text:
\section{Introduction}
\label{intro}
The certified reduced basis method is known as an efficient method for model order reduction of parametrized partial differential equations (see, e.g. \cite{haasdonk2017reduced,quarteroni2015reduced} , where also the terminology used in the present article is well-explained). The efficiency comes from the use of the Greedy search algorithm in the basis construction for the numerical approximation of the problem and a-posteriori control of the approximation error. The later serves not only for rigorous certification of the method, but also as the selection criterion in the Greedy selection process. This process provides incrementally better bases for the approximation and further significant speed-up in multi-query numerical simulations - relevant, for example, in the design, optimization and control contexts, through the use of RB surrogate models.

The reduced basis method was successfully applied to linear \cite{grepl2005posteriori,steih2012space,urban2014improved} and nonlinear \cite{grepl2012certified,yano2014space,yano2014Boussinesq} parabolic problems, where the spatial differential operator is coercive \cite{grepl2012certified,steih2012space} or inf-sup stable \cite{urban2014improved,yano2014space,yano2014Boussinesq}.  In general, there are two approaches for the reduced basis methods applied to unsteady problems: (1) first discretize, then estimate and reduce, (2) first estimate, then discretize and reduce. The approach (1) \cite{grepl2005posteriori,haasdonk2008reduced,grepl2012certified} is based on a time-marching problem in the offline phase and the error bounds or indicators are then stem from the structure of the discrete problem. The POD-Greedy procedure \cite{haasdonk2008reduced} is commonly used to construct the reduced-basis spaces and the Empirical Interpolation Method (EIM) \cite{barrault2004empirical,grepl2007efficient,maday2009general} is used to treat non-affine and nonlinear problems \cite{grepl2012certified}. The approach (2) starts from a weak space-time variational formulation (see, e.g. \cite{steih2012space,urban2014improved,yano2014space,yano2014Boussinesq}). The error bounds are then derived in the appropriate Bochner spaces with respect to the natural space-time norms. In this approach time is treated as a variable and thus it resembles the reduced-basis setting for elliptic problems \cite{rozza2007reduced}. The reduced-basis space is consequently constructed in the offline phase out of the space-time snapshots, obtained, for example, with the related Petrov-Galerkin discrete scheme.  However, the appropriate choice of the discrete spaces in the Petrov-Galerkin scheme results in a time-marching interpretation (see, e.g. \cite{urban2014improved,yano2014space}) of the discrete problem. In this way, the time-marching procedure allows to use the standard POD-Greedy approximation and to treat time as the parameter, which leads to the reduced-basis time-marching problem, but the error certification is accomplished  with the natural space-time norm error bound. We refer to \cite{glas2017two} for the detailed overview and comparison of these two approaches in the context of linear parabolic equations.

In this paper we treat quasilinear parabolic problems with the approach (2). We propose our $L^{2}(0,T;V)$ a-posteriori error bound, based on the space-time variational formulation of quasilinear parabolic PDEs with strongly monotone differential operators. We provide a Crank-Nicolson time-stepping interpretation of the discrete Petrov-Galerkin problem and consequently use the POD-Greedy procedure to construct the reduced-basis spaces of small dimension.

A time-marching interpretation also allows to treat the nonlinearity with the EIM in order to have offline-online decomposition for our problem available. Moreover, the parameter separability in time, achieved with the EIM, leads to a significant speed-up factor in the computational procedure. The error of the EIM is then also incorporated in the error bound.

Our work is motivated by the structure of the magnetoquasistatic approximation of Maxwell's equations (the eddy-current model). This equation finds its place in important applications, such as the computation of magnetic fields in the presence of eddy currents in electrical machines \cite{salon1995finite}. The development of fast and accurate simulation methods for such problems is of great importance in the optimization and design of electrical machines and other devices \cite{alla2019certified,ion2018robust}. Therefore there is a demand for reduced order models (see, e.g. \cite{Stykel2017}) of this quasilinear PDE, which can be further used as surrogates in the optimization procedure. Our approach is applicable to the 2-D magnetoquasistatic problem as well, and we present according numerical results.

\section{Space-Time Truth Solution}
\label{sec:2}
In this section we consider a space-time variational formulation of quasilinear parabolic partial differential equations, which we denote as the \textit{exact problem}. The corresponding discrete Petrov-Galerkin  approximation is called the \textit{truth} problem, as it is common in the RB setting. We assume that the solution to the exact problem can be approximated arbitrarily well by the discrete solution of the truth problem. We then neglect the corresponding approximation error.

\subsection{\textbf{Space-Time formulation}}
\label{sec:2.1}
Let $\Omega \subset \mathbb{R}^{d}$ be the spatial domain and $\mu \in \mathcal{D} \subset \mathbb{R}^{p}$, where $\mathcal{D}$ is a compact parameter set. Let  $V \subset H^{1}(\Omega) $  be a separable Hilbert space and $H:=L^{2}(\Omega)$. We denote by $ \langle \cdot,\cdot \rangle_{{V}}$, $\langle \cdot,\cdot \rangle_{H} $ and $ \lVert \cdot \rVert_{V}$, $\lVert  \cdot \rVert_{H}$ corresponding inner products and induced norms, respectively. To $V$ and $H$ we associate the Gelfand triple $V\hookrightarrow H \hookrightarrow V'$
 with duality pairing $\langle \cdot,\cdot \rangle_{V'V}$. The norm of $l\in V'$ is defined by $\Vert l\rVert_{V'}:=\underset{\psi \in V,\lVert \psi \rVert_{V}\neq 0}{\sup}\langle l,\psi \rangle_{V'V}/\lVert \psi \rVert_{V}$. We consider a parametrized quasilinear, bounded differential operator $A:V \times \mathcal{D} \rightarrow V'$ with induced quasilinear form
\begin{align}\label{2.1.1}
\langle A(u,\mu),v \rangle_{V'V}:=a[u](u,v;\mu)=\int_{\Omega} \nu(u(x);\mu)\nabla u \cdot \nabla v\ dx,
\end{align}
where the nonlinearity satisfies $\nu(\cdot;\mu) \in C^{1}(\mathbb{R})$. We assume that the forms \eqref{2.1.1} are strongly monotone on $V$  with monotonicity constants $m_{a}(\mu)>0$, i.e.
\begin{align}\label{2.1.2}
    a[v](v,v-w;\mu)-a[w](w,v-w;\mu) \geq m_{a}(\mu) \lVert v-w \rVert_{V}^{2}  \quad  \forall \, v,w\in V,
\end{align}
and Lipschitz continuous on $V$ with Lipschitz constants $L_{a}(\mu)>0$, i.e.
\begin{align}\label{2.1.3}
|a[u](u,v;\mu)-a[w](w,v;\mu)|\leq L_{a}(\mu) \lVert u-w \rVert_{V} \lVert v \rVert_{V} \quad  \forall \, u,w,v\in V.
\end{align}
In addition, we assume that these conditions hold uniformly:
\begin{align}
m_{a}:=\underset{\mu \in \mathcal{D}}{\inf}m_{a}(\mu)>0, \quad L_{a}:=\underset{\mu \in \mathcal{D}}{\sup}L_{a}(\mu) < \infty.
\end{align}

For given $(g(\cdot;\mu),u_{o}) \in L^{2}(I;V') \times H$ we consider the quasilinear parabolic initial value problem of finding $u(t):=u(t;\mu) \in V, t \in I$ a.e. on the time interval $I=(0,T]$, such that
\begin{align}\label{2.1.4}
  \dot{u}(t)+A(u(t),\mu)= g(t) \  \text{in} \ V', \ u(0)=u_{o} \ \text{in} \ H,
\end{align}
where $\dot{u}:=\frac{\partial u}{\partial t}$. We now define a space-time variational formulation of \eqref{2.1.4}. We use the trial space
\begin{align*}
\mathcal{X}:=W(0,T)=L^{2}(I;V)\cap H^{1}(I;V')=\{v \in L^{2}(I;V): v,\dot{v} \in  L^{2}(I;V') \}
\end{align*}
with the norm $\lVert w \rVert_{\mathcal{X}}^{2}:=\lVert \dot{w} \rVert_{L^{2}(I;V')}^{2}+\lVert w \rVert_{L^{2}(I;V)}^{2}$, and the test space  $\mathcal{Y}:=L^{2}(I;V) \times H$ with the norm $\lVert v \rVert_{\mathcal{Y}}^{2}:=\lVert v^{(1)} \rVert_{L^{2}(I;V)}^{2}+\lVert v^{(2)} \rVert_{H}^{2}$ for $v:=(v^{(1)},v^{(2)})$. The weak formulation of problem \eqref{2.1.4} reads: find $u:=u(\mu) \in \mathcal{X}$ such that
\begin{align}\label{2.1.5}
    B[u](u,v;\mu)=F(v;\mu), \quad \forall \, v\in \mathcal{Y},
 \end{align}
where
 \begin{align}\label{2.1.6}
    B[u](u,v;\mu):=& \int_{I} \langle \dot{u},v^{(1)} \rangle_{V'V}+ a[u](u,v^{(1)};\mu)dt+\langle u(0),v^{(2)} \rangle_{H}, \ \text{and} \\
    F(v;\mu):=& \int_{I}\langle g(\mu),v^{(1)}\rangle_{V'V}dt+\langle u_{o},v^{(2)}\rangle_{H}.
 \end{align}
Since $\mathcal{X} \hookrightarrow C(I;H)$, the initial value $u(0)$ is well-defined in $H$ \cite{zeidler2013linear}. We note that \eqref{2.1.2} implies coercivity of the quasilinear form $a[\cdot](\cdot,\cdot)$ and \eqref{2.1.3} implies hemicontinuity, i.e. the continuity of the mapping $s \rightarrow \langle A(u+sw),v \rangle_{V'V}$ for $s \in [0,1]$ and all $u,w,v \in V$. All together, the well-posedness of problem \eqref{2.1.5} follows, so that \eqref{2.1.5} admits a unique solution $u \in \mathcal{X}$, see e.g. \cite[Theorem 30.A]{zeidler2013nonlinear}.

\subsection{\textbf{Petrov-Galerkin Truth Approximation}}
\label{sec:2.2}
 From here onwards we omit the dependence on $\mu$ wherever appropriate. For the temporal discretization of \eqref{2.1.5} we use the time grid $0=t^{0}<t^{1}<...<t^{K}=T$ and set $I^{k}=(t^{k-1},t^{k}]$ for $k=1,...,K$. We set $\triangle t^{k}=t^{k}-t^{k-1}$ and define $\triangle t:=\max_{1\leq k \leq K} \triangle t^{k}$. For the spatial discretization we set $V_{h}=\text{span}\{\phi_{1},...,\phi_{\mathcal{N}_{h}}\}\subset V$, where $\dim {V_{h}}=\mathcal{N}_{h}$ and $h$ denotes the spatial discretization parameter. The functions $\phi_{i}$ will be defined in the numerical examples. With $\delta:=(\triangle t,h)$ we introduce the discrete trial space
\begin{align*}
\mathcal{X}_{\delta}:=\{u_{\delta} \in C^{0}(I;V),\restr{u_{\delta}}{I^{k}} \in \mathcal{P}_{1}(I^{k},V_{h}),\ k=1,...,K \} \subset \mathcal{X}
\end{align*}
and the discrete test space
\begin{align*}
\mathcal{Y}_{\delta}:=\{v_{\delta}\in L^{2}(I;V),\restr{v_{\delta}}{I^{k}} \in \mathcal{P}_{0}(I^{k},V_{h}),\ k=1,...,K \}\times V_{h} \subset \mathcal{Y}.
\end{align*}
With these choices of spaces the fully discrete truth approximation problem reads: find $u_{\delta}:=u_{\delta}(\mu) \in \mathcal{X}_{\delta}$, such that $u^{0}_{\delta}:=u_{\delta}(0)=P_{H}^{h}u_{o}$ and
\begin{align}\label{2.2.0}
    B[u_{\delta}](u_{\delta},v_{\delta};\mu)=F(v_{\delta};\mu) \quad \forall v_{\delta} \in \mathcal{Y}_{\delta},
\end{align}
where $P_{H}^{h}:H \rightarrow V_{h}$ denotes the $H$-orthogonal projection onto $V_{h}$. It follows as for \eqref{2.1.5} that problem \eqref{2.2.0} admits a unique solution $u_{\delta} \in \mathcal{X}_{\delta}$. The Petrov-Galerkin space-time discrete formulation \eqref{2.2.0} can be interpreted as Crank-Nicolson time-stepping scheme. Indeed, since the test space $\mathcal{Y}_{\delta}$ consists of piecewise constant polynomials in time, the problem can be solved via the following procedure for $k=1,...,K$:
\begin{align}\label{2.2.1}
     \int_{I^{k}}\langle \dot{u}_{\delta},v_{h} \rangle_{V'V}+a[u_{\delta}](u_{\delta},v_{h};\mu)dt=\int_{I^{k}}\langle g(\mu),v_{h} \rangle_{V'V}dt \quad \forall v_{h} \in V_{h}.
\end{align}
 Since the trial space $\mathcal{X}_{\delta}$ consists of piecewise linear and continuous polynomials in time with the values $u_{\delta}^{k}:=u_{\delta}(t^{k})$ and $u_{\delta}^{k-1}:=u_{\delta}(t^{k-1})$, we can represent $u_{\delta}$ on $I^{k}$ as the linear function
\begin{align}\label{2.2.2}
    u_{\delta}(t)=\frac{1}{\vartriangle t^{k}}\{(t^{k}-t)u_{\delta}^{k-1}+(t-t^{k-1})u_{\delta}^{k}\}, \ t\in I^{k}.
\end{align}
We use the representation \eqref{2.2.2} in \eqref{2.2.1}, test \eqref{2.2.1} against the basis functions $\phi_{i} \in V_{h} \ (i=1,...,\mathcal{N}_{h})$ and use the trapezoidal quadrature rule for the approximation of the appearing integrals. In this way we obtain the Crank-Nicolson time-stepping scheme, which for $k=1,...,K$ reads
\begin{align}\label{2.2.3}
    (u_{\delta}^{k}-u_{\delta}^{k-1},\phi_{i})_{H}+\frac{\vartriangle t^{k}}{2} \{ a[u_{\delta}^{k}](u_{\delta}^{k},\phi_{i};\mu)+ a[u_{\delta}^{k-1}](u_{\delta}^{k-1},\phi_{i};\mu)\}=\\ \nonumber
    =\frac{\vartriangle t^{k}}{2} \{\langle g(t^{k};\mu),\phi_{i}\rangle_{V'V}+\langle g(t^{k-1};\mu),\phi_{i}\rangle_{V'V}\}, \quad 1\leq i \leq \mathcal{N}_{h}.
\end{align}
Here we recall that the initial condition $u_{\delta}^{0}$ is obtained as an $H$-orthogonal projection of $u_{o}$ onto $V_{h}$. Given the ansatz $u_{\delta}^{k}=\sum_{i=1}^{\mathcal{N}_{h}} u_{i}^{k} \phi_{i}$ and defining $\mathbf{u}_{\delta}^{k}:=\{u_{i}^{k}\}_{i=1}^{\mathcal{N}_{h}} \in \mathbb{R}^{\mathcal{N}_{h}}$, the resulting nonlinear algebraic equations are then solved by applying Newton's method for finding the root $\mathbf{u}_{\delta}^{k}$ of
\begin{align}\label{2.2.4}
\mathbf{G}_{h}(\mathbf{u}_{\delta}^{k};\mu):&=\frac{1}{\vartriangle t^{k}}\mathbf{M}_{h}(\mathbf{u}_{\delta}^{k}-\mathbf{u}_{\delta}^{k-1})-\frac{1}{2}[\mathbf{g}_{h}^{k}(\mu)+\mathbf{g}_{h}^{k-1}(\mu)] \\ \nonumber
&+\frac{1}{2}[\mathbf{A}_{h}(\mathbf{u}_{\delta}^{k};\mu)\mathbf{u}_{\delta}^{k}+\mathbf{A}_{h}(\mathbf{u}_{\delta}^{k-1};\mu)\mathbf{u}_{\delta}^{k-1}],
\end{align}
where $\mathbf{M}_{h}:=\{\langle \phi_{i},\phi_{j} \rangle_{H}\}_{i,j=1}^{\mathcal{N}_{h}},  \mathbf{A}_{h}(\mathbf{u}_{\delta}^{k};\mu):=\{a[u_{\delta}^{k}](\phi_{i},\phi_{j};\mu)\}_{i,j=1}^{\mathcal{N}_{h}}\in \mathbb{R}^{\mathcal{N}_{h} \times \mathcal{N}_{h}}$ and $\mathbf{g}_{h}^{k}(\mu):=\{\langle g(t^{k};\mu),\phi_{i}\rangle_{V'V}\}_{i=1}^{\mathcal{N}_{h}}\in \mathbb{R}^{\mathcal{N}_{h}}$. The initial condition for \eqref{2.2.4} is given by $\mathbf{u}_{\delta}^{0}:=\{(u_{o},\phi_{i})_{H}\}_{i=1}^{\mathcal{N}_{h}}\in \mathbb{R}^{\mathcal{N}_{h}}$. The strong monotonicity of the quasilinear form $\eqref{2.1.2}$ guarantees that the equation \eqref{2.2.4} admits a unique root $\mathbf{u}_{\delta}^{k}$ for every parameter $\mu \in \mathcal{D}$.

The Newton's iteration for finding a root of \eqref{2.2.4} reads: starting with $\mathbf{u}_{\delta}^{k,(0)}$, for $z=0,1,...$ solve the linear system
\begin{align}\label{2.2.4b}
    \mathbf{J}_{h}(\mathbf{u}_{\delta}^{k,(z)};\mu)\delta \mathbf{u}_{\delta}^{k,(z)}=-\mathbf{G}_{h}(\mathbf{u}_{\delta}^{k,(z)};\mu)
\end{align}
to obtain $\delta \mathbf{u}_{\delta}^{k,(z)}$, and then update the solution $\mathbf{u}_{\delta}^{k,(z+1)}:=\mathbf{u}_{\delta}^{k,(z)}+\delta \mathbf{u}_{\delta}^{k,(z)}$. The system Jacobian matrix is given by
\begin{align}\label{2.2.5}
    \mathbf{J}_{h}(\mathbf{u}_{\delta}^{k};\mu)=\frac{1}{\vartriangle t^{k}}\mathbf{M}_{h}+\frac{1}{2}\mathbf{A}'_{h}(\mathbf{u}_{\delta}^{k};\mu),
\end{align}
where $\mathbf{A}'_{h}(\mathbf{u}_{\delta}^{k};\mu):=\{da[u_{\delta}^{k}](\phi_{i},\phi_{j};\mu)\}_{i,j=1}^{\mathcal{N}_{h}}\in \mathbb{R}^{\mathcal{N}_{h} \times \mathcal{N}_{h}}$. Here we assume the existence of the Fr\'echet derivative $A'(u;\mu) :V \times \mathcal{D} \rightarrow V'$ of the nonlinear operator $A(u;\mu)$ for every parameter $\mu \in \mathcal{D}$, which induces the corresponding bilinear form $\langle A'(u;\mu)v,w\rangle_{V'V}=da[u](v,w;\mu)$. We will specify it later for our examples. We note that $\mathbf{A}'_{h}(\mathbf{u}_{\delta}^{k};\mu)$ is positive definite, since $da[u](\cdot,\cdot;\mu)$ is coercive due to the strong monotonicity of $A$; therefore, the system \eqref{2.2.4b} admits a unique solution.

\section{The Reduced Basis method}
\label{sec:3}
In this section we introduce the reduced basis model and its numerical realization. Then we introduce our a-posteriori error bound and discuss its efficient evaluation.

\subsection{\textbf{Empirical interpolation of the nonlinearity}}
\label{sec:3.1}

We use the Empirical Interpolation Method (EIM) \cite{barrault2004empirical} to ensure the availability of an affine decomposition for the quasilinear form $a[u_{\delta}^{k}](\cdot,\cdot;\mu)$ for every parameter  $\mu \in \mathcal{D}$. We then need to find a parameter-separable (affine) counterpart $\nu_{M}(\cdot;\mu)$ of the nonlinear non-affine function $\nu(\cdot;\mu)$. For EIM nonlinearity approximation, we treat time as an additional parameter in the problem, thus we set $\mathbb{I}:=\{1,..., K\}$ as our discrete time set. We construct with the Algorithm \ref{alg:EIM algorithm} the nested sample sets $S_{M}^{\nu} \subset \mathcal{D}$ and $\mathbb{I}_{M}^{\nu}\subset \mathbb{I}$, where $S_{M}^{\nu}:=\{ \mu_{1}^{\nu} \in \mathcal{D},...,\mu_{M}^{\nu} \in \mathcal{D} \}$ and $\mathbb{I}_{M}^{\nu}:=\{ k_{1}^{M} \in \mathbb{I},...,k_{M}^{M} \in \mathbb{I} \}$, and associated approximation spaces $W_{M}^{\nu}:=\text{span}\{\nu(u_{\delta}^{k_{m}^{M}}(\cdot;\mu_{m}^{\nu});\mu_{m}^{\nu}),\ 1\leq m \leq M \}=\text{span}\{q_{1},...,q_{M}\}$. Algorithm \ref{alg:EIM algorithm} also provides the nested sets of interpolation points $T_{M}=\{x_{1}^{M},...,x_{M}^{M}\}, \ 1\leq M \leq M_{\text{max}}$. We build an affine approximation $\nu_{M}(u_{\delta}^{k}(x);\mu)$ of $\nu(u_{\delta}^{k}(x);\mu)$ for our time-marching scheme according to
\begin{align}\label{3.1.2}
\nu(u_{\delta}^{k}(x);\mu)\approx& \sum_{m=1}^{M}\varphi_{m}^{k}(\mu)q_{m}(x) \\ \nonumber
=& \sum_{m=1}^{M}(\mathbf{B}_{M}^{-1}\nu_{\mu}^{k})_{m}q_{m}(\hat{x}):= \nu_{M}(u_{\delta}^{k}(x);\mu),
\end{align}
where $\nu_{\mu}^{k}:=\{\nu(u_{\delta}^{k}(x_{m}^{M});\mu) \}_{m=1}^{M} \in \mathbb{R}^{M}$ and $\mathbf{B}_{M}\in \mathbb{R}^{M \times M}$ is the lower triangular interpolation matrix $(\mathbf{B}_{M})_{ij}=q_{j}(x_{i})$ with $(\mathbf{B}_{M})_{ii}=1 \ (i=1,...,M)$ by construction.

\begin{algorithm}[H]
  \caption{: EIM algorithm}
  \label{alg:EIM algorithm}
  \hspace*{\algorithmicindent} \textbf{Input:} Stopping tolerance $\epsilon_{EIM}$, max. number of iterations $M_{\text{max}}$, parameter set $\mathcal{D}$. \\
  \hspace*{\algorithmicindent} \textbf{Output:} Nested approximation spaces $\{W_{m}^{\nu}\}_{m=1}^{M}$,  nested interpolation points $\{T_{m}\}_{m=1}^{M}$. \\
  \begin{algorithmic}[1]
  \STATE{$ (\mu_{1}^{\nu},k_{1}^{M}) := \underset{(\mu,k) \in \mathcal{D} \times \mathbb{I}}{\mathrm{arg}\max} \ \lVert \nu(u_{\delta}^{k}(\cdot);\mu)\rVert_{L^{\infty}(\Omega)} $}
  \STATE{$S_{1}^{\nu}\times \mathbb{I}_{1}^{\nu}:=  \{\mu^{\nu}_{1}\}\times \{k_{1}^{M}\}$}
  \STATE{$r_{1}(x):=\nu(u_{\delta}^{k_{m}^{M}}(x;\mu^{\nu}_{m});\mu_{m}^{\nu})$}
  \STATE{$x_{1}^{M}:=\underset{x \in \Omega}{\mathrm{arg}\max}|r_{1}(x)|, \quad q_{m}:=r_{1}/r_{1}(x_{1}^{M})$}
  \STATE{$T_{1} := \{x_{m}^{M}\}, \quad Q_{1}:=\{q_{1}\},\quad W_{1}^{\nu}:= \text{span}(Q_{1})$}
  \WHILE{$2\leq m \leq M_{\text{max}}$ and $\delta_{m}^{max}>\epsilon_{EIM}$}
  \STATE{$ (\mu_{m}^{\nu},k_{m}^{M}) := \underset{(\mu,k) \in \mathcal{D} \times \mathbb{I}}{\mathrm{arg}\max} \ \lVert \nu(u_{\delta}^{k}(\cdot);\mu)- \nu_{m}(u_{\delta}^{k}(\cdot);\mu) \rVert_{L^{\infty}(\Omega)} $}
  \STATE{$ \delta_{m}^{max} := \underset{(\mu,k) \in \mathcal{D} \times \mathbb{I}}\max\ \lVert \nu(u_{\delta}^{k}(\cdot);\mu)- \nu_{m}(u_{\delta}^{k}(\cdot);\mu) \rVert_{L^{\infty}(\Omega)} $}
  \STATE{$S_{m}^{\nu}:= S_{m-1}^{\nu}\cup \{\mu^{\nu}_{m}\}, \quad \mathbb{I}_{m}^{\nu}:=\mathbb{I}_{m-1}^{\nu}\cup \{k_{m}^{M}\}$}
  \STATE{$r_{m}(x):=\nu(u_{\delta}^{k_{m}^{M}}(x;\mu^{\nu}_{m});\mu_{m}^{\nu})-\nu_{m}(u_{\delta}^{k_{m}^{M}}(x;\mu^{\nu}_{m});\mu_{m}^{\nu})$}
  \STATE{$x_{m}^{M}:=\underset{x \in \Omega}{\mathrm{arg}\max}|r_{m}(x)|, \quad q_{m}:=r_{m}/r_{m}(x_{m}^{M})$}
   \STATE{$T_{m} := T_{m-1}\cup \{x_{m}^{M}\}, \quad Q_{m}:=Q_{m-1}\cup\{q_{m}\}, \quad W_{m}^{\nu}:= \text{span}(Q_{m})$}
   \STATE{$m \leftarrow m+1$}
  \ENDWHILE
  \end{algorithmic}
\end{algorithm}

We then have the EIM approximation $\tilde{a}[\cdot](\cdot,\cdot;\mu)$ of the quasilinear form $a[\cdot](\cdot,\cdot;\mu)$, which admits the affine decomposition
 \begin{align}\label{3.1.3}
\tilde{a}[u_{\delta}^{k}](u_{\delta}^{k},v;\mu)=\sum_{m=1}^{M}\varphi_{m}^{k}(\mu) \tilde{a}_{m}(u_{\delta}^{k},v), \quad \tilde{a}_{m}(u_{\delta}^{k},v)=\int_{\Omega}q_{m} \nabla u_{\delta}^{k} \cdot \nabla v \ dx.
 \end{align}
We also assume the affine decomposition
 \begin{align}\label{3.1.3b}
     \langle g(t^{k};\mu),v \rangle_{V'V}= \sum_{q=1}^{Q_{g}}\theta_{g,q}^{k}(\mu)\langle g_{q},v \rangle_{V'V}
 \end{align}
for the right-hand side, where $\theta_{g,q}^{k} :\mathcal{D} \rightarrow \mathbb{R}$ are parameter-dependent functions and parameter-independent forms  $g_{q} : V \rightarrow \mathbb{R}$, $k=1,...,K$, $q=1,...,Q_{g}$. If \eqref{3.1.3b} is not available, the EIM procedure can be similarly applied.

\subsection{\textbf{Reduced basis approximation with the POD-Greedy method}}
\label{sec:3.2}

The idea of the reduced-basis approximation consists in replacing the ``truth" (high-dimensional) space $V_{h}$ in the definition of $\mathcal{X}_{\delta}$ and $\mathcal{Y}_{\delta}$ by a low-dimensional subspace $V_{N} \subset V_{h}$. With $V_{N}$ available we introduce the corresponding reduced trial space
\begin{align*}
\mathcal{X}_{\triangle t,N}:=\{u_{N}\in C^{0}(I;V),\restr{u_{N}}{I^{k}} \in \mathcal{P}_{1}(I^{k},V_{N}),\ k=1,...,K \}
\end{align*}
and the reduced test space
\begin{align*}
\mathcal{Y}_{\triangle t,N}:=\{v_{N}\in L^{2}(I;V),\restr{v_{N}}{I^{k}} \in \mathcal{P}_{0}(I^{k},V_{N}), \ k=1,...,K \} \times V_{N}.
\end{align*}
We construct $V_{N}:=\text{span}\{\xi_{1},...,\xi_{N}\} \subset V_{h}$ by the POD-Greedy procedure in Algorithm \ref{alg:RB-Greedy algorithm}, compare e.g. \cite{haasdonk2008reduced}. In our setting, the POD-Greedy alogorithm constructs iteratively nested spaces $V_{n},\ 1\leq n \leq N$ using an a-posteriori error estimator $\bigtriangleup(Y;\mu)$ (see the next section for details on a-posteriori error analysis), which predicts the expected approximation error for a given parameter $\mu$ in the space $Y:=\mathcal{Y}_{\triangle t,n}$. We want the expected approximation error to be less than the prescribed tolerance $\varepsilon_{RB}$.  We initiate the algorithm with the choice of the initial basis vector  $\xi_{1}:=u_{\delta}^{0} / \lVert u_{\delta}^{0} \rVert_{V}$; this choice is motivated by the assumption in Proposition 1. The snapshots  $u_{\delta}(\mu)$ for the procedure are provided by the parametrized  ``truth" approximation \eqref{2.2.0}. Next we proceed as stated in the following Algorithm \ref{alg:RB-Greedy algorithm}.
\begin{algorithm}[H]
  \caption{: POD-Greedy algorithm}
  \label{alg:RB-Greedy algorithm}
  \hspace*{\algorithmicindent} \textbf{Input:} Tolerance $\varepsilon_{RB}$, max. number of iterations $N_{\text{max}}$, $V_{1}=\text{span}\{\xi_{1}\}$, parameter set $\mathcal{D}$. \\
  \hspace*{\algorithmicindent} \textbf{Output:} RB spatial spaces $\{V_{n}\}_{n=1}^{N}$, RB trial spaces $\{\mathcal{X}_{\triangle t,n}\}_{n=1}^{N}$, RB test spaces $\{\mathcal{Y}_{\triangle t,n}\}_{n=1}^{N}$.
  \begin{algorithmic}[1]
   \WHILE { $2 \leq n \leq N_{\text{max}}$ and $\varepsilon_{n} :=\underset{\mu \in \mathcal{D}_{train}}{\max} \bigtriangleup(\mathcal{Y}_{\triangle t,n},\mu)> \varepsilon_{RB}$}
  \STATE {$[\varepsilon_{n}, \mu_{n}] \leftarrow \underset{\mu \in \mathcal{D}_{train}}{\mathrm{arg}\max}\bigtriangleup(\mathcal{Y}_{\triangle t,n-1},\mu)$}
  \STATE $e_{n}^{k}:=u_{\delta}^{k}(\mu_{n})-P_{V} u_{\delta}^{k}(\mu_{n}), \ k=1,...,K$
  \STATE $\xi_{n}:=\text{POD}_{1}(\{e_{n}^{k}\}_{k=1}^{K})$
  \STATE{$V_{n}:=V_{n-1} \bigoplus \text{span}\{\xi_{n}\}$}
  \STATE $\mathcal{X}_{\triangle t,n} \leftarrow \mathcal{X}_{\triangle t,n-1}, \quad \mathcal{Y}_{\triangle t,n} \leftarrow \mathcal{Y}_{\triangle t,n-1}$
  \STATE{$n \leftarrow n+1$}
  \ENDWHILE
  \end{algorithmic}
\end{algorithm}
\noindent
In Algorithm \ref{alg:RB-Greedy algorithm}, $P_{V}: V_{h} \rightarrow V_{n}$ denotes the $V$-orthogonal projection, and the operation $\text{POD}_{1}(\{e_{n}^{k}\}_{k=1}^{K})$ denotes the extraction of the dominant mode of  the Proper Orthogonal Decomposition (see, e.g. \cite{volkwein2013proper}). We also note that more modes can be extracted in every step of the algorithm: it reduces the offline computational time, but there is no guarantee that the produced basis will be of the smallest possible dimension.

The reduced-basis approximation of problem \eqref{2.2.0} reads: find $u_{N}:=u_{N}(\mu) \in \mathcal{X}_{\triangle t,N}$, such that  $u^{0}_{N}:=u_{N}(0)=P_{H}^{N}u_{o}$ and
\begin{align}\label{3.1.4}
    \tilde{B}[u_{N}](u_{N},v_{N};\mu)=\tilde{F}(v_{N};\mu) \quad \forall v_{N} \in \mathcal{Y}_{\triangle t,N},
\end{align}
where
\begin{align*}
     \tilde{B}[u_{N}](u_{N},v_{N};\mu)&=\int_{I} \langle \dot{u}_{N},v_{N}^{(1)} \rangle_{V'V}+ \tilde{a}[u_{N}](u_{N},v_{N}^{(1)};\mu)dt
     +\langle P_{H}^{N}u_{o},v_{N}^{(2)} \rangle_{H},\\
 \tilde{F}(v_{N};\mu):&= \int_{I}\langle g(\mu),v_{N}^{(1)}\rangle_{V'V}dt+\langle u_{\delta}^{0},v_{N}^{(2)}\rangle_{H},
\end{align*}
 and $P_{H}^{N}: V_{h}\rightarrow V_{N}$ denotes the $H$-orthogonal projection onto $V_{N}$. For mathematical convenience, we assume that the EIM approximation $\tilde{a}[\cdot](\cdot,\cdot;\mu)$ is sufficiently accurate in the sense that the form $\tilde{a}[\cdot](\cdot,\cdot;\mu)$ is strongly monotone with monotonicity constant $\tilde{m}_{a}(\mu):=m_{a}(\mu)-\epsilon_{a}>0$, where $\epsilon_{a} \in \mathbb{R}_{+}$ is small enough and is related to the EIM approximation error. Then it follows as for \eqref{2.1.5} that the problem $\eqref{3.1.4}$ admits a unique solution $u_{N}(\mu) \in \mathcal{X}_{\triangle t,N}$ for all $\mu \in \mathcal{D}$. However, in the EIM practice it is difficult to check this property a-priori, so that arguing the well-posedness of the upcoming discrete systems \eqref{discrete_root} and \eqref{rb_newton} in general is not possible.

 The problem \eqref{3.1.4} can be interpreted as the reduced-basis approximation of the Crank-Nicolson time-marching scheme with the EIM approximation of the nonlinearity, i.e.
\begin{align}\label{RB_scheme}
     (u_{N}^{k}-u_{N}^{k-1},v_{N}^{(1)})_{H}&+\frac{\vartriangle t^{k}}{2}\{ \tilde{a}[u_{N}^{k}](u_{N}^{k},v_{N}^{(1)};\mu)+\tilde{a}[u_{N}^{k-1}](u_{N}^{k-1},v_{N}^{(1)};\mu)\}\\ \nonumber
    &=\frac{\vartriangle t^{k}}{2} \{\langle g(t^{k};\mu),v_{N}^{(1)}\rangle_{V'V}+\langle g(t^{k-1};\mu),v_{N}^{(1)}\rangle_{V'V} \},
\end{align}
where the initial condition $u_{N}^{0}$ is obtained as an $H$-projection of $u_{\delta}^{0}$ onto $V_{N}$.  The resulting nonlinear algebraic equations are then solved with the RB counterpart of Newton's method by finding the root of
 \begin{align}\label{discrete_root}
\mathbf{G}_{N,M}(\mathbf{u}_{N}^{k};\mu)&=\frac{1}{\vartriangle t^{k}}\mathbf{M}_{N}(\mathbf{u}_{N}^{k}-\mathbf{u}_{N}^{k-1})-\frac{1}{2}[\mathbf{g}_{N}^{k}(\mu)+\mathbf{g}_{N}^{k-1}(\mu)] \\\nonumber
&+\frac{1}{2}[\mathbf{A}_{N,M}(\mu)\mathbf{u}_{N}^{k}+\mathbf{A}_{N,M}(\mu)\mathbf{u}_{N}^{k-1}],
 \end{align}
 where $\mathbf{M}_{N}:=\{\langle \xi_{i},\xi_{j} \rangle_{H}\}_{i,j=1}^{N}, \mathbf{A}_{N,M}(\mu):=\{\tilde{a}[u_{N}^{k}](\xi_{i},\xi_{j};\mu)\}_{i,j=1}^{N} \in \mathbb{R}^{N \times N}$ and $\mathbf{g}_{N}^{k}(\mu):=\{\langle g(t^{k};\mu),\xi_{i}\rangle_{V'V}\}_{i=1}^{N}\in \mathbb{R}^{N}$. The initial condition is given by $\mathbf{u}_{N}^{0}:=\{(u_{\delta}^{0},\xi_{i})_{H}\}_{i=1}^{N} \in \mathbb{R}^{N}$. The strong monotonicity of the quasilinear form $\eqref{3.1.3}$ guarantees that the equation \eqref{discrete_root} admits a unique root $\mathbf{u}_{N}^{k}$ for every parameter $\mu \in \mathcal{D}$.

 The Newton's iteration for finding a root of \eqref{discrete_root} reads: starting with $\mathbf{u}_{N}^{k,(0)}$, for $z=0,1,...$ solve the linear system
\begin{align}\label{rb_newton}
    \mathbf{J}_{N,M}(\mathbf{u}_{N}^{k,(z)};\mu)\delta \mathbf{u}_{N}^{k,(z)}=-\mathbf{G}_{N,M}(\mathbf{u}_{N}^{k,(z)};\mu)
\end{align}
to obtain $\delta \mathbf{u}_{N}^{k,(z)}$, and then update the solution $\mathbf{u}_{N}^{k,(z+1)}:=\mathbf{u}_{N}^{k,(z)}+\delta \mathbf{u}_{N}^{k,(z)}$. The system Jacobian matrix is given by
\begin{align}\label{rb_newton_matrix}
    \mathbf{J}_{N,M}(\mathbf{u}_{N}^{k};\mu)=\frac{1}{\vartriangle t^{k}}\mathbf{M}_{N}+\frac{1}{2}\mathbf{A}'_{N,M}(\mathbf{u}_{N}^{k};\mu).
\end{align}
If the mapping $\mu \mapsto \mathbf{A}'_{N,M}(\cdot;\mu)$ is bounded in $\mu \in \mathcal{D}$, then for $\vartriangle t^{k} \leq C(\mathcal{D})$, where $C(\mathcal{D})>0$ is some constant, the Jacobian matrix \eqref{rb_newton_matrix} is invertible. We will comment on the computation of the reduced parametrised counterpart $\mathbf{A}'_{N,M}(\mu):=\{d\tilde{a}[u_{N}^{k}](\xi_{i},\xi_{j};\mu)\}_{i,j=1}^{N} \in \mathbb{R}^{N \times N}$ of $\mathbf{A}'_{h}(\mathbf{u}_{\delta}^{k})$ in \eqref{rb_newton_matrix}. We have
\begin{align}\label{3.1.5}
    \tilde{a}[u_{N}^{k}](u_{N},\xi_{i};\mu)=\sum_{j=1}^{N}\sum_{m=1}^{M}\varphi_{m}^{k}(\mu)\tilde{a}_{m}(\xi_{j},\xi_{i})u_{N,j}^{k}, \quad 1\leq i \leq N.
\end{align}
With the EIM approximation of the nonlinearity it follows that
\begin{align}\label{3.1.6}
\sum_{s=1}^{M}(\mathbf{B}_{M})_{m,s}\varphi_{m,s}^{k}(\mu)&=\nu(u_{N}^{k}(x_{m}^{M};\mu);\mu), \quad 1\leq m\leq M \\ \nonumber
    &=\nu(\sum_{n=1}^{N}u_{N,n}^{k}\xi_{n}(x_{m}^{M});\mu),\quad 1\leq m\leq M.
\end{align}
Plugging \eqref{3.1.6} into \eqref{3.1.5} results in
\begin{align}\label{3.1.7}
   \tilde{a}[u_{N}^{k}](u_{N},\xi_{i};\mu)=\sum_{j=1}^{N}\sum_{m=1}^{M}\mathbf{D}_{i, m}^{N,M}(\mu)\nu(\sum_{n=1}^{N}u_{N,n}^{k}\xi_{n}(x_{m}^{M});\mu)u_{N,j}^{k}
\end{align}
with $\mathbf{D}^{N,M}(\mu)=\mathbf{A}_{N,M}(\mu) (\mathbf{B}_{M})^{-1} \in \mathbb{R}^{N \times M}$. Taking the derivative of \eqref{3.1.7} with respect to the components $u_{N,j}^{k}(\mu), \ 1\leq j \leq N$ , we derive the formula for  $\mathbf{A}'_{N,M}(\mu)=\mathbf{A}_{N,M}(\mu)+\mathbf{E}_{N,M}(\mu)$, where
\begin{align}\label{3.1.8}
   (\mathbf{E}_{N,M})_{i,j}=\sum_{s=1}^{N}u_{N,s}^{k} \sum_{m=1}^{M}\mathbf{D}_{i, m}^{N,M}(\mu)\partial_{1}\nu(u_{N}^{k}(x_{m}^{M});\mu)
\end{align}
We will give the exact form of $\partial_{1}\nu(u_{N}^{k}(x_{m}^{M});\mu)$ in the upcoming examples.

The proposed reduced numerical scheme contains parameter-separable matrices and thus allows offline-online decomposition. The offline phase (model construction) depends on expensive high-dimensional finite element simulations and thus on $\mathcal{N}$, but should be performed only once. However, the assembling of all the high-dimensional parameter-dependent quantities is computationally simplified due to the affine dependence on the parameters \eqref{3.1.3},\eqref{3.1.3b}. In the online phase (RB model simulation) the computational complexity scales polynomially in $N$ and $M$, independently of $\mathcal{N}$ and thus is inexpensive. The operation count associated with each Newton update of the residual $\mathbf{G}_{N,M}(\mathbf{u}_{N}^{k,(z)})$ in the online phase is $\mathcal{O}(N^{2}Q_{a}+N^{2}+M^{2}+NQ_{f_{o}})$ and the Jacobian $\mathbf{J}_{N,M}(\mathbf{u}_{N,M}^{k,(z)})$ is assembled at cost $\mathcal{O}(MN^{3})$ with the dominant cost of assembling $\mathbf{E}_{N,M}(\mu)$, and then inverted at cost $\mathcal{O}(N^{3})$.

\subsection{\textbf{Reduced basis certification}}
\label{sec:3.3}
An important ingredient of the reduced basis methodology is the verification of the error (certification of the reduced basis method). In the present work we provide an a-posteriori error bound, based on the residual, which allows quick evaluation. We denote by $R(\cdot;\mu) \in \mathcal{Y}'$  the residual of the problem, defined naturally as:
\begin{align}\label{3.2.1}
   R(v;\mu):=F(v;\mu)-\tilde{B}[u_{N}](u_{N},v;\mu)=\int_{I}\langle r(t;\mu),v \rangle_{V'V}dt  \quad \forall v \in \mathcal{Y}_{\delta}.
\end{align}
We have the following
\begin{prop}[A-posteriori Error Bound]
 Let $m_{a}(\mu)>0$ be a monotonicity constant from \eqref{2.1.2} and assume that $u^{0}_{\delta} \in V_{N}$. Then the error $e(\mu)=u_{\delta}(\mu)-u_{N}(\mu)$ of the reduced basis approximation is bounded by
 \begin{align}\label{3.2.2}
\lVert e(\mu)\rVert_{\mathcal{Y}}\leq \frac{1}{m_{a}(\mu)}(\lVert R(\cdot;\mu)\rVert_{\mathcal{Y}'}+\delta_{M}(\mu)\lVert u_{N}(\mu) \rVert_{L^{2}(I;V)})=:\bigtriangleup_{N,M}(\mu),
 \end{align}
where
 \begin{align}\label{3.2.3}
\delta_{M}(\mu)=\sup_{t \in I}\sup_{x \in \Omega}| \nu_{M}(u_{N}(x,t);\mu) - \nu(u_{N}(x,t);\mu)|
\end{align}
denotes the approximation error of the nonlinearity.
\end{prop}

\begin{proof}
Since in the case $e=0$ there is nothing to show, we assume that $e \neq 0$. We have $u_{\delta}^{0} \in V_{N}$ and $\restr{P_{H}^{N}}{V_{N}}=Id$, therefore $u_{N}^{0}:=P_{H}^{N}u_{\delta}^{0}=u_{\delta}^{0}$. It implies that $\lVert e(0)\rVert_{H}=0$, $\lVert e \rVert_{\mathcal{Y}}=\lVert e \rVert_{L^{2}(I;V)}$ and $\lVert R(\cdot;\mu)\rVert_{\mathcal{Y}'}=\lVert R(\cdot;\mu)\rVert_{L^{2}(I;V')}$. First we obtain the following estimate by applying Cauchy-Schwartz inequality:
\begin{align}\label{first_ineq}
\tilde{a}[u_{N}](u_{N},e;\mu)-a[u_{N}](u_{N},e;\mu)=\int_{\Omega}[\nu_{M}(u_{N};\mu)-\nu(u_{N};\mu)]\nabla u_{N}\cdot \nabla e \ dx \\ \nonumber
\leq \sup_{x \in \Omega}|\nu_{M}(u_{N}(x,\cdot);\mu) - \nu(u_{N}(x,\cdot);\mu)| \ \lVert u_{N} \rVert_{V}\lVert e \rVert_{V}.
\end{align}
Integrating \eqref{first_ineq} in $t$ and  applying the Cauchy-Schwartz inequality to the corresponding integral we get:
\begin{align*}
\int_{I} \tilde{a}[u_{N}](u_{N},e;\mu)-a[u_{N}](u_{N},e;\mu)dt \leq \delta_{M}(\mu) \lVert u_{N} \rVert_{L^{2}(I;V)}\lVert e\rVert_{\mathcal{Y}}.
\end{align*}
We then use the identity
\begin{align}\label{identity}
\int_{I} \langle \dot{e},e \rangle_{V'V}dt=  \frac{1}{2}\lVert e(T) \lVert_{H}^{2}-\frac{1}{2}\lVert e(0)\lVert_{H}^{2}
\end{align}
together with the strong monotonicity condition \eqref{2.1.2} and the estimate above to derive the bound:
\begin{align*}
 m_{a}(\mu) \lVert e \rVert_{\mathcal{Y}}^{2}\leq & \int_{I}  a[u_{\delta}](u_{\delta},e;\mu)-a[u_{N}](u_{N},e;\mu) dt+\frac{1}{2}\lVert e(T) \rVert_{H}^{2}\\
= \int_{I} \langle \dot{e},e \rangle_{V'V} dt+& \int_{I} a[u_{\delta}](u_{\delta},e;\mu)-a[u_{N}](u_{N},e;\mu) dt+\frac{1}{2}\lVert e(0) \rVert_{H}^{2}  \\
= \int_{I} \langle \dot{e},e \rangle_{V'V}dt+&\int_{I} a[u_{\delta}](u_{\delta},e;\mu)-\tilde{a}[u_{N}](u_{N},e;\mu)dt+\lVert e(0)\rVert_{H}^{2}\\
+ &\int_{I} \tilde{a}[u_{N}](u_{N},e;\mu)-a[u_{N}](u_{N},e;\mu)dt  \\
&\leq  \lVert R(\cdot;\mu) \rVert_{\mathcal{Y}'} \lVert e \rVert_{\mathcal{Y}}+\delta_{M}(\mu) \lVert u_{N} \rVert_{L^{2}(I;V)} \lVert e \rVert_{\mathcal{Y}},
\end{align*}
where we added and subtracted $\tilde{a}[u_{N}](u_{N},e;\mu)$ to get the definition of the residual \eqref{3.2.1}. Dividing both sides by $ \lVert e \rVert_{\mathcal{Y}} $ yields the result.
\end{proof}

We note that the assumption $u_{\delta}^{0} \in V_{N}$ implies that $\lVert e(0)\rVert_{H}=0$. We can guarantee this by choosing $\xi_{1}:=u_{\delta}^{0} / \lVert u_{\delta}^{0} \rVert_{V}$ as the initial basis for $V_{N}$ in the POD-Greedy procedure.

The dual norm of the residual $\rVert R(\cdot;\mu)\lVert_{\mathcal{Y}'}$ in \eqref{3.2.2} is not available analytically, but by using $\mathcal{Y}_{\delta}$ as the underlying test space, it becomes a computable quantity. The computation of $\rVert R(\cdot;\mu)\lVert_{\mathcal{Y}'}$ requires the knowledge of its Riesz representer $v_{\delta,R}(\mu) \in \mathcal{Y}_{\delta} $. Thanks to the Riesz representation theorem, on the discrete level it can be obtained from the equation
\begin{align}\label{3.2.4}
(v_{\delta,R}(\mu),v_{\delta})_{\mathcal{Y}}=R(v_{\delta};\mu) \quad \forall v_{\delta}\in \mathcal{Y}_{\delta}.
\end{align}
Since the test space $\mathcal{Y}_{\delta}$ consists of piecewise constant polynomials  in time, the problem \eqref{3.2.4} can be solved via the time-marching procedure for $k=1,...,K$ as follows:
\begin{align}\label{3.2.5}
 \int_{I^{k}}\langle v_{\delta,R}(t;\mu),v_{h} \rangle_{V}dt=\int_{I^{k}}\langle r(t;\mu),v \rangle_{V'V}dt \quad \forall v_{h} \in V_{h}.
 \end{align}
We note that $v_{R}^{k}(\mu):=\restr{v_{\delta,R}(\mu)}{I^{k}}$ is constant in time, hence the integration on the left-hand side of \eqref{3.2.5} is exact. For the right-hand side of \eqref{3.2.5} we represent $u_{N}(\mu) \in \mathcal{X}_{\triangle t, N}$ as the linear function  \eqref{2.2.2} on $I^{k}$ and use it as an input for the residual \eqref{3.2.1}. We then apply the trapezoidal quadrature rule for the approximate evaluation of the integral. The quadrature rule is chosen such that the quadrature error is of the size of the error of the truth Crank-Nicolson solution. We thus need to solve the following problems:
\begin{align}\label{3.2.6}
 \langle v_{R}^{k}(\mu),v_{h} \rangle_{V}= R^{k}(v_{h};\mu) \quad \forall v_{h} \in V_{h} \ (k=1,...,K),
\end{align}
where the right-hand side is given by
\begin{align}\label{3.2.7}
    R^{k}(v_{h};\mu)=\frac{1}{2}[\langle g(t^{k};\mu)+g(t^{k-1};\mu),v_{h}\rangle_{V'V}-\tilde{a}[u_{N}^{k}](u_{N}^{k},v_{h};\mu) \\ \nonumber
    -\tilde{a}[u_{N}^{k-1}](u_{N}^{k-1},v_{h};\mu)]  - \frac{1}{\triangle t^{k}}\langle u_{N}^{k}-u_{N}^{k-1},v_{h} \rangle_{H}.
\end{align}
Therefore the computation of the Riesz representer leads to a sequence of $K$ uncoupled spatial problems in $V_{h}$. The parameter separability structure of the residual
\begin{align*}
R^{k}(v_{h};\mu)=\sum_{q=1}^{Q_{R}}\theta_{R,q}^{k}(\mu)R_{q}(v_{h})
\end{align*}
is transferred by the linearity of the Riesz isomorphism to the parameter separability of its Riesz representer $v^{k}_{R}(\mu)$ together with the parameter dependent functions $\theta_{R,q}^{k}: \mathcal{D} \rightarrow \mathbb{R}$. Therefore, for $1\leq q \leq Q_{R}$ we have
\begin{align}\label{3.2.8}
v^{k}_{R}(\mu)=\sum_{q=1}^{Q_{R}}\theta_{R,q}^{k}(\mu)v_{R,q} \ \text{with} \ (v_{R,q},v_{h})_{V}=R_{q}(v_{h}) \quad \forall v_{h} \in V_{h}.
\end{align}
Finally we state the formulas for the residual norm  as well as the spatio-temporal norm of $u_{N}$. Since $\restr{v_{\delta,R}(\mu)}{I^{k}}$ is constant in time, the integration on $I^{k}$ is exact and we can compute the spatio-temporal norm of $v_{\delta,R}(\mu)$ as follows:
\begin{align*}
\lVert v_{\delta,R}(\mu) \rVert_{\mathcal{Y}}^{2}=\sum_{k=1}^{K} \triangle t^{k}\lVert v_{R}^{k}(\mu) \rVert_{V}^{2}=\sum_{k=1}^{K} \triangle t^{k}\Theta^{k}_{R}(\mu)^{T}\mathbf{G}_{R}\Theta^{k}_{R}(\mu),
\end{align*}
where $\mathbf{G}_{R}:=[\langle v_{R,q},v_{R,q'}\rangle]_{q,q'=1}^{Q_{R}}\in \mathbb{R}^{Q_{R}\times Q_{R}}$ and ${\Theta}^{k}_{R}(\mu):=[\theta_{R,q}^{k}(\mu)]_{q=1}^{Q_{R}}\in \mathbb{R}^{Q_{R}}$. The isometry of the Riesz isomorphism implies that $\lVert R(\cdot;\mu) \rVert_{\mathcal{Y}'}=\lVert v_{\delta,R}(\mu) \rVert_{\mathcal{Y}}$. Since $\restr{u_{N}(\mu)}{I^{k}}$ is a linear function in time, the trapezoidal quadrature rule on $I^{k}$ is exact. We then can compute the spatio-temporal norm $\lVert u_{N} \rVert_{\mathcal{Y}}$ of $u_{N} \in  \mathcal{X}_{\triangle t, N}$ according to
\begin{align*}
 \lVert u_{N} \rVert_{\mathcal{Y}}^{2}&= \sum_{k=1}^{K} \frac{\vartriangle t^{k}}{2}(\lVert u_{N}^{k} \rVert_{V}^{2}+\lVert u_{N}^{k-1} \rVert_{V}^{2})+\lVert u_{N}^{0} \rVert_{H}^{2} \\
& =\sum_{k=1}^{K} \frac {\vartriangle t^{k}}{2}[\mathbf{u}_{N}^{k \ T} \mathbf{K}_{N} \mathbf{u}_{N}^{k}+\mathbf{u}_{N}^{k-1 \ T}\mathbf{K}_{N} \mathbf{u}_{N}^{k-1}]+\mathbf{u}_{N}^{0 \ T} \mathbf{M}_{N} \mathbf{u}_{N}^{0},
\end{align*}
where $\mathbf{K}_{N}:=[\langle \xi_{i},\xi_{j} \rangle_{V}]_{i,j=1}^{N} \in \mathbb{R}^{N \times N}$. Since in our case the reduced basis is orthonormal in $V$, $\mathbf{K}_{N}$ is the identity matrix. The operation count in the online phase, associated with computation of the residual norm and the spatio-temporal norm on $\mathcal{Y}$ is correspondingly $\mathcal{O}(Q_{R}^{2}K)$ and $\mathcal{O}(N^{2}(K+1))$.

We note that our a-posteriori error bound takes into account the error of the nonlinearity approximation \eqref{3.2.3}. It is given by
 \begin{align}\label{3.2.9}
\delta_{M}(\mu) = \max_{k \in K}\max_{x \in \Omega}|\nu_{M}(u_{N}^{k}(x);\mu) - \nu(u_{N}^{k}(x);\mu)|.
\end{align}
Since the EIM approximation $\nu_{M}(\cdot;\mu)$ is constructed out of truth solutions, we assume that $N$ is chosen in such a way that $\nu_{M}(u_{N}^{k}(x);\mu)\approx \nu_{M}(u_{\delta}^{k}(x);\mu)$. We note that \eqref{3.2.9} requires the knowledge of $\nu(u_{N}^{k}(\mu);x;\mu)$ and thus one full evaluation of the nonlinearity for all $K$ time steps. Therefore the certification procedure is not completely mesh-independent.

\section{Examples and numerical results}
\label{sec:4}
In this section we consider examples of quasilinear parabolic PDEs with strongly monotone differential operators and apply the proposed reduced-basis techniques to these problems.

\subsection{\textbf{1-D magnetoquasistatic problem: analysis}}
\label{sec:4.1}
For the first numerical example we choose a 1-D magnetoquasistatic approximation of Maxwell's equations (see, e.g. \cite{bachinger2005numerical,salon1995finite}). Let $d=1$, $\Omega=(0,1)$ and $V:=H^{1}_{0}(\Omega) \hookrightarrow L^{2}(\Omega)=:H$. The norm on $V$ is $\lVert u\rVert_{V}^{2}:=\langle u',u' \rangle_{L^{2}}$, which is indeed a norm due to Poincare-Friedrichs inequality. We use the time interval $I=(0,0.2]$ and the parameter set $\mathcal{D}:=[1, 5.5] \subset \mathbb{R}$. For a parameter $\mu \in \mathcal{D}$, we want to find $u:=u(\mu)$ which solves
\begin{equation}\label{4.1.1}
\begin{aligned}[c]
\dot{u}-(\nu(|u'|;\mu)u')'& = g\\
u(t,x)&= 0\\
u_{o}(x)&=0
\end{aligned}
\begin{aligned}[c]
\quad&\text{on} \ I\times\Omega,\\
\quad &\forall \ (t,x) \in I\times \partial{\Omega},\\
\quad &\forall \ x \in \Omega.
\end{aligned}
\end{equation}
We here used $g(x,t):=12\sin(2 \pi x)\sin(2 \pi t)$ and define $\nu(s;\mu)=\exp{(\mu s^2)}+1$ as the reluctivity function.

We consider the quasilinear form for the weak formulation \eqref{2.1.5}, which here is given by
\begin{align}\label{4.1.2}
    a[u](u,v;\mu)=\int_{\Omega}\nu(|u'|;\mu)u'v'dx.
\end{align}
If the function $\nu(\cdot;\mu)\cdot : \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}_{0}^{+}$ is strongly monotone, i.e. if
\begin{align}\label{4.1.3}
 (\nu(s_{2};\mu)s_{2}-\nu(s_{1};\mu)s_{1})(s_{2}-s_{1})\geq m_{a}(\mu)(s_{2}-s_{1}), \quad \forall s_{2}, s_{1} \in \mathbb{R}_{0}^{+}
\end{align}
holds, then \eqref{4.1.2} satisfies the strong monotonicity condition \eqref{2.1.2}. Indeed, we set $s_{1}=w',  s_{2}=v'$ and integrating we get
\begin{align*}
    a[v](v,v-w)-a[w](w,v-w)&=\int_{\Omega} (\nu(v';\mu)v'-\nu(w';\mu)v')(v'-w')dx \\ \nonumber
    &\geq m_{a}(\mu) \int_{\Omega} (v'-w')^{2}dx=m_{a}(\mu) \lVert v-w \rVert_{V}^{2}.
\end{align*}
It is clear that the reluctivity function $\nu(s;\mu)$ in our example satisfies \eqref{4.1.3}. Furthermore, the monotonicity constant can be taken as $m_{a}=\underset{\mu \in \mathcal{D}}{\text{inf}}\ \underset{s \in \mathbb{R}_{+}}{\text{inf}} \ \nu(s;\mu)$, hence we have $m_{a}=2$ for our problem and the constant is parameter-independent. We also note that continuity of  $\nu(\cdot;\mu)$ implies hemicontinuity of \eqref{4.1.2} for every parameter $\mu \in \mathcal{D}$. Thus the weak formulation \eqref{2.1.5} of the PDE \eqref{4.1.1} admits a unique solution.

We specify the bilinear form $\langle A'(u;\mu)v,w\rangle_{V'V}=da[u](v,w;\mu)$ induced by the Fr\'echet derivative $A'(u;\mu) :V \times \mathcal{D} \rightarrow V'$ of the nonlinear operator $A(u;\mu)$. It is then used to compute the Jacobian matrix \eqref{2.2.5} for Newton method. In the present example we have
 \begin{align*}
da[u](v,w;\mu)= \int_{\Omega}\left(2\mu \ \nu'(| u'|;\mu) u' +\nu(| u'|;\mu)\right)v'w' \ dx.
\end{align*}
The derivative for the reduced-basis scheme in the formula \eqref{3.1.8}, thanks to the chain rule, is given by
\begin{align*}
\partial_{1}\nu(|u_{N}'^{k}(x_{m}^{M})|;\mu)=2\mu\nu'(|u_{N}'^{k}(x_{m}^{M})|;\mu) u_{N}'^{k}(x_{m}^{M})\xi_{j}'(x_{m}^{M}),
\end{align*}
where all the indices are according to \eqref{3.1.8}. However it was sufficient to drop the term  $\mathbf{E}_{N,M}(\mu)$ in $\mathbf{A}'_{N,M}(\mu)$ for our numerical experiments. This then corresponds to an inexact Newton-like method, which was applied in the numerical computations.

\subsection{\textbf{1-D magnetoquasistatic problem: numerical results}}
\label{sec:4.2}

The truth approximation is performed by the Petrov-Galerkin scheme, which is introduced in section \ref{sec:2}, where $V_{h}$ is the finite element space, composed of piecewise linear and continuous functions, defined on the partition of $\bar{\Omega}$ into $100$ equal subintervals and $\mathcal{N}_{h}=98$ nodes (excluding Dirichlet boundary nodes). For the time discretization we divide the interval $I$ into $K=200$ subintervals of length $\triangle t=10^{-3}$. We solve the problem with the Crank-Nicolson scheme \eqref{2.2.3}, while applying  Newton's method, described in section \ref{sec:2.2} on each time step for the numerical computation of the time snapshots. We iterate the Newton's method unless the norm of the residual \eqref{2.2.4} is less than the tolerance level, which we set to $10^{-8}$.

We generate the RB-EIM model as follows: we start from $\mathcal{D}_{train}^{EIM} \subset \mathcal{D}$  (a uniform grid of size 200) and compute truth solutions for each parameter in $\mathcal{D}_{train}^{EIM}$ to approximate the nonlinearity $\nu$  with its EIM counterpart $\nu_{M}$. We set $M_{max}=8$ as the maximal dimension of the EIM approximation space. Next we run the POD-Greedy procedure with $M=M_{\text{max}}$ and obtain $N_{\text{max}}=5$ for $\varepsilon_{RB}=10^{-5}$, where $\mathcal{D}_{train}$ is a uniform grid over $\mathcal{D}$ of size 400.  For the POD-Greedy procedure and method certification we use the estimator \eqref{3.2.2}. We solve the problem with the reduced Crank-Nicolson scheme \eqref{RB_scheme}, while applying RB Newton's method, described in section \ref{sec:3.2} on each time step for the numerical computation of the time snapshots. We iterate the Newton's method unless the norm of the residual \eqref{discrete_root} is less than the tolerance level, which we set to $10^{-8}$.

Next we introduce a test sample $\mathcal{D}_{test} \subset \mathcal{D}$ of size 200 (uniformly random sample from $\mathcal{D}$), the maximum of the estimator $\max\bigtriangleup_{N,M}:=\underset{\mu \in \mathcal{D}_{test}}{\max}\bigtriangleup_{N,M}(\mu)$, the ``truth norm" error and its maximum over the test sample
\begin{align*}
\varepsilon^{true}_{N,M}(\mu):=\lVert u_{\delta}(\mu)-u_{N}(\mu) \rVert_{\mathcal{Y}}, \quad \max \varepsilon^{true}_{N,M}:= \underset{\mu \in \mathcal{D}_{test}}{\max} \varepsilon^{true}_{N,M}(\mu).
\end{align*}
Once the reduced-basis model is constructed ($N_{\text{max}}=7,M_{\text{max}}=8)$, we verify the convergence with $N$  of $\max \bigtriangleup_{N,M}$ and $\max \varepsilon^{true}_{N,M}$ on a test sample $\mathcal{D}_{test}$ and plot in Fig.\ref{fig:1} the $N$-$M$ convergence curves for different values of $M$. We can see that the estimator in Fig.\ref{fig:1}(b) reaches the desired tolerance level $\varepsilon_{RB}=10^{-5}$ for $(N_{\text{max}},M_{\text{max}})=(5,8)$.

Next we investigate the influence of the EIM approximation error in the estimation process. We can split the estimator \eqref{3.2.2} into two parts: the reduced-basis and the nonlinearity approximation error estimation contributions
\begin{align*}
\bigtriangleup^{RB}_{N,M}(\mu):=\frac{1}{m_{a}}{\lVert R(\cdot;\mu)\rVert_{\mathcal{Y}'}} \ \text{and} \ \bigtriangleup^{EI}_{N,M}(\mu):=\frac{\delta_{M}(\mu)}{m_{a}}\lVert u_{N}(\mu) \rVert_{\mathcal{Y}}.
\end{align*}
We then set
\begin{align}\label{4.2.1}
\bigtriangleup^{RB}_{N,M}:=\underset{\mu \in \mathcal{D}_{test}}{\max} \bigtriangleup^{RB}_{N,M}(\mu), \quad \bigtriangleup^{EI}_{N,M}:=\underset{\mu \in \mathcal{D}_{test}}{\max} \bigtriangleup^{EI}_{N,M}(\mu).
\end{align}
In Fig.\ref{fig:2}(a) we plot $\bigtriangleup^{RB}_{N,M}$  and $\bigtriangleup^{EI}_{N,M}$  for $1 \leq N \leq 5$ and $M=4$, $M=8$: we can see that $M$ has nearly no influence on $\bigtriangleup^{RB}_{N,M}$, but we observe the ``plateau" in $\bigtriangleup^{EI}_{N,M}$, which limits the convergence of the estimator \eqref{3.2.2} with increasing $N$. The separation points, or ``knees", of the $N$-$M$-convergence curves then reflect a (close-to) balanced contribution of both error terms.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5 \textwidth]{Fig1a.jpg}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{Fig1b.jpg}}
\caption{ (a): Convergence with $N$ of $\max \varepsilon^{true}_{N,M}$ for different values of $M$ on the test set, 1-D example. (b): Convergence with $N$ of $\max\bigtriangleup_{N,M}$ for different values of $M$ on the test set, 1-D example.}%
\label{fig:1}%
\end{figure}

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{Fig2a.jpg}}
\subfloat[]{\includegraphics[width=0.5 \textwidth]{Fig2b.jpg}}
\caption{ (a): The dependence of $\bigtriangleup^{RB}_{N,M}$ and $\bigtriangleup^{EI}_{N,M}$ contributions with $N$ for fixed values of $M$. (b): The reluctivity function $\nu(|u_{N}'(x)|;\mu)$ and its EI-approximation ($M=8$) $\nu_{M}(|u_{N}'(x)|;\mu)$ for the parameter $\mu=5.5$ at $t=0.2$ (b).}%
\label{fig:2}%
\end{figure}

In Table \ref{tab:1} we present, as a function of $N$ and $M$, the values of $\max\bigtriangleup_{N,M}$, $\bigtriangleup^{RB}_{N,M}$, $\bigtriangleup^{EI}_{N,M}$,  $\varepsilon^{true}_{N,M}$ and mean effectivities  $\bar{\eta}_{N,M}:=\frac{1}{|\mathcal{D}_{test}|}\sum_{\mu \in \mathcal{D}_{test}} \eta_{N,M}(\mu)$, where $\eta_{N,M}(\mu):=\bigtriangleup_{N,M}(\mu)/\lVert u_{\delta}(\mu)-u_{N}(\mu) \rVert_{\mathcal{Y}}$. We note that the tabulated $(N,M)$ values correspond roughly to the ``knees" of the $N$-$M$-convergence curves. We can see that the effectivities are lower bounded by 1 and are of moderate size, thus the error estimator is reliable and there is no significant overestimation of the true error.

We then plot (see Fig. \ref{fig:2}(b)) the reluctivity function $\nu(|u_{N}'(x)|;\mu)$ and its EI approximation $\nu_{M}(|u_{N}'(x)|;\mu)$ for the parameter $\mu=5.5$ at $t=0.2$; we can see that there is no visible difference between the original function and its EIM counterpart. Although the problem at hand is merely chosen to illustrate the methodology, we report on the average CPU time for comparison. The finite element method takes $\approx 0.47$ sec to obtain the solution, and the RB method ($N_{\text{max}}, M_{\text{max}}$), which takes $\approx 0.08/0.10$ sec without and with the a-posteriori certification and results in the speed-up factor of 5.87/4.70\footnote{All the computations are performed in MATLAB on Intel Xeon(R) CPU E5-1650 v3, 3.5 GHz x 12 cores, 64 GB RAM}. The offline phase requires less than 10 minutes for our implementation.
\begin{table}
\begin{tabular}{l l l l l l l l}
\hline\noalign{\smallskip}
  $N$ & $M$ &$\max\bigtriangleup_{N,M}$ & $\bigtriangleup^{RB}_{N,M}$ & $\bigtriangleup^{EI}_{N,M}$ & $\max \varepsilon^{true}_{N,M}$ & $ \bar{\eta}_{N,M}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
  2  & 2  & 6.10 E-03 & 5.60 E-03 & 7.60 E-04 & 1.60 E-03 & 4.00 \\
  3  & 4  & 5.62 E-04 & 5.05 E-04 & 1.12 E-04 & 1.32 E-04 & 5.82 \\
  5  & 8  & 6.25 E-06 & 4.47 E-06 & 1.81 E-06 & 1.79 E-06 & 4.58 \\
\noalign{\smallskip}\hline
\end{tabular}
\caption{Performance of the 1D RB-EIM magnetoquasistatic approximation of Maxwell's equations on the test set}
\label{tab:1}
\end{table}

\subsection{\textbf{2-D magnetoquasistatic problem: analysis}}
\label{sec:4.3}

As second example we consider a 2-D magnetoquasistatic problem for modelling of eddy currents in a steel pipe\footnote{http://www.femm.info/wiki/TubeExample}. Let $\bar{\Omega}=\bar{\Omega}_{1}\bigcup \bar{\Omega}_{2}$ be a circular cross-section of the steel pipe with radius $r_{2}$, where $\Omega_{1}$ is the conducting domain (iron) and $\Omega_{2}$ is the non-conducting domain of radius $r_{1}$. The wire is represented by the part with the radius $r_{0}$ and the complementary part is the air gap (see Fig.\ref{fig:3}(a)). We assume that the magnetic reluctivity function and the electric conductivity function have different structure on conducting and non-conducting domains, respectively, i.e.
\begin{equation*}
\begin{aligned}[c]
  \nu(x,s) =
  \begin{cases}
                                   \nu_{1}(s), \  \text{for $x \in \Omega_{1}$}, \\
                                   \nu_{2},  \  \text{for $x \in \Omega_{2}$} \\
  \end{cases}\ \text{and}
\end{aligned}
\quad
\begin{aligned}[c]
  \sigma(x) =
  \begin{cases}
                                   \sigma_{1}>0, \  \text{for $x \in \Omega_{1}$}, \\
                                   \epsilon>0,  \  \text{for $x \in \Omega_{2}$}, \\
  \end{cases}
\end{aligned}
\end{equation*}
where $\nu_{2},\sigma_{1}>0$ denote constants. We assume that the reluctivity function satisfies
\begin{align}\label{4.3.1}
   0 < \nu_{\text{LB}} \leq \nu(x,s)\leq \nu_{UB}, \quad  \forall x\in \Omega, \ s \in \mathbb{R}^{+}_{0},
\end{align}
where $\nu_{\text{LB}}$ and $\nu_{UB}$ are accessible constants. We note that the air-gap and the coils in the steel pipe are electrically non-conductive, i.e. $\sigma(\xi)=0$ for $\xi \in \Omega_{2}$. However, we introduce a regularization parameter $\epsilon=10^{-8}$ as a value of $\sigma$ for the non-conducting domain. This allows us to consider a pure parabolic problem instead of a parabolic-elliptic system with differential-algebraic structure (see, e.g. \cite{Stykel2017}). We set $\mu:=\sigma_{1}$  and define the parameter set $\mathcal{D}=[5\cdot 10^6, 10^7]$ and the time interval $I=(0,0.02]$. We thus have a parametrized quasilinear parabolic equation
\begin{equation}\label{4.3.2}
\begin{aligned}[c]
\sigma(x;\mu)\dot{u}-\nabla \cdot(\nu(x,\lvert \nabla u\rvert)  \nabla  u)&=g \quad  \\
u(t,x)&= 0\\
u_{o}(x)&=0
\end{aligned}
\begin{aligned}[c]
\quad&\text{on} \ I\times\Omega,\\
\quad &\forall \ (t,x) \in I\times \partial{\Omega},\\
\quad &\forall \ x \in \Omega.
\end{aligned}
\end{equation}
The right-hand side is the electric-flux density
\begin{equation*}
  g(x,t) =
  \begin{cases}
                                   \frac{I_{e}(t)}{2\pi r_{0}}, \  \text{for $x \in \Omega_{1}$}, \\
                                   0,  \  \text{for $x \in \Omega_{2}$}, \\
  \end{cases}
\end{equation*}
where $I_{e}(t)=100\cdot \sin(100\pi t)$ is the electric current.

We consider the quasilinear form for the weak formulation \eqref{2.1.5}, which here is given by
\begin{align}\label{4.3.3}
    a[u](u,v;\mu)=\int_{\Omega}\nu(x,|\nabla u|;\mu)\nabla u \cdot \nabla v \ dx.
\end{align}
In practical applications, the nonlinear reluctivity function is often defined through magnetization curves or $|B|$-$|H|$ curves. The underlying physical properties of ferromagnetic materials determine the $|B|$-$|H|$ curve. These curves are naturally strongly monotone and, in practice, their analytical form is unknown. Instead, only a finite number of discrete points $(|H_{k}|,|B_{k}|), k=1,...,K_{c}$ with $|H_{k}|$,$|B_{k}|$  denoting the magnitude of the magnetic field (measured in ampere/meter) and magnetic flux (measured in tesla), is given from the real life measurements. In order to reconstruct a continuous, monotone $|B|$-$|H|$ curve, monotonicity-preserving interpolation with cubic splines is applied \cite{heise1994analysis}. We define a mapping $g_{1}:\mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}$ which determines the magnetization curve via $|B|=g_{1}(|H|)$. An example of a $|B|$-$|H|$ curve, based on the measurements of a ferromagnetic material, which is used in our problem, is given in Fig.\ref{fig:3}(b); the real life measurements were provided by \cite{Schops2020}. The mapping $s \mapsto \nu_{1}(s)s, s\in \mathbb{R}_{+}$ then denotes the inverse $g_{1}^{-1}$ of $g_{1}$ and thus also is strongly monotone. The nonlinear reluctivity function $\nu_{1}:\mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}$ then is given by $\nu_{1}(s):=g_{1}^{-1}(s)/s,s\in \mathbb{R}_{+}$; it is required that $\nu_{1} \in C^{1}(\mathbb{R}^{+}_{0};\mathbb{R}^{+})$ and the spline approximation technique guarantees this property. If $g_{1}^{-1}(s)=\nu_{1}(s)s$ satisfies the strong monotonicity condition \eqref{4.1.3}, then the mapping $\mathbf{s} \mapsto \nu_{1}(|\mathbf{s}|)\mathbf{s}, \mathbf{s} \in \mathbb{R}^{2}$ is strongly monotone with monotonicity constant $\nu_{LB}$ and Lipschitz continuous with Lipschitz constant $\nu_{UB}$. The mapping $\mathbf{s}\mapsto \nu_{2}\mathbf{s},  \mathbf{s} \in \mathbb{R}^{2}$ is linear, therefore $\mathbf{s}\mapsto \nu(x,|\mathbf{s}|)\mathbf{s}$ is strongly monotone for all $x \in \Omega$. The form \eqref{4.3.3} then is strongly monotone with the monotonicity constant $\nu_{LB}$ and Lipschitz continuous with the Lipschitz constant $3\nu_{UB}$ (see \cite{heise1994analysis} for the corresponding proofs). Hence the weak formulation \eqref{2.1.5} of the PDE \eqref{4.3.2} admits a unique solution.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.48\textwidth]{tube_pic.jpg}}
\subfloat[]{\includegraphics[width=0.50 \textwidth]{Fig3b.jpg}}
\caption{ (a): Geometry of the computational domain: the wire (dark grey), the air gap (white), the iron (bright grey). (b): Example of a $|B|$-$|H|$ curve, approximated with cubic splines from measured data points.}%
\label{fig:3}%
\end{figure}

We specify the bilinear form $\langle A'(u;\mu)v,w\rangle_{V'V}=da[u](v,w;\mu)$ induced by the Fr\'echet derivative $A'(u;\mu) :V \times \mathcal{D} \rightarrow V'$ of the nonlinear operator $A(u;\mu)$. It is then used to compute the Jacobian matrix \eqref{2.2.5} for Newton method. With
\begin{equation*}
n[u] =
  \begin{cases}
                                   \frac{\nabla u}{|\nabla u|}, \  \text{for} \  \nabla u \neq 0, \\
                                   0,  \  \text{for} \ \nabla u = 0, \\
  \end{cases}
\end{equation*}
we have
 \begin{align*}
da[u](v,w;\mu)=\int_{\Omega}\nu'(x,|\nabla u|;\mu)(n[u] \cdot  \nabla w)(\nabla u \cdot \nabla v)+\nu(x,|\nabla u|;\mu)\nabla v \cdot \nabla w \ dx,
\end{align*}
and the derivative for the reduced-basis scheme in the formula \eqref{3.1.8}, thanks to the chain rule, is given by
\begin{align*}
\partial_{1}\nu(x;|\nabla u_{N}^{k}(x_{m}^{M})|;\mu)=\nu'(x;|\nabla u_{N}^{k}(x_{m}^{M})|;\mu) n[u_{N}^{k}](x_{m}^{M})\cdot\nabla\xi_{j}(x_{m}^{M}),
\end{align*}
where all the indices are according to \eqref{3.1.8}.

In this example, the monotonicity constant $m_{a}(\mu)$ is not available analytically. As it was mentioned earlier in the discussion on $|B|$-$|H|$ curves, we can choose $\nu_{LB}>0$ as our monotonicity constant.  However, since for each parameter $\mu \in \mathcal{D}$ there holds
\begin{align}\label{4.3.4}
m_{a}(\mu):=\underset{k \in K}{\min} \ \underset{x \in \Omega}{\min} \ \nu_{1}(|\nabla u_{N}^{k}(x)|;\mu)\geq \nu_{LB},
\end{align}
and the computation of \eqref{4.3.4} only requires one full evaluation of the nonlinearity, which already has been performed to evaluate \eqref{3.2.9}, we here use $m_{a}(\mu)$ as our constant for the estimation.

\subsection{\textbf{2-D magnetoquasistatic problem: numerical results}}
\label{sec:4.4}
The truth approximation is performed by the Petrov-Galerkin scheme, which is introduced in section \ref{sec:2}, where $V_{h}$ is the finite element space, composed of piecewise linear and continuous functions, defined on a triangle mesh containing $4374$ triangles and $\mathcal{N}_{h}=2107$ nodes (excluding Dirichlet boundary nodes). For the time discretization we divide the interval $I$ into $K=200$ subintervals of length $\triangle t=10^{-4}$. The nonlinear reluctivity function $\nu_{1}$ is reconstructed from the real $B-H$ measurements using monotonicity-preserving cubic spline interpolation and  $\nu_{2}$ value is chosen as the reluctivity of air. We then solve the problem with the Crank-Nicolson scheme \eqref{2.2.3}, while applying Newton's method, described in section \ref{sec:2.2}, on each time step for the numerical computation of the time snapshots. We iterate the Newton's method unless the norm of the residual \eqref{2.2.4} is less than the tolerance level, which we set to $10^{-8}$.

We generate the RB-EIM model as follows: we start from $\mathcal{D}_{train}^{EIM} \subset \mathcal{D}$  (a uniform grid of size 200) and compute truth solutions for each parameter in $\mathcal{D}_{train}^{EIM}$ to approximate the nonlinearity $\nu_{1}$  with the EIM counterpart $\nu_{1}^{M}$. We set $M_{\text{max}}=44$ as the maximal dimension of the EIM approximation space. Next we run the POD-Greedy procedure with $M=M_{\text{max}}$ and obtain $N_{\text{max}}=14$ for $\varepsilon_{RB}=10^{-4}$, where $\mathcal{D}_{train}$ is a uniform grid over $\mathcal{D}$ of size 400.  For the POD-Greedy procedure and the method certification we use the estimator \eqref{3.2.2}. The monotonicity constant is evaluated as in \eqref{4.3.4}. We solve the problem with the reduced Crank-Nicolson scheme \eqref{RB_scheme}, while applying RB Newton's method, described in section \ref{sec:3.2}, on each time step for the numerical computation of the time snapshots. We iterate the Newton's method unless the norm of the residual \eqref{discrete_root} is less than the tolerance level, which we set to $10^{-8}$.

Then we verify the convergence with $N$ of $\max \varepsilon^{true}_{N,M}$ (Fig. \ref{fig:4}(a)) and $\max \bigtriangleup_{N,M}$ (Fig. \ref{fig:4}(b)) on a test sample $\mathcal{D}_{test}$ (a uniformly random sample of size 200) for different values of $M$. We can see  that the estimator in Fig.\ref{fig:3}(b) reaches the desired tolerance level $\varepsilon_{RB}=10^{-4}$ for $(N_{\text{max}},M_{\text{max}})=(14,44)$. We note that the convergence is not monotone at some points due to the EIM interpolation of the non-polynomial nonlinearity behind the problem. We can also see from Fig.\ref{fig:4}(a) that increasing $M$ above 20 has nearly no impact on the convergence of the truth norm error, but the estimator in Fig.\ref{fig:4}(b) still shows a considerable decrease with increasing $M$. Indeed, in Fig.\ref{fig:5}(a) we plot $\bigtriangleup^{RB}_{N,M}$  and $\bigtriangleup^{EI}_{N,M}$ as defined in \eqref{4.2.1} for $1 \leq N \leq 14$ and $M=20$, $M=44$: we can see that $M$ has nearly no influence on $\bigtriangleup^{RB}_{N,M}$, but we can observe the ``plateau" in $\bigtriangleup^{EI}_{N,M}$, which limits the convergence of the estimator \eqref{3.2.2} with increasing $N$. We also plot the values of $\bigtriangleup_{N,M}(\mu)$ and $\varepsilon^{true}_{N,M}(\mu)$ and the truth error for $(N_{\text{max}},M_{\text{max}})$ for every parameter $\mu \in \mathcal{D}_{test}$ in Fig.\ref{fig:5}(b).

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.49\textwidth]{Fig4a.jpg}}
\subfloat[]{\includegraphics[width=0.51 \textwidth]{Fig4b.jpg}}
\caption{(a): Convergence with $N$ of $\max \varepsilon^{true}_{N,M}$ for different values of $M$ on the test set, 2-D example. (b): Convergence with $N$ of $\max\bigtriangleup_{N,M}$ for different values of $M$ on the test set, 2-D example.}%
\label{fig:4}%
\end{figure}

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{Fig5a.jpg}}
\subfloat[]{\includegraphics[width=0.5 \textwidth]{Fig5b.jpg}}
\caption{(a): The dependence of $\bigtriangleup^{RB}_{N,M}$ and $\bigtriangleup^{EI}_{N,M}$ contributions with $N$ for fixed values of $M$. (b): Values of $\varepsilon^{true}_{N,M}$ and $\max \vartriangle_{N,M}$ for $(N_{\text{max}},M_{\text{max}})=(14,44)$ on the test set.}
\label{fig:5}%
\end{figure}

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.42\textwidth]{Fig6a.jpg}}
\qquad
\subfloat[]{\includegraphics[width=0.42\textwidth]{Fig6b.jpg}} \\
\subfloat[]{\includegraphics[width=0.42\textwidth]{Fig6c.jpg}}
\qquad
\subfloat[]{\includegraphics[width=0.42\textwidth]{Fig6d.jpg}}\\
\caption{The truth magnetic flux density $|\nabla u_{\delta}|$ for $\mu=10^7$  at (a) $t=0.01$, (b) $t=0.02$. The reduced-basis magnetic flux density $|\nabla u_{N}|$ for $\mu=10^7$ at (c) $t=0.01$, (d) $t=0.02$.}
\label{fig:6}
\end{figure}

In Table \ref{tab:2} we present, as a function of $N$ and $M$, the values of $\max\bigtriangleup_{N,M}$, $\bigtriangleup^{RB}_{N,M}$, $\bigtriangleup^{EI}_{N,M}$,  $\max \varepsilon^{true}_{N,M}$ and the mean effectivities $\bar{\eta}_{N,M}$. We note that the tabulated $(N,M)$ values correspond roughly to the ``knees" of the $N$-$M$-convergence curves (see example 1 for the terminology and definitions). We can see that the effectivities are lower bounded by 1, but the values are relatively large. We conject that this is related to the structure of the nonlinearity and the effectivities are proportional to $C\cdot \nu_{UB}/ \nu_{LB}$, where $C$ is some constant.

\begin{table}
\begin{tabular}{l l l l l l l l l}
\hline\noalign{\smallskip}
  $N$ & $M$ &$\max\bigtriangleup_{N,M}$ & $\bigtriangleup^{RB}_{N,M}$ & $\bigtriangleup^{EI}_{N,M}$ & $\max\varepsilon^{true}_{N,M}$ & $ \bar{\eta}_{N,M}$  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
  6  & 16 & 1.60 E-03 & 6.68 E-04 & 1.40 E-03 & 2.15 E-05 &  98.06 \\
  9  & 20 & 4.34 E-04 & 1.94 E-04 & 3.67 E-04 & 5.40 E-06 &  89.84  \\
  11 & 36 & 1.76 E-04 & 1.38 E-04 & 1.04 E-04 & 4.68 E-06 &  64.28  \\
  14 & 44 & 6.97 E-05 & 4.63 E-05 & 5.81 E-05 & 1.93 E-06 &  48.27  \\
\noalign{\smallskip}\hline
\end{tabular}
\caption{Performance of 2-D RB-EIM model on the test set}
\label{tab:2}
\end{table}

In Fig.\ref{fig:6} we show the truth finite element magnetic flux density $|\nabla u_{\delta}(x,t,\mu)|$ and the corresponding reduced magnetic flux density $|\nabla u_{N}(x,t,\mu)|$ for $\mu=10^7$ and $t=0.01$ and $t=0.02$. We observe that flux densities look very similar. Next we compare the average CPU time required for both the finite element method, which takes $\approx 70$ sec to obtain the solution, and the RB method with $(N_{\text{max}}, M_{\text{max}})=(14,44)$, which takes $\approx 1.80/2.42$ without and with the a-posteriori certification and results in the speed-up factors (rounded) of 39 and 29, respectively. The offline phase requires the knowledge of the truth finite-element solutions for the EIM approximation step. Since 200 truth solutions were generated in the consecutive order, it takes $\approx$ 4 hours. The generation of these truth solutions could be performed in parallel which would reduce the offline time. The POD-Greedy sampling takes $\approx$ 40 minutes for our implementation.

\section{Conclusion}
\label{sec:5}
In this paper we propose the space-time reduced-basis method for quasilinear parabolic PDEs. We think that our space-time formulation combined with the chosen Petrov-Galerkin discretization provides an elegant approach to treat these kind of problems. We present a new a-posteriori error bound and use it for the reduced basis construction with the POD-Greedy procedure.  The developed methodology is applied to the magnetoquasistatic approximation of Maxwell's equations and numerical results confirm a good speed-up factor, which supports the validity of this approach. The reduced-basis methods developed in the paper will further be extended to treat more complicated industrial problems. It will further have a significant impact on the PASIROM project\footnote{http://www.pasirom.de/}, where the surrogate reduced-basis models are planned to be used in the optimization of electrical machines.

\section*{Acknowledgements}
Both authors acknowledge the support of the collaborative research project PASIROM funded
by the German Federal Ministry of Education and Research (BMBF) under grant no. 05M2018.

redpajama_set_name: RedPajamaArXiv


text:
10.01.2006 Feature Article
Mojo Odyssey: From Kufuor's. Nkrumah's & Rawlings' Closet
By Ebby Koney
Mojo Odyssey: FROM KUFUOR'S, NKRUMAH'S, & RAWLINGS' CLOSET.
Our suspicious nature is so well embedded in our body politic that we tend to ignore genuine compatriots who draw attention to mistakes of our rulers. We ascribe ill-motive as their rationale. Strangely, it is usually a different story, when foreigners speak. We tend to listen to them more than we condemn them.
With that in focus, it is perhaps proper at the beginning of this piece to refer to the World Bank Country Director, Mr. Mat Karlsson's advice on Human gold (Ghana's potent Human Resource material) and Agriculture as one of the avenues for alleviation of poverty, as a 'gift' one man can give a whole nation. Ayekoo, Mr. Karlsson!
Kufuorites have shown a CULTURE OF INTOLERANCE to criticism of their hero's erring ways. Sorry to disappoint them this time: Kufuorites would have to vent their spleen on a non-Ghanaian who dissects their kingpin's foibles. It may be a surprise, if they actually do not pour invectives on folks who bring Karlsson's message to them! Mr. Kufuor has till 2008 to undo the serious damage he has done to his own reputation.
Thayer Watkins of Silicon Valley USA looked into Kwame Nkrumah's closet and aired out some 'dirty laundry'. Edgar Cooke, an economist peeked at Jerry John Rawlings umbrella and got wet on a dry day, and Mats Karlsson, World Bank Country Director, like a Zoo Keeper, raised the elephant's tail and got splashed with mess, or something unpleasant.
Dear Reader, our task here is simple. We intend to observe what these people saw and follow up with our own random thoughts. We will peek into 'hidden boxes' of our prominent leaders opened! by Thayer Watkins, Edgar Cooke and Mats Karlsson to see whether we agree with their conclusions. We keep mulling why Ghana has fallen behind other nations who won independence just around the same time as Ghana's? Has it not been easy to blame military coups as cause for our economic instability? Has it not been easy to blame the Colonialist, the white man and Dear God, for our seeming perpetual state of poverty? Has it not been difficult for us to be honest about disastrous decisions fostered on us as a nation by people we entrusted our destinies with? Analyzing the errant ways of our ! leaders that caused significant mistakes and seeing what could have been done differently, would help future leaders make better choices.
3 past and present leaders who have been Presidents of Ghana for 5 or more years feature in our discussion on some of their failings. Some random thoughts: Measure their failure by each drop of fresh water from the Volta and other rivers that flow wastefully through un-irrigated arable lands into the Atlantic Ocean. Measure their success by how they have dealt with or attempted to deal with, resources and opportunities to turn the Accra Plains, Afram Plains into the bread basket of Africa…..before PRAISE or CONDEMNATION!
ON JOHN AGYEKUM KUFUOR: NPP 2001-PRES! ENT (Term ends 2008)
"TO WHOM MUCH IS GIVEN, MUCH IS EXPECTED" says the Good Book. Between 2001 and present, the Government of Ghana, under John Kufuor has received so much money from International Sources with not much to show for. In real dollar terms, Kufuor has received from "development partner's inflows from more than $600 million in 2002 and projected to hit $1.2 billion by the end of next year", according to the World Bank Country Director, Mr. Mats Karlsson. The World Bank says John Kufuor has no reason to complain about economic difficulties because Ghana has all the resources for accelerated development at its disposal. "I think we ar! e at a stage where there are no excuses for Ghana. You've got your debt relief, you've got HIPC and you've got a double of development partner inflows over the past three years."
The hydra-headed nation-killer, CORRUPTION and INEFFICIENCY, have been identified as the causes of Kufuor's economic distress. "It is part of the corruption issue and I don't want to mince words on this: it is still a real issue," Mr. Karlsson laments. This leaves Mr. Kufuor with mud on his face in rejecting charges that there is corruption in his administration. He has under his command the awesome machinery of State Investigatory Power! s, the Police CID, SFO, BNI, CHRAG, yet he wants ordinary folks to investigate myriad cases on corruption and present him with dossiers.
The question of Kufuor's inefficiency relates "to whether the projects for which development partner's monies were being released were actually being seen on the ground". "If every school is supposed to have a set of textbooks, go to the schools and check whether they have the textbooks. What could be the answer?" intones Mat Karlsson. Karlsson says the same questions could go for the other sectors such as roads, health and agriculture, adding that even when the investments ! had been well implemented, "Are they being maintained and managed?"
He said development partners could not go beyond making financial resources available since the responsibility of strategizing, implementing and monitoring were functions of the country receiving the funds; else the bank and other development partners would be seen as stepping beyond bounds.
Dear Kukrudu Kufuorites, Can you defend KUFUOR? Tell us what he WILL do as a lame duck president and remove doubts of the World Bank Country Director. Kufuor has till 2008 to make history and Kufuorites should not insult Mr. Mat Karlsson, for, "methinks, he doth speaketh truth"!
ON KWAME NKRUMAH: CPP 1957 TO 1966
Thayer Watkins in an essay faulted Kwame Nkrumah for the following economic fumbles:
"A sugar plant at Asutsuare was built without a water system and remained idle for a year before this flaw was corrected. A tomato and mango canning plant was built at Wenchi with a capacity to process 5,000 tons of tomatoes and 7,000 tons of mangoes each year. After it was built at a cost 80 percent above budget and ready to begin operations the authorities discovered there were hardly any mango trees in the area of the p! lant and it would take seven years for newly planted mango trees to start bearing.
There was to be a cattle hide-leather-shoe complex. The slaughter house was sited in the Upper Region at Bolgatanga, a not unreasonable decision since the north is the cattle-raising area of Ghana. However the markets for cattle in pre-Nkrumah times were not in the north, probably for good reason.
A tannery for turning the hides into leather was sited in the south at Aveyime. The plant in the north could not supply enough hides so the tannery had to import hid! es. The leather to be produced at Aveyime was to go to a footwear factory in Kumasi. After the excessive transporting of the raw materials and final product around the country the Ghanaian consumers were not willing to buy because they turned up to be shoddy products."
Dear Nkrumahists, can you Defend NKRUMAH? Was Nkrumah sabotaged by the West who disliked his romance with the Eastern Bloc? The little I can say for the great African Pioneer is this: His main goal was for Africa to Unite. His mantra was 'Seek ye first the political kingdom and all other things shall be added". Our Ghanaian tradition is to be capable of taking responsibility for a household before you set one up. Did we need to be economi! cally viable with our 'economic kingdom' before setting up our 'political Kingdom'? The egg and the chicken, which comes first? In setting up all these factories, Nkrumah might have wanted to get people to work, set Ghana up one step ahead of other newly emerging Independent States and bring in Raw Materials from other African States to the extent Ghana could not produce them herself. Mangoes abound in Sierra Leone and The Gambia, Cows spread across Mali, Burkina Faso etc. When Valco was set up, Kaiser transported Aluminum Ingots from Tema to Jamaica where they had their plant. Why couldn't Nkrumah transport raw materials from other countries into Wenchi to foster African Trade?
One troubling note on the tannery: Did Nkrumah factor in the Ghanaian palatal love for "Wele" (burnt cow skin) in Okro stew? "Wele" Cow Skin, not 'gracing' the cooking pot containing pea-nut butter soup is certainly an object of traditional artifacts, which at times cause war between tribes! So which other cow-skin was Nkrumah counting on to feed a shoe factory? Probably, from outside the shores of Ghana! Nkrumah ought to have ens! ured he had trade agreements with other nations to feed those factories with raw materials, if Ghana herself could not.
ON JERRY JOHN RAWLINGS: NDC 1993-2001 Edgar Cooke, an Economist dissects JJ Rawlings' financial sector administration thus:
"The Financial Sector had its fair share of scandals and bank closures. The Bank for Housing and Construction was liquidated. Ghana Cooperative Bank and Meridian BIAO (1) were likewise liquidated.National Investment Bank had cases of embezzlement to the tune of millions of Cedis. One! such case reported in the press had the Accountant and a Deputy Managing Director dismissed.
A scandal in January 1997 at Ghana Commercial Bank and involving the Managing Director of A-Life supermarket chain, who owns 4% of GCB put to test and raised suspicions about the Bank of Ghana's regulatory capacity. The scandal involved the use of a generous overdraft facility and Cheque cashing privileges to cash over 75 billion Cedis of GCB and two other banks.
Interest rates were high. These high interest rates were attributed to excessive government borrowing from the public. These rates fell from their 1990 high in 1991 but started climbing again in 1994 peaking at an all time high in 1997. In 2000, the prime rate stood at 45%, 91 day Treasury bill rates was 42.72% and that of the 182 day Trea! sury Bills and 1 year notes were 37.70% and 42.80% respectively.…… manufacturing and domestic trade, used up 47% of the total loans and advances (excluding cocoa marketing). Since 2001, the Manufacturing Sector's share has decreased from 31% to 25% while that of domestic trade has increased from 11% to 23%. The falling share of the manufacturing sector is of great concern and which is amplified in the performance of the industry sector in GDP. The increased lending to domestic trade and its unproductiveness affected the industrial sector. The BOG failed to mop up excess liquidity in the system. Currency outside the banks increased from its 1990 figure of 20% to 41% in 2001."
Dear Akatamansonians, can you defend RAWLINGS? The little I can say for Rawlings is that, perhaps in his "dotage" he gave free pass to Yaw Osafo Marfo, current NPP Minister with alleged Presidential aspirations, who supervised the collapse of Bank for Housing and Construction as its Managing Director prior to wading into politics. It is unfathomable to think that in the early days of the Revolution, an A Life Executive would dream of misusing Banking privileges when Rawlings' own mentor Rear Admiral Amedume had been executed for borrowing 50, 000 Cedis from a Bank! Where had the Eagle flown to, when greedy businessmen and corrupt public officers like the Mamponghene's wife splashed egg on Rawlings' face? Rawlings' political life started with a BANG, but ended with a whimper.
Rawlings should not be faulted for trying to adapt to the universal truth: For a country to develop it has to industrialize. Rawlings can point to the f! act that he was trying to resuscitate the manufacturing sector which was the largest user of the loans and advances followed closely by domestic trade. Rawlings can surely attest to the fact that there was an increase in manufacturing during his period 1990 to 2001.
Mojo Odyssey pondering on what might have been for Ghana! Views expressed by the author(s) do not necessarily reflect those of GhanaHomePage.
Ebby Koney, © 2006
The author has 22 publications published on Modern Ghana. Column Page: EbbyKoney
Disclaimer: "The views/contents expressed in this article are the sole responsibility of the author(s) and do not neccessarily reflect those of Modern Ghana. Modern Ghana will not be responsible or liable for any inaccurate or incorrect statements contained in this article."
Reproduction is authorised provided the author's permission is granted.
More from Author (22)
Historical Odyssey #5: Agric. Policies Short Changing Ghana
Historical Agricultural Odyssey 4
Historical Odyssey 3: In Search Of Innovative Agricultural Policies
Historical Odyssey: Agricultural Policies Chapter Two
Historical Odyssey of our Agricultural Policies
HOT 🔥
Can President Kufuor Become Another Mojo Man Of Ghana?
Mojo Odyssey: Rawlings' Charismatic Authority ...
Rawlings Mojo, Obed Asamoah & 2 Other Unc Wisemen
Historical Odyssey 3: In Search Of Innovative Agricultural
Mojo Odyssey: The Nature Of Media Barons In Ghana Politics
Mojo Odyssey: Costly Presidential Mistakes

redpajama_set_name: RedPajamaCommonCrawl


Opens a command prompt at a location determined by the currently active window.
For Explorer windows this will be the location in view (if the location is accessible by the command prompt, e.g. not Control Panel).
For the desktop this will be the user's Desktop folder (if running NT).
In all other cases this will be the window's working directory.


The business leadership organization Identity Clark County (ICC) has appointed Lance Barrett of Barrett & Co. PLLC to its Board of Directors, and named Ron Frederiksen of RSV Building Solutions as Director Emeritus.
Barrett, a CPA, is founder and managing partner of Barrett & Co. a Camas-based public accounting firm he created in 1986 that specializes in construction, manufacturing, transportation and real estate industries. Barrett brings more than 30 years of public and corporate accounting experience across the Portland-Vancouver USA metro area. Active in the business and accounting industry, Barrett was honored by his peers in 2011 as North American advisor of the year. He has degrees in business and accounting from Southern Oregon State College.
Frederiksen is Chairman of RSV Building Solutions, a highly-regarded area DesignBuild construction firm that supports small- to mid-sized businesses and nonprofits in defining and building suitable workspaces. He has been an active investor and leader in Identity Clark County since the organization was created in 1993.
Frederiksen joins Elie Kassab of Prestige Development as the organization's first Directors Emeritus, a special honor recognizing long-time business and community leadership through Identity Clark County.


We will try our hardest to include a variety of events in a variety of locations for our members, but if you have any suggestions or ideas for us, we would love to hear them! Please send us an email at info@seesawplaydates.com. Also, we hope you are making connections with other members in the app. It's our goal to have you create playdates with one another through the app at your favorite local spots. If we can help facilitate these playdates in any way, please let us know!
Caregivers and children ages birth - 18 months are welcome.
Caregivers and their children ages birth - 35 months are welcome for the free interactive story time.
Caregivers and their children ages birth - 35 months are welcome. This half hour will get you moving and grooving to the music! Bang, rattle, swish and twirl to songs, rhymes and stories!


text:
Home » Media Center » Newsletter » 2011 Q3 Fall » Investor Concern for Fracking Risk Grows
Fracking releases known toxins into air and water across the U.S.
In the past year, The New York Times, the Wall Street Journal, Reuters, and the documentary "Gasland" have focused a national spotlight on the human and environmental health dangers of the controversial drilling technique, commonly known as "fracking." Investors are paying attention.
In the 2011 proxy season, investors clearly voiced their concern about the controversial drilling practice to companies like ExxonMobil, Chevron, and Ultra Petroleum. As You Sow's resolutions asking those companies for transparency around fracking policies and safety saw an increase in support from shareholders. At Ultra Petroleum, the support doubled to 42% and a first-time resolution at oil and gas giant Chevron received an astounding 41% of investors' votes. As a typical environmental resolution would only expect about 7%, these vote results demonstrate the real concern of individual and mainstream institutional investors.
"These are really extraordinary votes," said Senior Strategist Michael Passoff. "It took the shareholder campaign on climate change more than a decade to achieve this level of support."
As You Sow and a coalition of NGO's are currently conducting dialogues with several companies engaged in fracking to discuss key performance indicators to ensure the health and safety of the communities where the drilling is occurring. Notably, one company disclosed that a reduction in the amount of toxic chemicals in their fracking fluids improved their bottom line.
In the 2012 proxy season, we will be re-filing with these oil and gas giants to prevent fracking from damaging the health of communities across the United States.


Climate change is one of the major challenges Benin is currently facing to ensure food security. Benin's economy is primarily based on agriculture that is mainly rain-fed; this program is helping vulnerable communities to fight effectively against the negative effects of climate change.
Supporting vulnerable communities to cope with climate change.


text:
24 Hour Global Contact +44 207 754 3555
About MS Risk
Why Hire MS Risk?
MS Risk Blog
COVID 19 – Highlighting Weak Land Borders in Vietnam and Thailand
At the time of writing both Thailand and Vietnam have kept COVID 19 under control relative to other countries in … Read the rest
Iran's New Shahid Roudaki Warship: Does This Mean Conflict Is on the Horizon?
On 19 November 2020, the Islamic Republic of Iran and its paramilitary Islamic Revolutionary Guards Corps (IRGC) added an auxiliary … Read the rest
Moldavia Towards EU and Russia's Reaction
The newly elected Moldavian President Maia Sandu's policies, internally and externally, are expected to be challenged by Russia's interest in … Read the rest
Visit the MS Risk Blog
24 Hour Global Contact
http://msrisk.com/wp-content/uploads/2020/12/Ghana-Security-Update-9-Dec-2020.pdf"> Security Update – Presidential and Parliament Elections in Ghana (9 December 2020)
Ghana, a beacon of a well-functioning democracy in a region that has recently experienced political instability, held its presidential and parliamentary elections on Monday 7 December 2020 in a vote that is expected to be a tight race between two veteran leaders. Incumbent Nana Akufo-Addo is seeking re-election for a second term in office and faces a challenge from former President John Mahama. Ghanaians also voted in parliamentary elections on Monday for 275 members of parliament.
http://msrisk.com/wp-content/uploads/2020/12/Ghana-Security-Update-9-Dec-2020.pdf">Download PDF Report
http://msrisk.com/wp-content/uploads/2020/11/Security-Advisory-Ethiopia-18-November-2020.pdf"> Security Update – Ethiopia (18 November 2020)
Fighting continues in the Tigray region, with Ethiopian troops making advances on the Shire area and on the regional capital Mekele.
http://msrisk.com/wp-content/uploads/2020/11/Security-Advisory-Ethiopia-18-November-2020.pdf">Download PDF Report
Ethiopian PM announces final offensive as calls for dialogue are ignored.
Over two weeks of violence in the northern Tigray region are now further threatening the stability of the wider Horn of Africa region, with neighbouring Eritrea targeted in recent days as Sudan deals with a growing refugee crisis.
http://msrisk.com/wp-content/uploads/2020/11/Security-Advisory-Ethiopia-13-November-2020.pdf"> Security Advisory – Ethiopia (13 November 2020)
Violence continues in the northern Tigray region of Ethiopia, with reports emerging that hundreds have been killed on both sides of the conflict.
Virtual Security Director
Ragnall House, South Suite
18 Peel Road, Douglas
IM1 4LZ, Isle of Man
info@msrisk.com
© 2021 MS Risk and the MS Risk wordmark are registered trade-marks of MS Risk Limited, registered in The Isle of Man, company number 007435V.


At the end of January 2019 in Atyrau subsidiary of "Kazaeronavigatsia" RSE held a seminar-meeting "Prophylaxy of influenza diseases". Expert of Atyrau Department of public health in transport held a mini-lecture on the main methods of prevention of influenza and SARS and the measles epidemic that broke out in the country. According to the forecast of experts, the traditional increase in sickness should be expected in the end of January – beginning of February. The only scientifically based method of fighting influenza and measles is prevention –vaccination. At the end of the lecture were distributed brochures on "Prophylaxy of influenza and SARS".


text:
title: Kelvin Hotel

listing-type: accommodation
island: south
region: southland
destination: Southland

order: a
image-path: kelvin-hotel.jpg
image-alt: Kelvin Hotel
refer-url: http://www.kelvinhotel.co.nz/
---
Centrally located Invercargill accommodation offering tastefully decorated hotel rooms, superior suites and two bedroom apartments.

redpajama_set_name: RedPajamaGithub


text:
Canadian Perspectives, Autumn 2020
We're pleased to share with you two feature articles from our new issue of Canadian Perspectives magazine. Canadian Perspectives is a perk of membership with the Council and this issue features articles on some of the most pressing issues facing our communities, country and planet. You'll also read about inspiring ways we are coming together to build community power and a better future for all.
Big Oil bailouts are undermining climate action: Campaigner Dylan Penner recalls earlier this year when the federal government was floating the idea of providing massive bailouts for Big Oil.
Ontario Government an example of why we need a National Strategy on Long-term Care: Canadian Perspectives' Editor, Jan Malek, writes about the Ontario government's failure to protect seniors in long-term care from the COVID-19 pandemic.
The most remarkable thing about all of this amazing work is that the Council does it without any donations from governments, corporations or political parties. It's the generous contributions and activist energy of people like you across the country who keep this unique and beautiful people-powered movement of ours going strong. Thank you!
The Council believes political literacy is crucial to regaining control of our communities and our country. We encourage you to copy and share these resources with others.
Like what you read? Need more perspective? Join our growing citizens' movement and get a free subscription to Canadian Perspectives.
HERE ARE JUST 5 REASONS WHY YOU SHOULD BECOME A MEMBER OF THE COUNCIL OF CANADIANS »
For more than 35 years, the Council has fought for progressive change based on fair and sustainable trade, clean water, climate justice, democracy and stronger public health care. We connect more than 150,000 Members and supporters together from coast-to-coast-to-coast with a powerful collective voice with which we hold governments accountable and drive transformative change.
We work in solidarity and partnership
We work in solidarity with Indigenous Peoples, racialized and non-status people and those who face social, economic and environmental injustices. We also partner with and support local and national sister organizations doing vital work on shared issues of concern.
We work for justice and democracy
Most governments support an economic system that puts unlimited growth above the vital needs of people and the planet. We believe in the need for transformative change based on the principles of justice and democracy in the absence of true leadership by governments.
We are independent, non-partisan and people-powered
Our work is sustained by individual contributions from our Members and donors. We do not accept donations from corporations, governments or political parties. This financial independence allows our advocacy work to be 100% independent, direct and uncompromising.
We are democratic
We are a grassroots organization built on democratic principles. These principles provide an open membership and give all Members a voice and vote in our organization. We are accountable to the groups we work with and to the frontline activists and communities we support.
As a Member, you will continue to drive the transformative change we need in Canada and beyond – but you are also granted benefits, including:
Voting rights at the Council of Canadians Annual Membership Meetings.
Access to training and skills-building programs.
Opportunities to connect with Council of Canadians chapters in your area.
Access to exclusive content, such as the Council's Canadian Perspectives magazine.
Updates on all the Council of Canadians' actions, campaigns and financial reports.
Joining is fast and easy. Simply indicate "Yes, I want to be a Member" on our online membership enrolment form and make a donation of $1 or more to cover your annual membership fee.


FlexPro 2017 is the most extensive update in the 25-year history of this analysis software from Weisang.
The new FlexPro 2017 with a fully redesigned Microsoft Office-like interface excels not only in its ease of use, but also in its expert ability to analyze series measurements and data archives.
The new Microsoft Office-like ribbon makes FlexPro 2017 very easy to use. It highlights important commands and dynamically inserts optional tools. That improves the software's layout and ease of use.
Choose objects and style elements from neatly arranged galleries.
The cursor tools are displayed as soon as you activate the cursors.
You can now adjust the design of your diagrams by changing the diagram style and color scheme.
The color scheme determines the color of the curves.
Just choose a table style sheet from the gallery.
Use conditional formatting to improve the readability of data tables. Use rules to highlight extreme values or duplicates. Visualize the relative quantity of data using data bars or color scales. One look at a table with this formatting is enough to give you a general overview of your data.
Use data bars or manage the text and background color depending on the data.
The FlexPro Data Explorer option indexes measurement data archives on the server or your hard disk. FlexPro 2017 calculates characteristic quantities during indexing. You can then use these quantities for subsequent queries.
Use the new FlexPro 2017 index server to grant your entire team access to your company's measurement data archive (Microsoft SQL Server license required).
Long-term measurements are often stored as a sequence of files, each of which contains a small time interval of the measured data. When querying such data, FlexPro 2017 automatically concatenates them and provides the complete time range as a single data set.
With file indexing in client/server mode, an instance of FlexPro carries out the indexing and stores the results in a Microsoft SQL Server database.
When analyzing test series, the number of data sets to evaluate and results to present vary. FlexPro 2017 makes it easy for you to present this type of measured data.
The dynamic diagrams and tables automatically adjust to the number of data sets to display.
The number of curves and table cells varies with the number of analyzed units under test.
Improved automatic data format recognition.
Easy to use interface with found elements highlighted in color.
Support for different character encodings.
Import data arranged in rows.
Support for time zones and languages when decoding date values.
Filtering of headers from the data stream.
Share import schemas with colleagues.
FlexPro automatically detects names, units and data in the text file.
New intuitive interface with the look and feel of MS Office.
Diagram and table format templates.
File indexing in client/server mode (optional).
A new analysis object to calculate instantaneous amplitude, frequency and phase.
Improved and additional FPScript functions.
Multichannel index operation in FPScript.
Try FlexPro out on a real-world project: discover all the features of FlexPro for free over 30 days.


Study the map and describe one cause of the French and Indian War?
From the Diagram, list two causes of the French and Indian War?
Remember, rivalries in Europe always spill over into the Colonies.
Mission: To rid the Ohio Valley of the French Invaders.
Braddock's Dilemma: Tried to fight a European style war in the wilderness of Pennsylvania.
Braddock's force is routed and retreats in disarray. During the battle on July 9th Braddock is mortally wounded. Braddock dies and is buried in the middle of the road he built and the remainder of his army marches over him to hide the grave from the French and their allies.
Failed to secure Ft. Duquesne.
Washington to the rescue again.
Prussia v France and its Allies in Europe.
Britain fought France in the Caribbean, India, and North America.


the Drugs Controller General of India (dcgi) is preparing guidelines for linking the drug regulatory authority and the Indian patent office. The system called 'patent-registration linkage' will deny marketing approval to generic drug manufacturers for drugs patented in India. Public health organizations warn that the system will reduce access to cheap, life-saving drugs and goes against the Indian Patent Act, 1970.
At present, a generic manufacturer gets marketing approval from dcgi even before the patent for the drug is expired. The Bolar provision in the Indian Patent Act allows this. If there is a violation of patent rights, the patent holder can appeal in the court. But under patent linkage, even if the patent is wrongly granted--which is often the case--the generic manufacturer cannot get marketing approval. It will have to wait for the patent to expire. There will be a time lag between the expiry of the patent and market approval of the generic drug and this will delay access to cheap drugs for patients.
The linkage system also implies that dcgi will have to cross check if any patent has been issued in India for a drug before considering the marketing request. "dcgi's office has neither the capacity nor the mandate to judge violations of patent law. dcgi's mandate as a government official is to ensure that the drugs available in the market are safe and of acceptable quality. Why is the dcgi becoming the patent police?" asked Leena Mengahaney of Mdecins sans Frontires (msf). International public health organizations, including msf, fear that the patent linkage system will affect supply of cheap drugs to countries in Africa because India is the source of generics for other developing nations.
Patent linkages are not required under the Trade and Related Aspects of the Intellectual Property Rights.