, we can come to the conclusion that MGMsFEM serves as a flexible tool as one can adjust the numbers of offline and online basis functions to achieve relatively high accuracy without adding some computation costs.

	Figure \ref{fig:scompare1} and Figures \ref{fig:pc_b1}-\ref{fig:pc_b4} present the water-cut curves for case 1 and case 2 separately. There is only one producer while four in the latter case. For the water-cut, it is referred to the fraction of water $q_w/q_t$ in the produced fluid, where $q_t=q_w+q_o$. Besides, it is a function of time.  $q_w$ and $q_o$ are the flow rates of water and oil at the production edge of the model. Specifically, $q_w=\int_{\partial\Omega^{\text{out}}} f(S)v\cdot ndS$ and $q_t=\int_{\partial\Omega^{\text{out}}} v\cdot ndS$, where $\partial\Omega^{\text{out}}$ is the outer flow boundary. We make a comparison among four cases: MFEM, MMsFEM with n=5,10,20 and MGMsFEM with 4+2 basis and n=20. One may observe that the MGMsFEM best approximate the reference case while the MMsFEM offers relatively lower accuracy, where the accuracy increases with smaller coarse-mesh sizes. However, it is worth mentioning that the water-cut lines are almost identical between the best two approximations, MGMsFEM and MMsFEM with $n=5$.

	Figure \ref{fig:scompare1}-\ref{fig:scompare3} show saturation dynamics at three time instants, 50, 500, 2000 respectively, where water is injected from four sides. In each figure, (a),(b),(c) correspond to solutions resulted from MFEM, MMsFEM with $n=10$ and MGMsFEM with $n=20$ and $4+2$ basis in each local neighborhood. It is evident that solution by MGMsFEM can approximate the reference case excellently while some minor differences could be detected from MFEM and MMsFEM. However, in addition to the fact that the overall shapes of saturation maps are similar between reference and two approximations, most of the fine detail is retained, which shows the computational efficiency of these two approximation methods. Figure \ref{fig:scompareb1}-\ref{fig:scompareb3} show the dynamics of saturation in the other circumstance, where the injector is in the center. Similarly results can be attained in this case. Consequently, these two methods offer relatively good alternatives for reference solution from MFEM, which is consistent with aforementioned error results.  Summarizing these cases, numerical results convincingly show that one can use single-phase flow computation to
	solve the two-phase flow model to get relatively accurate approximation.\\

	\begin{figure}
		\centering
		\includegraphics[scale=0.5]{Figures/modelspe10-eps-converted-to.pdf}
		\caption{SPE10 model in log10 scale.}
		\label{fig:model}
	\end{figure}
	\begin{table}[H]
		\centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
			\hline
			Method    & $n$ &Dof   & $T_{\text{setup}}(s)$ &$T_{\text{sim}}(s)$ &$e_s$  \tabularnewline
			\hline
			MFEM&/   &4188400&/&122879.0 &/ \tabularnewline\hline
			MMsFEM&20    &439 &3114.4& 3363.4&0.091  \tabularnewline\hline
			MMsFEM&10   &3868 &3338.7&6570.9 &0.084  \tabularnewline\hline
			MMsFEM&5    &32368&4128.0&28505.6 &0.060  \tabularnewline\hline
			MGMsFEM(1+0)&20    & 439&1514.5&3823.4 &0.393 \tabularnewline\hline
			MGMsFEM(1+0)&10    & 3868&625.9&7966.3 &0.311 \tabularnewline\hline
			MGMsFEM(1+0)&5    & 32368&3093.4&31715.5 &0.168 \tabularnewline\hline
			MGMsFEM(6+0)&20    &1974&1669.6&5451.3 &0.103 \tabularnewline\hline
			MGMsFEM(8+0)&20   & 2588&1718.4&6059.4 &0.080 \tabularnewline\hline
			MGMsFEM(2+1)&20    &1053 &2846.4&3741.4 & 0.112\tabularnewline\hline
			MGMsFEM(2+2)&20    & 1360&3992.2&4175.7 & 0.064\tabularnewline\hline
			MGMsFEM(4+2)&20   & 1974&4118.8&4997.4 &0.047 \tabularnewline\hline
			MGMsFEM(2+2)&10    & 12304&1767.0&8258.0 &0.046 \tabularnewline\hline

		\end{tabular}
		\caption{Computation time and error comparison of different methods for case 1.}
		\label{time1}
	\end{table}

	\begin{table}[H]
		\centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
			\hline
			Method    & $n$ &Dof& $T_{\text{setup}}(s)$ &$T_{\text{sim}}(s)$ &$e_s$  \tabularnewline
			\hline
			MFEM&/ &4188400 &/&119944.2 &/ \tabularnewline\hline
			MMsFEM&20  &439 &2930.3&3756.7 &0.096  \tabularnewline\hline
			MMsFEM&10  &3868& 3069.3&6689.9 &0.082  \tabularnewline\hline
			MMsFEM&5   &32368& 4180.2&31195.6 &0.063  \tabularnewline\hline
			MGMsFEM(1+0)&20   &439&1539.1&4126.2&0.353  \tabularnewline\hline
			MGMsFEM(1+0)&10   &3868& 637.6&7971.9&0.297  \tabularnewline\hline
			MGMsFEM(1+0)&5   &32368& 3146.1&32127.2&0.167  \tabularnewline\hline
			MGMsFEM(6+0)&20    &1974&1708.9&5108.2 &0.110 \tabularnewline\hline
			MGMsFEM(8+0)&20    &2588&1727.8&5961.6 &0.086 \tabularnewline\hline
			MGMsFEM(2+1)&20    &1053&2884.2&4017.7 &0.122 \tabularnewline\hline
			MGMsFEM(2+2)&20    &1360&4076.4&4540.6 &0.077 \tabularnewline\hline
			MGMsFEM(4+2)&20    &1974&4178.8&5203.4 & 0.059\tabularnewline\hline
			MGMsFEM(2+2)&10    &12304&1771.3&8851.4 &0.051 \tabularnewline\hline
		\end{tabular}
		\caption{Computation time and error comparison of different methods for case 2.}
		\label{time2}
	\end{table}
	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{eave1}
		\caption{Relative saturation errors as functions of time for case 1.}
		\label{fig:e_1}
	\end{figure}
	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{pc1}
		\caption{Comparison of water-cut between MMsFEM and MGMsFEM for case 1.}
		\label{fig:pc_1}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine1}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_1}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_1}}
		\caption{Comparison of saturation obtained in three methods at time instant 50 for case 1.}
		\label{fig:scompare1}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine2}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_2}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_2}}
		\caption{Comparison of saturation obtained in three methods at time instant 500 for case 1.}
		\label{fig:scompare2}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine3}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_3}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_3}}
		\caption{Comparison of saturation obtained in three methods at time instant 2000 for case 1}
		\label{fig:scompare3}
	\end{figure}

	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{eave2}
		\caption{Relative saturation errors as functions of time for case 2}
		\label{fig:e_2}
	\end{figure}
	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{pcb1}
		\caption{Comparison of water-cut between MMsFEM and MGMsFEM for case 2 at producer 1.}
		\label{fig:pc_b1}
	\end{figure}

	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{pcb2}
		\caption{Comparison of water-cut between MMsFEM and MGMsFEM for case 2 at producer 2.}
		\label{fig:pc_b2}
	\end{figure}

	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{pcb3}
		\caption{Comparison of water-cut between MMsFEM and MGMsFEM for case 2 at producer 3.}
		\label{fig:pc_b3}
	\end{figure}

	\begin{figure}[H]
		\centering
		\includegraphics[trim={1cm 6cm 1cm 7cm},clip,width=3.8in]{pcb4}
		\caption{Comparison of water-cut between MMsFEM and MGMsFEM for case 2 at producer 4.}
		\label{fig:pc_b4}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine_b1}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_1b}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_b1}}
		\caption{Comparison of saturation obtained in three methods at time instant 50 for case 2.}
		\label{fig:scompareb1}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine_b2}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_2b}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_b2}}
		\caption{Comparison of saturation obtained in three methods at time instant 500 for case 2.}
		\label{fig:scompareb2}
	\end{figure}

	\begin{figure}[H]
		\centering
		\subfigure[reference]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sfine_b3}}
		\subfigure[MMsFEM with n=10]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{slim_n10_3b}}
		\subfigure[MGMsFEM with n=20 and 4+2 basis functions]{
			\includegraphics[trim={1cm 7.5cm 1cm 7cm},clip,width=3.0in]{sms_n20_4_2_b3}}
		\caption{Comparison of saturation obtained in three methods at time instant 2000 for case 2.}
		\label{fig:scompareb3}
	\end{figure}
	\section{Discussions and conclusions}
	In the paper, we compared two multiscale methods for the simulations of two-phase models with permeability field which is highly heterogeneous and of high contrast. The main difference between these two methods is the construction of multiscale basis functions. For MGMsFEM, we construct the offline multiscale space via a set of independent spectral problems, which can be implemented in parallel to save time. For the online multiscale space, we enrich the offline space by some residual driven online basis functions, which may be solved in some oversampled regions. The global information contained in the online basis functions enhances the accuracy remarkably. For the MMsFEM with limited global information, it is modified from the standard MMsFEM. In this method, no local spectral problem is implemented. In addition, for MMsFEM, we incorporate global information in the computation of multiscale basis functions. In particular, we need to solve an independent single-phase problem in advance to get information about the global flow pattern from the initial velocity field. This procedure is not included in MGMsFEM but it serves as an effective tool to improve the performance of the MMsFEM.

	In terms of the most important factor, i.e. accuracy, we can see that these two methods can both give satisfactory outcomes. MMsFEM can retain most of the small-scale information from the reference solutions. Besides, it can give relatively good approximations of the concerned quantities, such as the water-cut and saturation. Another advantage of this method is that the multiscale space is smaller that that of MGMsFEM since only one multiscale basis function is incorporated in each local neighborhood, which results in higher efficiency. However, a shortcoming is that the performance of the method is dependent on the similarity of the initial velocity field and the actual one in simulation. If there is a big mismatch between these two cases, there is no guarantee that the MMsFEM will give a satisfying result. Moreover, since single basis function is required in each local neighborhood for MMsFEM, we need to seek other ways to improve the performance. On the other hand, there are more flexibility for the MGMsFEM since we can change the number of offline and online basis functions in each local region. Besides, we can also adjust the number of oversampling layers as we construct the online basis functions. Consequently, MGMsFEM is a more flexible method than the MMsFEM and one may easily find a combination of the concerned parameters to obtain a better approximation. However, there are still some drawbacks for MGMsFEM. First, as one may verify from the numerical results, in somes cases only offline multiscale space is not sufficient to obtain a good approximation especially when the permeability field is of high heterogeneity such as $\kappa_3$ in this paper. In our experiments, MMsFEM can offer better results than MGMsFEM with a much bigger offline space. Consequently, one may seek to enrich the offline space with online basis functions, which require further computation. Besides, since multiple basis functions are used in each local neighborhood, the final system is bigger than MMsFEM, which give rise to larger computational cost. One also needs to construct a snapshot space and reduce its dimension to obtain the offline space, which is not required in MMsFEM.

	Consequently, the choice between these two methods is dependent on the actual problem. For example, if it is easy to obtain the required information of the initial velocity field or it is well-prepared, then MMsFEM may be considered as a good alternative. On the other hand, if it is demanding to obtain the required global information and offline space in MGMsFEM is sufficient to give a satisfying result, then MGMsFEM may be advantageous over the other one. However, when MMsFEM can not provide sufficient accuracy, one may apply MGMsFEM combined with online enrichment if the offline space is not sufficient. Thus, we can see that both MMsFEM and MGMsFEM possess some advantages, which lead to remarkable performances in some specific cases. However, both methods have some limitations. It is worth mentioning that in the following work we can combine these two methods in some way to pursue better outcomes than applying a single method.

	\section*{Acknowledgments}

	The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14304719 and 14302018) and CUHK Faculty of Science Direct Grant 2019-20.

redpajama_set_name: RedPajamaArXiv


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Discover new shows
Ars Nova at Greenwich House
About Ars Nova at Greenwich House
Ars Nova exists to discover, develop and launch singular theater, comedy and music artists who are in the early stages of their careers. We were founded as a playground and a launch pad for emerging artists to create new work, with a dual mission to build the next generation of audiences for live entertainment by prioritizing a zeitgeist-forward aesthetic and marrying it with accessible ticket prices. Today, Ars Nova is known for its "pop-culture-savvy experimentation, with a hipness that sets it apart" (New York Times). Ars Nova presents over 100 unique live events each year, supporting over 500 emerging theater, comedy, and music artists. With our feverish bounty of programming, we are the stomping ground and launching pad for visionary, adventurous artists of all stripes. As NYC's premiere development hub, we create an environment where the weird kids thrive and where no idea is too crazy to be possible. These collaborations result in pirate-puppet-rock-odysseys, electro-pop-operas, farces-in-burlesque, nights of mind-bending magic, plays performed mostly in silence, immersive Korean-pop music factory experiences, and everything and anything in between.

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text:
How's Your Political I.Q.?: Oct. 15-19
John Gizzi
1. Rep. Jo Ann Davis (R.-Va.) died last week at age 59. How many sitting U.S. representatives have died this year?
2. Sen. Pete Domenici (R.-N.M.) recently became the fifth Republican senator to announce he will not seek re-election in '08. How many of the other four can you name?
3. Domenici has served in the Senate since 1972. Who is the only Republican senator who is senior to Domenici?
4. Domenici is one of only two Republican senators elected from New Mexico since 1932. Can you name the other?
5. What is the only race for office Domenici ever lost?
1.Two. Democrat Juanita Millender-McDoald of California, who died on April 22, and Republican Paul Gillmor of Ohio, who died on September 5.
2. Wayne Allard (Colo.), Larry Craig (Idaho), Chuck Hagel (Neb.) and John Warner (Va.).
3. Ted Stevens of Alaska, who has served in the Senate since 1968.
4. Former astronaut Harrison "Jack" Schmidt, who served from 1976-82. (Republican Gov. Edwin Meacham appointed himself to the Senate in 1963 but was defeated for a full term the following year.)
5. He ran unsuccessfully for governor of New Mexico in 1970.
Big Labor, MoveOn.org Have SCHIP Hit List
Liberal Screechers After Ann (Again)
Written By John Gizzi
John Gizzi has come to be known as �??the man who knows everyone in Washington�?� and, indeed, many of those who hold elected positions and in party leadership roles throughout the United States. With his daily access to the White House as a correspondent, Mr. Gizzi offers readers the inside scoop on what�??s going on in the nation�??s capital. He is the author of a number of popular Human Events features, such as �??Gizzi on Politics�?� and spotlights of key political races around the country. Gizzi also is the host of �??Gizzi�??s America,�?� video interviews that appear on HumanEvents.com. Gizzi got his start at Human Events in 1979 after graduating from Fairfield University in Connecticut and then working for the Travis County (Tex.) Tax Assessor. He has appeared on hundreds of radio and TV shows, including Fox News Channel, C-SPAN, America's Voice,The Jim Bohannon Show, Fox 5, WUSA 9, America's Radio News Network and is also a frequent contributor to the BBC -- and has appeared on France24 TV and German Radio. He is a past president of the Georgetown Kiwanis Club, past member of the St. Matthew's Cathedral's Parish Council, and secretary of the West End Friends of the Library. He is a recipient of the William A. Rusher Award for Journalistic Excellence and was named Journalist of the Year by the Conservative Political Action Conference in 2002. John Gizzi is also a credentialed correspondent at the World Bank and the International Monetary Fund. He has questioned two IMF managing directors, Dominique Strauss-Kahn and Christine LaGarde, and has become friends with international correspondents worldwide. John�??s email is JGizzi@EaglePub.Com


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Q: End of 3-manifold Let $M$ be an irreducible, orientable, open 3-manifold with finitely generated fundamental group, this gives us a Scott's Core $C\hookrightarrow M$ so that the inclusion is an homotopy equivalence and such that the ends of $M$ are in 1-1 correspondence with the components of $M\setminus int(C)$.
Let $\mathcal E$ be an end and $\partial_{\mathcal E} C$ be the boundary component facing it. I want to show that the inclusion $\partial_{\mathcal E} C\hookrightarrow Z$, for $Z$ the component of $M\setminus int(C)$ containing $\mathcal E$, is an isomorphism on $H_1$.
I can get surjectivity but I fail to get injectivity.

A: $H_i(M,C)=0$ by assumption that the inclusion of $C$ is a homotopy equivalence. Now use excision on the interior of $C$ to see that $H_i(M \ \text{int }C,\partial C)=0$. But $(M \setminus \text{int } C,C)$ is the disjoint union of the spaces you're interested, and (relative) homology is additive under disjoint union, hence all of your groups are zero.
(If one is pedantic they say "You need the set you're excising to have closure contained in $C$! But just take a tubular neighborhood of the boundary etc.)

A: This is an answer to the follow-up question you asked in the comments to Mike's answer.
Theorem. Suppose that $C\subset M$ is a Scott compact core of an open irreducible 3-manifold $M$. Then for each component $E\subset M- int(C)$ (informally known as an "end" of $M$), for the (necessarily connected) boundary surface  $S=\partial E$, the inclusion $i: S\to E$ induces an isomorphism $i_*$ of fundamental groups.
Proof.

*

*First, let us check the harder part, the injectivity of $i_*$. (It is harder since it uses some nontrivial results of 3-dimensional topology.)  I will work in the PL category. Suppose that $i_*$ is not injective. Then, by the Loop Theorem, there exists a simple loop $L\subset S$ bounding a properly embedded disk $D_1\subset E$ such that $L$ does not bound a disk in $S$. However, the inclusion $C\to M$ is $\pi_1$-injective. Therefore,  the loop $L$ as above, is null-homotopic in $C$. By the Dehn Lemma, since $L$ is simple, it bounds a properly embedded disk $D_2\subset C$. (This is a historic accident that one of the results I use is called a theorem and the other is called a lemma, both are hard, with the first correct proof due to Papakyriakopoulos.) Since $M$ is irreducible, the 2-sphere $D_1\cup D_2$ bounds a 3-ball $B\subset M$.  Let $S':= S\cap B$. This subsurface of $S$ has exactly one boundary component, namely, $L$.  I claim that $S'$ is is the 2-disk, which will contradict the fact that $L$ does not bound a disk in $S$. Suppose not. Then the interior of $S'$ contains two simple loops $a, b$ which intersect transversally in exactly one point. Therefore, one of these loops, say, $a$, is homologically nontrivial in $C$. But $a$ is contained in the 3-ball $B\subset M$ and, hence, is homologically trivial in $M$.  (Let me know if you want to see a proof.) Therefore, the inclusion map $C\to M$ is not a homotopy-equivalence. A contradiction. Thus, $i_*$ is injective.

*I will now prove the surjectivity of $i_*$. Let $C'$ denote $(M - E) \cup S$. Since the inclusion $C\to M$ is $\pi_1$-surjective, so is the inclusion $C'\to M$. By  the Seifert - van Kampen theorem, the fundamental group of $M$ is the pushout of the diagram
$$
\pi_1(C')\leftarrow \pi_1(S)\rightarrow \pi_1(E).
$$
If $i_*$ is not surjective, the homomorphism $\pi_1(C')\to \pi_1(M)$ cannot be surjective either: This is a consequence of the group-theoretic fact that if you have a group amalgam
$$G=G_1 \star_{G_3} G_2$$
and $G_3\ne G_2$ then $G_1\ne G$. A contradiction.  qed

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She was descended from the masters of the universe. To hold her, he challenged the gods.
Best selling author Janet Morris began writing in 1976 and has since published more than 30 novels, many co-authored with her husband Chris Morris or others. She has contributed short fiction to the shared universe fantasy series Thieves World, in which she created the Sacred Band of Stepsons, a mythical unit of ancient fighters modeled on the Sacred Band of Thebes. She created, orchestrated, and edited the Bangsian fantasy series Heroes in Hell, writing stories for the series as well as co-writing the related novel, The Little Helliad, with Chris Morris. She wrote the bestselling Silistra Quartet in the 1970s, including High Couch of Silistra, The Golden Sword, Wind from the Abyss, and The Carnelian Throne. This quartet had more than four million copies in Bantam print alone, and was translated into German, French, Italian, Russian and other languages. In the 1980s, Baen Books released a second edition of this landmark series. The third edition is the Author's Cut edition, newly revised by the author for Perseid Press. Most of her fiction work has been in the fantasy and science fiction genres, although she has also written historical and other novels. Morris has written, contributed to, or edited several book-length works of non-fiction, as well as papers and articles on nonlethal weapons, developmental military technology and other defense and national security topics.
Wind from the Abyss 5 out of 5 based on 0 ratings. 3 reviews.
Finally I can get the Silistra series for my e-reader!
Love the Silistra books. This third one rocks.
Wind from the Abyss Paperback by Janet Morris A superb read and thoroughly enjoyable. Excellent!


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Temperature affects human migration, new research shows
Saturday, January 20, 2018 - 11:30am
Writer Adam Wernick
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In 2015, more than a million refugees and migrants sought safety in Europe. Those numbers have dropped as European countries have worked to shut down access routes.
If global warming gas emissions continue at the present pace, the number of asylum-seekers to Europe could increase by nearly 200 percent, according to a new study.
Unrest, war and terrorism have boosted the number of desperate people fleeing parts of the Middle East and Africa, but new research from Columbia University economist Wolfram Schlenker shows a warming planet may also be a culprit.
The research, which appears in the journal, Science, links higher temperatures in agricultural regions with the flood of people seeking asylum in the European Union. If current temperature trends continue, the EU can expect an additional 600,000 or more refugees begging to enter each year — nearly twice as many as those who currently seek asylum.
The researchers examined the immigration numbers to EU destination countries and looked at 103 source countries for which they had both immigration and temperature data, Schlenker explains. They used satellite scans to identify agricultural areas and found that these areas are responsible for 93 percent of all asylum applications to the EU.
The paper, Schlenker says, is in two parts. The first links year-to-year changes in weather to changes in asylum applications over a period of 15 years, starting with 2000. It asks a straightforward question: Was it hotter than average or colder than average and, as a result, did asylum applications to the European Union from that country increase or decrease?
"Around 20 degrees Celsius, which is 68 degrees Fahrenheit, we see that asylum applications seem to be lowest. If you're hotter or colder, they tend to increase," Schlenker explains. "This implies that if you're a country which currently has a temperature that is higher than this optimal 20 degree Celsius level, you would see an increase with warming. If you are in a country that currently has a temperature that is lower than the 20 degrees Celsius optimum, you would actually see a decrease from warming."
The second part of the paper uses this statistical relationship to predict what could happen to asylum applications by the end of this century. Under a "business-as-usual" scenario — continued use of fossil fuels and high population growth — asylum applications to the EU are predicted to increase by 188 percent.
While in scientific research there's a big difference between correlation and causation, Schlenker feels confident that the first part of the paper reveals a causal relationship. "Those weather shocks, whether you're hotter than normal or colder than normal, are random and exogenous," he explains. "So, that's not just a correlation. I'm pretty sure this is a causal relationship."
The second part, which uses the observed statistical relationship and fast-forwards 80 years into the future, makes an assumption that events essentially remain constant, but "you might argue that in 80 years things might change," Schlenker says. But they could change in either direction.
"On the one hand, people might adapt [and] become better at withstanding higher temperatures," he explains. "Therefore, we might expect that in the future you see less sensitivity to temperature than we observe now. On the other hand, you might argue that right now we're basically measuring the effect of [an occasional] bad outcome … while in the future, with climate change, we would observe [these outcomes] years in a row, [which] might lead to further unraveling."
Despite the potentially tragic outcomes his research predicts, Schlenker remains optimistic.
"There are myriad impacts from climate change in various countries," he says. "I also really believe in human ingenuity. I really believe that humans have ways to combat climate change, and so far, I feel like the best estimate is that it's not that costly, considering what the potential benefits might be. So, I personally feel that with human action, with human engineering and new developments, it might very well still be feasible to achieve something, but what we need is the political willingness to engage in that."
This article is based on an interview that aired on PRI's Living on Earth with Steve Curwood.
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In fall of 2019, over eighty organizations and thousands of individuals across Baltimore came together to pass the Elijah Cummings Healing City Act and build a movement for change. Now we are taking the movement live in a weekly Facebook Live series.
Truth & Healing is an ongoing citywide conversation about how Baltimore can heal from its enduring trauma. We hope to elevate conversations around mental health, racisim, youth activism, anti-violence work, and more. We will center the voices of Baltimoreans from all walks of life who are working to heal our city.
Tune in weekly as we focus on moving Baltimore forward, together.
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Our youth are taking to the streets by the thousands to demand an end to the racial, social, and economic inequities and injustices that have traumatized generations. Now is the time to show them we hear them, and we are with them. Baltimore should be increasing funding for youth and trauma-related services as the Healing City Act mandates--Not spending less.
Take action now to ensure that Baltimore's legacy will be one we can be proud of. Our youth demand nothing less, and we need to stand with them.
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On December 21, 2019, Destiny was senselessly gunned down in front of her young daughter in her salon. To honor her legacy, barbers, beauticians and community members have created the Destiny's Dream Scholarship fund to support students in the Mervo Cosmetology program.
The scholarship will award $1,000 to five students each year to help cover expenses for uniforms, board certifications and the beauty kit each student needs to purchase to participate in the program. Please help fund this scholarship in Destiny's memory. Together, we can heal our city and work to break the cycle of trauma and violence that stole Destiny from us.
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During what is arguably the most challenging period in our lifetime, it is more important than ever for Baltimoreans to unite. We face a deadly virus that threatens to tear us apart. Necessary distancing leaves our most vulnerable residents isolated without access to emotional support. In this moment of crisis, we refuse to turn our backs on our neighbors.
The Baltimore Neighbors Network provides a virtual community of volunteers to offer hope, comfort and social solidarity. We also enlist pro bono clinicians to help address the looming mental health crisis that COVID-19 will create. Baltimore's greatest asset is our people. WE are the medicine.
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On February 7 & 9, 2020, the united efforts of dozens of organizations and over 80 individuals culminated in a weekend full of community building, healing and solidarity in our commitment to building a racially, socially and economically just and thriving Baltimore for ALL.
On Friday, hundreds of students led discussions and panels on trauma and healing at Morgan State University. On Sunday, hundreds of neighbors from across the city met at Coppin State University in a Celebration of Solidarity with free hair services, a resource fair, conversation, music and entertainment, before marching in unity to Frederick Douglass High School for the historic signing of the Elijah Cummings Healing City Act. (All Photos Courtesy of John Waire)
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The Power of Honesty
01 Dec The Power of Honesty
There are tons of sayings that you've probably heard over the course of your lifetime. Some of them you listen and take to heart, and others you just tend to ignore. Some of my favorites have to do with truth and honesty. I'm sure that you've heard the saying, "honesty is the best policy." You're also probably familiar with, "the truth shall set you free!" I know these two phrases were definitely some of those things that I heard a lot when I was growing up. I used to hate them though when I was younger because I lied quite a bit. It was never anything big, but on occasion I would tell a fib whenever it suited my needs. If I could get away with it then I would do it. And sometimes I told a lie so much that I started to actually believe the lie that I had been telling. But the thing is, even though I was able to get away with it, it was still taxing. Having to keep a story straight takes a lot of work. If you mess up a detail one time then it could end up throwing your whole story off, just like it did in the storyline of a play I saw recently…
Now, I know I've told you guys about my love of musicals, so bear with me here while I talk about this one. I won't go into too much detail about it, just in case you want to see it, but I will give you enough information so that you can get the gist of the lesson from the play.
*Spoilers Ahead*
Dear Evan Hansen is about this kid in high school who tells a lie. He tells himself that he's doing it to help out some people who could use some cheering up. And to be fair there was probably a small part of him that was telling the lie because he genuinely wanted to do some good for someone else, but most of the reason had to do with him wanting to be liked, and I can't fault him for that. High school is tough, especially when you're a loner, so it made sense that he wanted people to like him.
When he first tells the lie, there's just a handful of people who know the story that he's made up, but by the end of the first act everyone in his school and hundreds of thousands of people across the country have also heard this story (thanks to the power of the internet) that he's fabricated as well.
When we come back from the first act we find out that his lie does end up doing some good. By keeping the fib going he's able to raise awareness about a serious issue, but halfway through the second act the fib ends up being too much for him to bear. People start finding holes in his story and it becomes harder and harder for him to keep lying. Eventually he ends up telling the truth to the people he was trying to help in the beginning, and by the end of the play you can definitely see why telling the truth is so important. He was only hurting the people that he tried to help by keeping this secret from them. When he lied to them he allowed them to live in some fantasy world, and while it was nice for a while, knowing the truth was able to provide them with the healing that they longed for. The protagonist was even able to feel better after the initial fallout that occurred after they found out that he had been lying to them for months. So in this case (as it is in most cases). Honesty was the best policy, and the truth really did set him free.
*Spoilers Over*
When you tell the truth it's freeing. You feel better because you don't have to worry about keeping the story straight and the other people around you will appreciate your honesty as well. When all of the information is out there on the table everybody benefits because they know what they're going to get. You deserve that, and so does the other person.
Positive thinker, tell the truth. It may not seem like the smart thing to do in the moment, but in the long run you'll be happy that you did!
decisions happiness honesty positivity truth


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Home » 3D , Byron Howard , Chris Williams , comedy , Disney , John Travolta , Miley Cyrus , PG » "Bolt" Review
"Bolt" Review
Title: Bolt
Director: Chris Williams, Byron Howard
Starring: John Travolta, Miley Cyrus
Studio: The Walt Disney Company
Genre(s): Comedy
Rated: PG (For some mild action and peril)
Welcome back Disney. Far too long have you been in Pixar's shadows. The last big hit you had was with "Lilo & Stitch," a movie that stood out in a sea of duds and lessor attempts, and that was back in 2002. That movie was based off an idea by Chris Sanders, a longtime animator who hit box office gold. Now, after "Home on the Range," "Chicken Little," and several other poor attempts to reclaim your former glory, you have brought us "Bolt," a movie about a dog who run faster then a speeding bullet, is more powerful then a locomotive, and able to take down armies with a single bark. Or, at least, he'd like to think so. Though Bolt (John Travolta) spends his days hanging out with his "person" Penny (Miley Cyrus) and helping her fight the evil Green Eyed Scientist, he's really filming a hit TV show called (what else) Bolt.
But this is no ordinary TV show. It's kind of like an animated version of "The Truman Show," except in this case the show producers try to keep up the illusion to the dog that the show is real. When Penny is kidnapped in a cliffhanger ending though, it's up to Bolt to go save her. Or so he thinks. As you probably know, Bolt has no powers. So instead of making a serious progress to save Penny, he spends his time being overly annoying to a black cat named Mittens, who he believes knows where Penny is (because, you know, cats are linked to mad scientists). Their adventure leads them across America, discovering many eccentric creatures such as wise talking pigeons and a hamster in a rolling ball named Rhino. Though short of an original story, "Bolt" gets high marks for taking an average sounding screenplay and making it work.
The idea was thought up by aforementioned Chris Sanders, who was supposed to direct the film as well, until John Lasseter read an early draft of the script and said it was "too quirky for it's own good." Thus Sanders was let go, and Disney had to produce the film with a couple of new comers. And while I would have been very interested to see what Sanders would have done with the film (the cat in his version had an eye patch, which just made her look cool), I'm happy with the final result. This is actually a good example of how effective movies can be. Here we have one of the silliest ideas for a movie all year, and yet it's written with enough wit and heart that we come to grow invested in it. I'm sure John Travolta could be making movies that make him more money, but his performance of Bolt is filled with the right amount of innocence and braveness that makes him so likely.
We know he's supposed to be annoying, but you've got to love him for his loyalty to his person. Mittens the cat also gets points for having a sympathetic back story, while Rhino provides the token best friend role. On the final end of the spectrum is Penny, played by Miley Cyrus. Truth be told, I don't think Cyrus will ever be a star, and her getting top billing for this role only emphasizes that fact. It's not that Penny is a poor character per se, but she's the least interesting thing is this movie, and is more of a McGuffen then anything else. Hiring a regular high school girl to voice this character would have wielded the same results for a fraction of the price. But now I'm nick picking, as this doesn't matter too much in the long run. The point is "Bolt" is great family fun and one of the more pleasant surprises of the year. Just make sure that, as an added bonus, you see this movie in Digital 3D, for a more three-dimensional atmosphere.
Parents have to worry about only a little bit of violence, and even that is limited to explosions and a fire near the end. Should be fine for ages 5 and up.
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Home > News > Rickshaws for refugees win USD$1 million Hult Prize
Rickshaws for refugees win USD$1 million Hult Prize
Written by Pauline Chiou, a Hult contributing writer. Photography courtesy of AP Images.
The intersection of entrepreneurship, innovation and social impact is happening in a most unlikely place: the largest slum in the world. Pakistan's Orangi Town is a slum of 2.5 million people. It's the proving ground of a start-up that recently won the $1 million Hult Prize, an annual global business competition tackling the world's biggest challenges. This year's theme is to dramatically improve the lives of refugees.
Roshni Rides captured USD $1 million in seed funding as the 2017 Hult Prize winner. The start-up is already shaking up the existing transportation network in the slum. The four founders are New Jersey college students who traveled to Pakistan and studied the rhythm of life in Orangi Town.
"This is the largest slum in Asia. They are displaced, urbanized refugees," says Hasan Usmani, a first-generation Pakistani American and recent Rutgers University graduate. "All four of us on the team come from parents who are immigrants, so this is for them."
Here's how Roshni Rides works: The start-up pays rickshaw drivers a monthly salary to drive on a set route with fixed fares. For passengers, the system eliminates price gouging. For drivers, it creates a steadier flow of passenger volume.
Former US President Bill Clinton presents at the Hult Prize Finals (Jason DeCrow/Hult Prize Foundation via AP Images)
Supply chain management in the slum
The founders of Roshni Rides are all supply chain management majors at Rutgers Business School. They put those streamlining skills to work in the slum.
"The whole business is about creating efficiency," says CEO Gia Farooqi." A rickshaw driver now picks up six to eight customers a day, but two million rickshaws are on the road. That's crazy! The rickshaws are underutilized. So we tried to answer why there's the mismatch."
Here were the challenges:
Uneven supply. The team discovered most rickshaw drivers congregate around urban centers where they can charge higher fares and get faster turnover. This means a glut of rickshaws in a handful of areas while the rest of the sprawling slum is left with fewer transportation options.
Personal safety. The bus system is overcrowded and dominated by men. According to Roshni Rides, 89% of Pakistani women don't take a job that requires a commute mainly because of safety. The team sees rickshaw transportation as a solution for women.
Affordability. Price gouging by rickshaw drivers is common outside the urban centers. Roshni Rides lowered their rickshaw fares to 80 rupees (US $0.76) per ride. They put the drivers on a fixed salary so the drivers have guaranteed income that's 27% higher than a normal month's wage. The drivers are also given devices to swipe pre-paid fare cards. This reduces the need for cash.
The Roshni Rides team from Rutgers University celebrate their win (Mark Von Holden/Hult Prize Foundation via AP Images)
The global view: A growing crisis
Former President Bill Clinton announced the winner of the Hult Prize at the United Nations in September. "This year's topic is particularly important because we have 65 million displaced people in the world—the largest number since World War II," said Clinton. "This toll of involuntary migration is enormous."
According to the United Nations, 55% of the world's refugees come from South Sudan, Afghanistan and Syria. 20 people are displaced from their homes every minute because of conflict or persecution. Pakistan and Turkey are among the biggest host countries for refugees.
The UN Refugee Agency reports that an unprecedented 65.6 million people around the world have been forced from home. Among them are nearly 22.5 million refugees, over half of whom are under the age of 18. Infographic courtesy of UNHCR.
Whether in a refugee camp or a slum, it's all about survival. "They are in a spiral," explains Bill McDonough, architect and co-author of the book "Cradle to Cradle" on sustainable living. "Sustainability has three aspects: economic, social, and environmental. Graceful and creative things in life happen in peace. If you're in a war, you're not planning a garden."
"Sustainability has three aspects: economic, social, and environmental. Graceful and creative things in life happen in peace."
-Bill McDonough, architect and co-author of Cradle to Cradle
The judges' view: Passion is a given. Show financial discipline.
The Hult Prize judges evaluated the business plans of six global finalist teams.
The ideas ranged from transportation upgrades to a digital jobs platform for refugees. The choice was so difficult that it took three rounds of voting to determine a winner. The deciding factor: financial viability.
"Social entrepreneurs often lose sight that they have to run a business. You can't rely on donations," says judge Chuck Kane, chairman of One Laptop Per Child. "If you're not profitable or just break even, you impact fewer and fewer people."
"It was hard to actually know where USD$1 million would be most transformative," says another judge Premal Shah, President of the microfinance non-profit, Kiva.org. "What I look for is pilot traction."
The pilot that Roshni Rides conducted in Pakistan helped them reshape their product and business plan. The original idea was a network of solar-powered rickshaws. The team quickly saw the cost challenges of converting existing rickshaws into a solar vehicle so they scrapped the idea. The team also mapped out ten stops for customers on a rickshaw shuttle route. After conducting surveys, the team learned the majority of passengers wanted to go to the same few places (e.g. market, schools, hospital, employment center) so they whittled down the route to these locations.
"We have a lot of data from our pilot program to go through," says Farooqi. "We have a scaling plan and we want to expand to the rest of South Asia. We want to be in Bangladesh, India, and Sri Lanka by Year 5."
Former US President Bill Clinton poses for selfies with the crowd at the end of an inspiring night (Jason DeCrow/Hult Prize Foundation via AP Images)
If you would like to find out more about Hult's global business programs, download a brochure here.
Pauline Chiou is a contributing writer for Hult International Business School. She currently covers global business, news, and politics for CNN International as a freelance journalist based in New York. She spent nine years in Asia as an anchor for CNN in Hong Kong and CNBC in Singapore. Follow Pauline on Twitter @PaulineChiou.
Hult offers a range of highly skills-focused and employability-driven business school programs including a range of MBA options and a comprehensive one year Masters in International Business. To find out more, take a look at our blog How the Hult Prize turned me into a social entrepreneur. Download a brochure or get in touch today to find out how Hult can help you to learn about the business world, the future, and yourself.


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More Wins in Ohio Stadium for Wi-Fi
Erin DeVilbiss | December 6, 2019
This season, Ohio State's Wireless Expansion Project had as many wins in Ohio Stadium as the Buckeyes did.
On August 31, 2019, Ohio State played its first game against Florida Atlantic and introduced the new stadium Wi-Fi to the public. This network is the nation's first Wi-Fi6 stadium network. It placed Ohio State among the top ten stadiums for Wi-Fi usage during the first game of the season.
As the season progressed, Ohio State hit even more impressive numbers (both on and off the field).
"The peak of the season was on October 5, 2019, against Michigan State wherein the stadium observed 25.8TB of data consumption, 18.3TB of which was on the Wi-Fi network," said Ryan Holland, Associate Director of Wireless and Network Engineering at Ohio State. "This amount of data consumption placed Ohio State as the highest stadium usage of all time, exceeding that of Super Bowl 53."
Verizon's first implementation of Passpoint technology was enabled for the Miami game, which allowed "Verizon devices to connect without user action, resulting in a dramatic increase in number of connecting clients," Holland explained.
"The last game of the season against Penn State saw the highest concurrent client count with 52,400 devices using Wi-Fi simultaneously. It was this game that we observed the highest adoption of Wi-Fi, observing nearly 82% of tickets scanned resulting in a device connected to Wi-Fi," said Holland.
The infrastructure supporting these impressive amounts of data consumption and Wi-Fi usage in the stadium include:
1,750 Wi-Fi Access Points
58 Network Switches
47 Miles of Cabling
34 Data Closets
13 Network Servers and Appliances
As the Buckeyes continue to make improvements and changes as they play, so will Ohio State's Enterprise Networking team. There have been plenty of lessons learned, like making sure wireless access points are sufficiently sealed so they don't take on water damage and encouraging hungry squirrels to eat things other than new cabling (which took down several access points).
Enterprise Networking is already working on its next steps. We can't wait to be back in The Shoe!
wi-fi, OSU Wireless, osuwireless, Wireless Expansion Project, Ohio Stadium