text:
\subsection{PH-preserving MOR techniques}%
Traditional methods which directly retain the pH form in the reduction process are based on Galerkin projections, see, e.g., \cite{Antoulas2020, MehU22} for surveys. The original state $x(t)$ is approximated by ${x(t)\approx V x_r}$,
where the columns of ${V\in\R^{n\times r}}$ form a basis for a suitably chosen subspace of dimension $r$. For instance, in the pH-ODE case with $\pE= I_n$, the ROM coefficient matrices are computed by
\begin{equation*}
	\begin{array}{lll}
		\rJ=U^\T\pJ U, & \rG = U^\T\pG, & \rQ = V^\T\pQ V, \\ \rR=U^\T\pR U, & \rP = U^\T\pP, & \rN=\pN, \quad \rS=\pS,
	\end{array}
\end{equation*}
where ${U = QV(V^\T QV)^{-1}}$, which clearly enforces structure preservation. In \cite{Gugercin2012} an adaptation (IRKA-PH) of the well-known iterative rational Krylov algorithm (IRKA) was proposed that iteratively updates $V$ to fulfill a subset of $\cH_2$ optimality conditions via tangential interpolation of the original transfer function. While IRKA does not ensure stability (or even passivity) of the ROM a priori, it leads to (locally) $\cH_2$ optimal models upon convergence. IRKA-PH, on the other hand, preserves the pH structure and thus produces passive ROMs which, however, generally only fulfill a subset of the $\cH_2$ optimality conditions. Consequently, no $\cH_2$ optimality is achieved in general. The matrix $V$ can also be chosen in order to approximate the Dirac structure of the original model, resulting in the effort- and flow-constraint method \cite{Polyuga2012}. The extension of IRKA-PH and Dirac structure approximation to pH-DAEs is addressed in \cite{HauMM19,BeaGM21}.

\subsection{Passivity-preserving MOR techniques}%
Positive real balanced truncation (PRBT) for ODEs is a well-studied MOR method, see  \cite{GugA04} and the references therein for a survey. The method is based on computing a minimal solution $X_{\min}$ of \eqref{eq:KYP_ineq} (in the sense of the Loewner ordering in symmetric matrices) and a minimal solution $Y_{\min}$ of the \emph{dual KYP inequality}
\begin{equation*}
    \mat{-AY - YA^\T & B-YC^\T \\ B^\T-C Y & D+D^\T} \geq 0.
\end{equation*}
The solutions $X_{\min}$ and $Y_{\min}$ are then used to transform the system to a positive-real balanced realization where the transformed minimal solution of the KYP inequalities $\hat{X}_{\min}$ and $\hat{Y}_{\min}$ are equal and diagonal, i.\,e.,  $\hat{X}_{\min} = \hat{Y}_{\min} = \text{diag}(\eta_1,\ldots,\eta_n)$. Then the states corresponding to the small positive real characteristic values $\eta_i$ can be truncated. Finally, PRBT admits an a priori error bound in the gap metric that is derived in \cite{GuiO14}.

Another passivity-preserving MOR technique is  achieved via spectral factorization \cite{BreU21}. It initially requires a factorization
\begin{equation}\label{eq:specfac}
	\mathcal{W}(X)=\mat{L&M}^\T\mat{L&M}
\end{equation}
with $L\in\R^{k\times n}$, $M\in\R^{k\times m}$ and where $\text{rank}\mat{L&M}$ is as small as possible. If the pair $(A,B)$ is \emph{stabilizable} and $X = X_{\min}$ is the minimal solution of $\mathcal{W}(X) \ge 0$, then the resulting spectral factor
system $(A,B,L,M)$
may be reduced via traditional (unstructured) MOR techniques, such as IRKA or balanced truncation (BT), and the passive ROM is obtained from the reduced spectral factor.

For all passivity-preserving reduction methods, the passive ROM may eventually be transformed back to a pH representation by applying Theorem \ref{thm:KYP}.

\section{Introduction}

\file{elsarticle.cls} is a thoroughly re-written document class
for formatting \LaTeX{} submissions to Elsevier journals.
The class uses the environments and commands defined in \LaTeX{} kernel
without any change in the signature so that clashes with other
contributed \LaTeX{} packages such as \file{hyperref.sty},
\file{preview-latex.sty}, etc., will be minimal.
\file{elsarticle.cls} is primarily built upon the default
\file{article.cls}.  This class depends on the following packages
for its proper functioning:

\begin{enumerate}
\item \file{natbib.sty} for citation processing;
\item \file{geometry.sty} for margin settings;
\item \file{fleqn.clo} for left aligned equations;
\item \file{graphicx.sty} for graphics inclusion;
\item \file{txfonts.sty} optional font package, if the document is to
  be formatted with Times and compatible math fonts;
\item \file{hyperref.sty} optional packages if hyperlinking is
  required in the document;
\item \file{endfloat.sty} optional packages if floats to be placed at
 end of the PDF.
\end{enumerate}

All the above packages (except some optional packages) are part of any
standard \LaTeX{} installation. Therefore, the users need not be
bothered about downloading any extra packages.  Furthermore, users are
free to make use of \textsc{ams} math packages such as
\file{amsmath.sty}, \file{amsthm.sty}, \file{amssymb.sty},
\file{amsfonts.sty}, etc., if they want to.  All these packages work in
tandem with \file{elsarticle.cls} without any problems.

\section{Major Differences}

Following are the major differences between \file{elsarticle.cls}
and its predecessor package, \file{elsart.cls}:

\begin{enumerate}[\textbullet]
\item \file{elsarticle.cls} is built upon \file{article.cls}
while \file{elsart.cls} is not. \file{elsart.cls} redefines
many of the commands in the \LaTeX{} classes/kernel, which can
possibly cause surprising clashes with other contributed
\LaTeX{} packages;

\item provides preprint document formatting by default, and
optionally formats the document as per the final
style of models $1+$, $3+$ and $5+$ of Elsevier journals;

\item some easier ways for formatting \verb+list+ and
\verb+theorem+ environments are provided while people can still
use \file{amsthm.sty} package;

\item \file{natbib.sty} is the main citation processing package
  which can comprehensively handle all kinds of citations and
works perfectly with \file{hyperref.sty} in combination with
\file{hypernat.sty};

\item long title pages are processed correctly in preprint and
  final formats.

\end{enumerate}

\section{Installation}

The package is available at author resources page at Elsevier
(\url{http://www.elsevier.com/locate/latex}).
It can also be found in any of the nodes of the Comprehensive
\TeX{} Archive Network (\textsc{ctan}), one of the primary nodes
being
\url{http://tug.ctan.org/tex-archive/macros/latex/contrib/elsarticle/}.
Please download the \file{elsarticle.dtx} which is a composite
class with documentation and \file{elsarticle.ins} which is the
\LaTeX{} installer file. When we compile the
\file{elsarticle.ins} with \LaTeX{} it provides the class file,
\file{elsarticle.cls} by
stripping off all the documentation from the \verb+*.dtx+ file.
The class may be moved or copied to a place, usually,
\verb+$TEXMF/tex/latex/elsevier/+,
or a folder which will be read
by \LaTeX{} during document compilation.  The \TeX{} file
database needs updation after moving/copying class file.  Usually,
we use commands like \verb+mktexlsr+ or \verb+texhash+ depending
upon the distribution and operating system.

\section{Usage}\label{sec:usage}
The class should be loaded with the command:

\begin{vquote}
 \documentclass[<options>]{elsarticle}
\end{vquote}

\noindent where the \verb+options+ can be the following:

\begin{description}

\item [{\tt\color{verbcolor} preprint}]  default option which format the
  document for submission to Elsevier journals.

\item [{\tt\color{verbcolor} review}]  similar to the \verb+preprint+
option, but increases the baselineskip to facilitate easier review
process.

\item [{\tt\color{verbcolor} 1p}]  formats the article to the look and
feel of the final format of model 1+ journals. This is always single
column style.

\item [{\tt\color{verbcolor} 3p}] formats the article to the look and
feel of the final format of model 3+ journals. If the journal is a two
column model, use \verb+twocolumn+ option in combination.

\item [{\tt\color{verbcolor} 5p}] formats for model 5+ journals. This
is always of two column style.

\item [{\tt\color{verbcolor} authoryear}] author-year citation style of
\file{natbib.sty}. If you want to add extra options of
\file{natbib.sty}, you may use the options as comma delimited strings
as arguments to \verb+\biboptions+ command. An example would be:
\end{description}

\begin{vquote}
 \biboptions{longnamesfirst,angle,semicolon}
\end{vquote}

\begin{description}
\item [{\tt\color{verbcolor} number}] numbered citation style. Extra options
  can be loaded with\linebreak \verb+\biboptions+ command.

\item [{\tt\color{verbcolor} sort\&compress}] sorts and compresses the
numbered citations. For example, citation [1,2,3] will become [1--3].

\item [{\tt\color{verbcolor} longtitle}] if front matter is unusually long, use
  this option to split the title page across pages with the correct
placement of title and author footnotes in the first page.

\item [{\tt\color{verbcolor} times}] loads \file{txfonts.sty}, if
available in the system to use Times and compatible math fonts.

\item [{\tt\color{verbcolor} reversenotenum}] Use alphabets as
author--affiliation linking labels and use numbers for author
footnotes. By default, numbers will be used as author--affiliation
linking labels and alphabets for author footnotes.

\item [{\tt\color{verbcolor} lefttitle}] To move title and
author/affiliation block to flushleft. \verb+centertitle+ is the
default option which produces center alignment.

\item [{\tt\color{verbcolor} endfloat}] To place all floats at the end
of the document.

\item [{\tt\color{verbcolor} nonatbib}] To unload natbib.sty.

\item [{\tt\color{verbcolor} doubleblind}] To hide author name,
affiliation, email address etc. for double blind refereeing purpose.

\item[] All options of \file{article.cls} can be used with this
  document class.

\item[] The default options loaded are \verb+a4paper+, \verb+10pt+,
  \verb+oneside+, \verb+onecolumn+ and \verb+preprint+.

\end{description}

\section{Frontmatter}

There are two types of frontmatter coding:
\begin{enumerate}[(1)]
\item each author is
connected to an affiliation with a footnote marker; hence all
authors are grouped together and affiliations follow;
\pagebreak
\item authors of same affiliations are grouped together and the
relevant affiliation follows this group.
\end{enumerate}

An example of coding the first type is provided below.

\begin{vquote}
 \title{This is a specimen title\tnoteref{t1,t2}}
 \tnotetext[t1]{This document is the results of the research
    project funded by the National Science Foundation.}
 \tnotetext[t2]{The second title footnote which is a longer
    text matter to fill through the whole text width and
    overflow into another line in the footnotes area of the
    first page.}
\end{vquote}

\begin{vquote}
\author[1]{Jos Migchielsen\corref{cor1}%
  \fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}

\author[2]{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}

\author[3]{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}

 \cortext[cor1]{Corresponding author}
 \fntext[fn1]{This is the first author footnote.}
 \fntext[fn2]{Another author footnote, this is a very long
   footnote and it should be a really long footnote. But this
   footnote is not yet sufficiently long enough to make two
   lines of footnote text.}
 \fntext[fn3]{Yet another author footnote.}

 \affiliation[1]{organization={Elsevier B.V.},
                 addressline={Radarweg 29},
                 postcode={1043 NX},
                 city={Amsterdam},
                 country={The Netherlands}}

 \affiliation[2]{organization={Sayahna Foundation},
                 addressline={JWRA 34, Jagathy},
                 city={Trivandrum}
                 postcode={695014},
                 country={India}}
\end{vquote}

\begin{vquote}

 \affiliation[3]{organization={STM Document Engineering
                               Pvt Ltd.},
                 addressline={Mepukada, Malayinkil},
                 city={Trivandrum}
                 postcode={695571},
                 country={India}}
\end{vquote}

The output of the above \TeX{} source is given in Clips~\ref{clip1} and
\ref{clip2}. The header portion or title area is given in
Clip~\ref{clip1} and the footer area is given in Clip~\ref{clip2}.

\deforange{blue!70}
\src{Header of the title page.}
\includeclip{1}{130 612 477 707}{1psingleauthorgroup.pdf
\deforange{orange}

\deforange{blue!70}
\src{Footer of the title page.}
\includeclip{1}{93 135 499 255}{1pseperateaug.pdf
\deforange{orange}

Most of the commands such as \verb+\title+, \verb+\author+,
\verb+\affiliation+ are self explanatory.  Various components are
linked to each other by a label--reference mechanism; for
instance, title footnote is linked to the title with a footnote
mark generated by referring to the \verb+\label+ string of
the \verb=\tnotetext=.  We have used similar commands
such as \verb=\tnoteref= (to link title note to title);
\verb=\corref= (to link corresponding author text to
corresponding author); \verb=\fnref= (to link footnote text to
the relevant author names).  \TeX{} needs two compilations to
resolve the footnote marks in the preamble part.
Given below are the syntax of various note marks and note texts.

\begin{vquote}
  \tnoteref{<label(s)>}
  \corref{<label(s)>}
  \fnref{<label(s)>}
  \tnotetext[<label>]{<title note text>}
  \cortext[<label>]{<corresponding author note text>}
  \fntext[<label>]{<author footnote text>}
\end{vquote}

\noindent where \verb=<label(s)>= can be either one or more comma
delimited label strings. The optional arguments to the
\verb=\author= command holds the ref label(s) of the address(es)
to which the author is affiliated while each \verb=\affiliation=
command can have an optional argument of a label. In the same
manner, \verb=\tnotetext=, \verb=\fntext=, \verb=\cortext= will
have optional arguments as their respective labels and note text
as their mandatory argument.

The following example code provides the markup of the second type
of author-affiliation.

\begin{vquote}
\author{Jos Migchielsen\corref{cor1}%
  \fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}
 \affiliation[1]{organization={Elsevier B.V.},
                 addressline={Radarweg 29},
                 postcode={1043 NX},
                 city={Amsterdam},
                 country={The Netherlands}}

\author{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}
 \affiliation[2]{organization={Sayahna Foundation},
                 addressline={JWRA 34, Jagathy},
                 city={Trivandrum}
                 postcode={695014},
                 country={India}}

\author{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}
 \affiliation[3]{organization={STM Document Engineering
                               Pvt Ltd.},
                 addressline={Mepukada, Malayinkil},
                 city={Trivandrum}
                 postcode={695571},
                 country={India}}
\end{vquote}

\vspace*{-.5pc}

\begin{vquote}
\cortext[cor1]{Corresponding author}
\fntext[fn1]{This is the first author footnote.}
\fntext[fn2]{Another author footnote, this is a very long
  footnote and it should be a really long footnote. But this
  footnote is not yet sufficiently long enough to make two lines
  of footnote text.}
\end{vquote}

The output of the above \TeX{} source is given in Clip~\ref{clip3}.

\deforange{blue!70}
\src{Header of the title page..}
\includeclip{1}{119 563 468 709}{1pseperateaug.pdf
\deforange{orange}

Clip~\ref{clip4} shows the output after giving \verb+doubleblind+ class option.

\deforange{blue!70}
\src{Double blind article}
\includeclip{1}{124 567 477 670}{elstest-1pdoubleblind.pdf
\deforange{orange}

\vspace*{-.5pc}
The frontmatter part has further environments such as abstracts and
keywords.  These can be marked up in the following manner:

\begin{vquote}
 \begin{abstract}
  In this work we demonstrate the formation of a new type of
  polariton on the interface between a ....
 \end{abstract}
\end{vquote}

\vspace*{-.5pc}
\begin{vquote}
 \begin{keyword}
  quadruple exiton \sep polariton \sep WGM
 \end{keyword}
\end{vquote}

\noindent Each keyword shall be separated by a \verb+\sep+ command.
\textsc{msc} classifications shall be provided in
the keyword environment with the commands
\verb+\MSC+. \verb+\MSC+ accepts an optional
argument to accommodate future revisions.
eg., \verb=\MSC[2008]=. The default is 2000.\looseness=-1

\subsection{New page}
Sometimes you may need to give a page-break and start a new page after
title, author or abstract. Following commands can be used for this
purpose.

\begin{vquote}
  \newpageafter{title}
  \newpageafter{author}
  \newpageafter{abstract}
\end{vquote}

\begin{itemize}
\leftskip-2pc
\item [] {\tt\color{verbcolor} \verb+\newpageafter{title}+} typeset the title alone on one page.

\item [] {\tt\color{verbcolor} \verb+\newpageafter{author}+}  typeset the title
and author details on one page.

\item [] {\tt\color{verbcolor} \verb+\newpageafter{abstract}+}
typeset the title,
author details and abstract \& keywords one one page.

\end{itemize}

\section{Floats}
{Figures} may be included using the command, \verb+\includegraphics+ in
combination with or without its several options to further control
graphic. \verb+\includegraphics+ is provided by \file{graphic[s,x].sty}
which is part of any standard \LaTeX{} distribution.
\file{graphicx.sty} is loaded by default. \LaTeX{} accepts figures in
the postscript format while pdf\LaTeX{} accepts \file{*.pdf},
\file{*.mps} (metapost), \file{*.jpg} and \file{*.png} formats.
pdf\LaTeX{} does not accept graphic files in the postscript format.

The \verb+table+ environment is handy for marking up tabular
material. If users want to use \file{multirow.sty},
\file{array.sty}, etc., to fine control/enhance the tables, they
are welcome to load any package of their choice and
\file{elsarticle.cls} will work in combination with all loaded
packages.

\section[Theorem and ...]{Theorem and theorem like environments}

\file{elsarticle.cls} provides a few shortcuts to format theorems and
theorem-like environments with ease. In all commands the options that
are used with the \verb+\newtheorem+ command will work exactly in the same
manner. \file{elsarticle.cls} provides three commands to format theorem or
theorem-like environments:

\begin{vquote}
 \newtheorem{thm}{Theorem}
 \newtheorem{lem}[thm]{Lemma}
 \newdefinition{rmk}{Remark}
 \newproof{pf}{Proof}
 \newproof{pot}{Proof of Theorem \ref{thm2}}
\end{vquote}

The \verb+\newtheorem+ command formats a
theorem in \LaTeX's default style with italicized font, bold font
for theorem heading and theorem number at the right hand side of the
theorem heading.  It also optionally accepts an argument which
will be printed as an extra heading in parentheses.

\begin{vquote}
  \begin{thm}
   For system (8), consensus can be achieved with
   $\|T_{\omega z}$
   ...
     \begin{eqnarray}\label{10}
     ....
     \end{eqnarray}
  \end{thm}
\end{vquote}

Clip~\ref{clip5} will show you how some text enclosed between the
above code\goodbreak \noindent looks like:

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newtheorem}}
\includeclip{2}{1 1 453 120}{jfigs.pdf}
\deforange{orange}

The \verb+\newdefinition+ command is the same in
all respects as its\linebreak \verb+\newtheorem+ counterpart except that
the font shape is roman instead of italic.  Both
\verb+\newdefinition+ and \verb+\newtheorem+ commands
automatically define counters for the environments defined.

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newdefinition}}
\includeclip{1}{1 1 453 105}{jfigs.pdf}
\deforange{orange}

The \verb+\newproof+ command defines proof environments with
upright font shape.  No counters are defined.

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newproof}}
\includeclip{3}{1 1 453 65}{jfigs.pdf}
\deforange{orange}

Users can also make use of \verb+amsthm.sty+ which will override
all the default definitions described above.

\section[Enumerated ...]{Enumerated and Itemized Lists}
\file{elsarticle.cls} provides an extended list processing macros
which makes the usage a bit more user friendly than the default
\LaTeX{} list macros.   With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list counter
type and its attributes.

\begin{vquote}
 \begin{enumerate}[1.]
 \item The enumerate environment starts with an optional
   argument `1.', so that the item counter will be suffixed
   by a period.
 \item You can use `a)' for alphabetical counter and '(i)' for
   roman counter.
  \begin{enumerate}[a)]
    \item Another level of list with alphabetical counter.
    \item One more item before we start another.
\end{vquote}

\deforange{blue!70}
\src{List -- Enumerate}
\includeclip{4}{1 1 453 185}{jfigs.pdf}
\deforange{orange}

Further, the enhanced list environment allows one to prefix a
string like `step' to all the item numbers.

\begin{vquote}
 \begin{enumerate}[Step 1.]
  \item This is the first step of the example list.
  \item Obviously this is the second step.
  \item The final step to wind up this example.
 \end{enumerate}
\end{vquote}

\deforange{blue!70}
\src{List -- enhanced}
\includeclip{5}{1 1 313 83}{jfigs.pdf}
\deforange{orange}

\section{Cross-references}
In electronic publications, articles may be internally
hyperlinked. Hyperlinks are generated from proper
cross-references in the article.  For example, the words
\textcolor{black!80}{Fig.~1} will never be more than simple text,
whereas the proper cross-reference \verb+\ref{tiger}+ may be
turned into a hyperlink to the figure itself:
\textcolor{blue}{Fig.~1}.  In the same way,
the words \textcolor{blue}{Ref.~[1]} will fail to turn into a
hyperlink; the proper cross-reference is \verb+\cite{Knuth96}+.
Cross-referencing is possible in \LaTeX{} for sections,
subsections, formulae, figures, tables, and literature
references.

\section[Mathematical ...]{Mathematical symbols and formulae}

Many physical/mathematical sciences authors require more
mathematical symbols than the few that are provided in standard
\LaTeX. A useful package for additional symbols is the
\file{amssymb} package, developed by the American Mathematical
Society. This package includes such oft-used symbols as
$\lesssim$ (\verb+\lesssim+), $\gtrsim$ (\verb+\gtrsim+)  or
$\hbar$ (\verb+\hbar+). Note that your \TeX{}
system should have the \file{msam} and \file{msbm} fonts installed. If
you need only a few symbols, such as $\Box$ (\verb+\Box+), you might try the
package \file{latexsym}.

Another point which would require authors' attention is the
breaking up of long equations.  When you use
\file{elsarticle.cls} for formatting your submissions in the
\verb+preprint+ mode, the document is formatted in single column
style with a text width of 384pt or 5.3in.  When this document is
formatted for final print and if the journal happens to be a double column
journal, the text width will be reduced to 224pt at for 3+
double column and 5+ journals respectively. All the nifty
fine-tuning in equation breaking done by the author goes to waste in
such cases.  Therefore, authors are requested to check this
problem by typesetting their submissions in final format as well
just to see if their equations are broken at appropriate places,
by changing appropriate options in the document class loading
command, which is explained in section~\ref{sec:usage},
\nameref{sec:usage}. This allows authors to fix any equation breaking
problem before submission for publication.
\file{elsarticle.cls} supports formatting the author submission
in different types of final format.  This is further discussed in
section \ref{sec:final}, \nameref{sec:final}.

\enlargethispage*{\baselineskip}

\subsection*{Displayed equations and double column journals}

Many Elsevier journals print their text in two columns. Since
the preprint layout uses a larger line width than such columns,
the formulae are too wide for the line width in print. Here is an
example of an equation  (see equation 6) which is perfect in a
single column preprint format:

In normal course, articles are prepared and submitted in single column
format even if the final printed article will come in a double column
format journal. Here the problem is that when the article is typeset by
the typesetters for paginating and fit within the single column width,
they have to break the lengthy equations and align them properly. Even
if most of the tasks in preparing your proof is automated, the equation
breaking and aligning requires manual judgement, hence this task is manual.
When there comes a manual operation that area is error prone. Author
needs to check that equation pretty well.

However if authors themselves break the equation to the single column
width typesetters need not want to touch these area and the proof authors
get will be without any errors.

\setlength\Sep{6pt}
\src{See equation (6)}
\deforange{blue!70}
\includeclip{4}{105 500 500 700}{1psingleauthorgroup.pdf}
\deforange{orange}

\noindent When this document is typeset for publication in a
model 3+ journal with double columns, the equation will overlap
the second column text matter if the equation is not broken at
the appropriate location.

\vspace*{6pt}
\deforange{blue!70}
\src{See equation (6) overprints into second column}
\includeclip{3}{59 421 532 635}{elstest-3pd.pdf}
\deforange{orange}
\vspace*{6pt}

\noindent The typesetter will try to break the equation which
need not necessarily be to the liking of the author or as it
happens, typesetter's break point may be semantically incorrect.
Therefore, authors may check their submissions for the incidence
of such long equations and break the equations at the correct
places so that the final typeset copy will be as they wish.

\section{Bibliography}

Three bibliographic style files (\verb+*.bst+) are provided ---
\file{elsarticle-num.bst}, \file{elsarticle-num-names.bst} and
\file{elsarticle-harv.bst} --- the first one can be used for the
numbered scheme, second one for numbered with new options of
\file{natbib.sty}. The third one is for the author year
scheme.

In \LaTeX{} literature, references are listed in the
\verb+thebibliography+ environment.  Each reference is a
\verb+\bibitem+ and each \verb+\bibitem+ is identified by a label,
by which it can be cited in the text:

\verb+\bibitem[Elson et al.(1996)]{ESG96}+ is cited as
\verb+\citet{ESG96}+.

\noindent In connection with cross-referencing and
possible future hyperlinking it is not a good idea to collect
more that one literature item in one \verb+\bibitem+.  The
so-called Harvard or author-year style of referencing is enabled
by the \LaTeX{} package \file{natbib}. With this package the
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\verb+\citep[e.g.][Ch. 2]{Gea97}+ produces (e.g. Governato et
al., 1997, Ch. 2).
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In the numbered scheme of citation, \verb+\cite{<label>}+ is used,
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In addition to the above standard \verb+.bst+ files, there are 10
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\section[Graphical ...]{Graphical abstract and highlights}
A template for adding graphical abstract and highlights are available
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\pagebreak
Please refer below to see how to code them.

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\section{Final print}\label{sec:final}

The authors can format their submission to the page size and margins
of their preferred journal.  \file{elsarticle} provides four
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\lmrgn=3em
\begin{description}
\item [\texttt{1p}:] $1+$ journals with a text area of
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7.78in, single column style only.

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style.

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Following pages have the clippings of different parts of
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\begin{comment}
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{\bf [Specimen single column article -- Click here]}
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{\bf [Specimen double column article -- Click here]}
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~\hfill $\Box$

\end{document}

\section{}
\label{}

\section{}
\label{}

\section{}
\label{}

\section{Introduction}%
\label{sec:introduction}
\input{Introduction}

\section{Preliminaries}%
\label{sec:preliminaries}
\input{Prelims}

\section{A new optimization-based approach}%
\label{sec:our_method}

\input{pauls_approach}
\input{tims_approach}

\section{Numerical examples}%
\label{sec:numerical_examples}

\input{Experimental_setup.tex}

\input{Results.tex}

\section{Conclusion}%
\label{sec:conclusion}

We have presented two optimization-based methods for structure-preserving MOR of pH-DAEs. These make it possible to compute accurate ROMs with respect to either the $\hinf$ or the $\htwo$ norm. The main benefits compared to state-of-the-art methods are the simplified treatment of the algebraic equations, which can be incorporated into the parameterized ROM in a structure-preserving way without increasing its state dimension. Furthermore, our methods are data-driven, such that no transformations to the FOM system matrices are required. Nonetheless, we have shown, how transformations can be applied in order to obtain an accurate estimate of the feedthrough, which is essential in the $\htwo$ case. Finally, our numerical experiments show that the optimization-based methods often lead to a higher accuracy (especially in the $\hinf$ norm).

We are currently investigating the application of our method to higher index pH-DAEs. These may have improper transfer functions, which are not currently supported in our parameterization.

\bibliographystyle{elsarticle-num}

\subsection{$\hinf$ approximation}%
\label{sub:ourmethod_hinf}

The algorithm presented in \cite{Schwerdtner2020} to obtain a good $\hinf$ approximation of a pH-FOM with transfer function $\pHtf$ using a parametrized low-order system with transfer function $\pHtfr( \cdot, \theta)$ is based on minimizing the objective function
\begin{align}
  \begin{split}
  \pushleft{\loss(\theta;\pHtf,\gamma,\mathcal{S}) :=} \\ \pushright{\frac{1}{\gamma}\sum\limits_{s_i\in \mathcal{S}}
  \sum\limits_{j=1}^{m}{\left({\left[\sigma_j \left(\pHtf(s_i)-\pHtfr(s_i,\theta)\right)-\gamma\right]}_+\right)}^2}
\end{split}
  \label{eq:loss}
\end{align}
with respect to $\theta$, where
\begin{align*}
  {[ \cdot ]}_+:  \R \rightarrow [0,\infty), \quad x \mapsto
  \begin{cases}
    x & \text{if } x\ge 0,\\
    0 & \text{if } x<0,
  \end{cases}
\end{align*}
for decreasing values of $\gamma > 0$. Here $\mathcal{S} \subset \ri \R$ is a set of sample points, at which the original and reduced transfer functions are evaluated and $\sigma_j(\cdot)$ denotes the $j$-th singular value of its matrix argument.

The justification for using
$\loss$
as a surrogate for the $\hinf$ error is that $\loss(\cdot;H,\gamma,\mathcal{S})$ attains its global minimum (at zero), when all singular values of the error transfer function at all sample points $s_i$ are below the threshold~$\gamma$. Therefore, if the sample points are chosen appropriately, then $\loss(\theta;\pHtf,\gamma,\mathcal{S}) = 0$ is an indication for ${{\|\pHtf - \pHtf_r(\cdot,\theta)\|}_{\cH_\infty} \le \gamma}$. In \cite{Schwerdtner2021}, an adaptive sampling procedure is introduced which ensures that the sample points are appropriately distributed along the imaginary axis based on the given FOM and status of the optimization.

In this work, we use Algorithm~\ref{alg:bisection} to determine an $\hinf$ approximation via a bisection procedure, which determines the minimal value for $\gamma$ (up to a relative tolerance $\epsilon_1$) at which a minimization of $\loss$ with respect to $\theta$ terminates at zero. The tolerance $\epsilon_2$ that is used in line $6$ of the algorithm is the maximum value of $\loss$, which is still numerically interpreted as zero, such that $\gamma$ is reduced in the subsequent bisection step. The sample points are updated after each bisection step because the adaptive sampling update rule introduced in~\cite{Schwerdtner2021} depends on the current value of $\gamma$.

\begin{algorithm}[tbh]
  \LinesNumbered
  \SetAlgoLined
  \DontPrintSemicolon
  \SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
  \Input{FOM transfer function $\pHtf\in\mathcal{RH}_{\infty}^{\enu \times \enu}$,
  initial ROM transfer function $\pHtfr( \cdot, \theta_0) \in \rhinf^{\enu \times \enu}$ with parameter $\theta_0 \in \R^{n_\theta}$,
  initial sample point set $\mathcal{S} \subset \mathrm{i}\R$,
  upper bound $\gamma_{\rm u} > 0$,
  bisection tolerance $\varepsilon_1 > 0$,
  termination tolerance $\varepsilon_2 > 0$
}
  \Output{Reduced pH-ODE of order $r$}
  Set $j:=0$ and $\gamma_{\rm l}:=0$.\;
  \While{$(\gamma_{\rm u}-\gamma_{\rm l})/(\gamma_{\rm u}+\gamma_{\rm l}) > \varepsilon_1$}{
    Set $\gamma:=(\gamma_{\rm u}+\gamma_{\rm l})/2$.\;
    Update the sample set $\mathcal{S}$ using~\cite[Alg.~3.1]{Schwerdtner2021}. \;
    Solve the minimization problem $\alpha := \min_{\theta\in \R^{n_\theta}}\loss(\theta;\pHtf,\gamma,\mathcal{S})$ with minimizer $\theta_{j+1} \in \R^{n_\theta}$, initialized at $\theta_j$. \;
  \eIf{$\alpha > \varepsilon_2$}{
    Set $\gamma_{\rm l}:=\gamma$.\;
    }{
    Set $\gamma_{\rm u}:=\gamma$.\;
  }
  Set $j:=j+1$.
  }
  Construct the ROM with $\theta_j$ as in Lemma~\ref{lem:PHParam}.
  \caption{SOBMOR-$\hinf$}
  \label{alg:bisection}
\end{algorithm}

The benefits of using this approach instead of directly minimizing the $\hinf$ norm were discussed in detail in~\cite[Remark~3.3]{Schwerdtner2020}. The main reasons for using~\eqref{eq:loss} instead of the $\hinf$ norm are the differentiability of~$\loss$ with respect to~$\theta$, the local convergence of the method, and the prohibitive computational costs as well as reliability issues of the $\hinf$ norm computation (for the large-scale error system) inside an optimization loop.

\subsection{$\htwo$ approximation}%
\label{sub:ourmethod_htwo}
In order to obtain a finite $\htwo$ error, the polynomial parts of the FOM and the ROM transfer function must be equal. When using projection-based methods on ODE models this \emph{feedthrough matching} is automatic. In the DAE case, this is typically obtained by preserving the algebraic part, i.\,e., by including the null-space of the $E$-matrix in the projection matrices. For systems with multiple algebraic constraints this is undesirable, and a reduction of the subsystem corresponding to the  algebraic constraints may be necessary, see, for instance, \cite{MehS05}.

Another remedy (used for pH-DAEs with index one in \cite{BeaGM21}) is to compute the polynomial part $\pHtf_{\rm p}(s) \equiv \mathcal{D}_0$ of the FOM before the reduction and include it in the feedthrough terms $\rS$ and $\rN$ of the ROM, which we will use here as well. The direct computation of $\mathcal{D}_0$ (see Section \ref{sec:preliminaries}) may, however, require transformations of the FOM. Alternatively, $\mathcal{D}_0$ can also be estimated by sampling the transfer function of the FOM at sufficiently large $s\in\C$ in an iterative manner as proposed in \cite{Schwerdtner2020a}. We then decompose it in its symmetric and skew-symmetric part, respectively, i.\,e.,
\begin{align}
  S_0 &:= \frac{1}{2}\left(\mathcal{D}_0^\T+\mathcal{D}_0\right), \label{eq:S0} \\
  N_0 &:= \frac{1}{2}\left(\mathcal{D}_0^\T-\mathcal{D}_0\right). \label{eq:N0}
\end{align}
If the ROM is parameterized as in Lemma~\ref{lem:PHParam} then the $\cH_2$ error ${\Vert \pHtf - \pHtfr(\cdot,\theta)\Vert}_{\cH_2}$ is only well-defined if we have that
\begin{align}\label{eq:SN_match}
	\rS(\theta)  &= S_0,\quad \rN(\theta) = N_0,
\end{align}
since otherwise $\pHtf-\pHtfr(\cdot,\theta) \notin \mathcal{RH}_{2}^{m \times m}$. Consequently,  we first have to fix all parameters in  $\theta_N$ such that ${\rN(\theta) = N_0}$. Since we indirectly parameterize $\rS(\theta)$ via $\theta_W$, we first analyze which parameters in $\theta_W$ have an impact on $\rS(\theta)$. For this, consider a separation of ${\theta_W \in \R^{n_W}}$ into ${\theta_W =: \mat{\theta_{W_1}^{\T}, \theta_{W_2}^{\T}}^\T}$, where ${\theta_{W_1} \in \R^{n_W - m(m+1)/2}}$ and ${\theta_{W_2} \in \R^{m(m+1)/2}}$. Then we can decompose
\begin{align}
	\vtu(\theta_W) = \mat{\Xi_1 & \Xi_2 \\ 0 & \Xi_3},
\end{align}
where the matrices $\Xi_1, \,\Xi_2$ depend only on $\theta_{W_1}$, and $\Xi_3$ depends only on~$\theta_{W_2}$. Consequently, ${\rS(\theta) = \Xi_3\Xi_3^\T}$ only depends on~$\theta_{W_2}$ and we can set $\theta_{W_2}$ such that ${\rS(\theta) = S_0}$. The remaining parameters $\theta_{W_1}$ may still be subject to optimization and it holds that ${W(\theta) \geq 0}$ for all ${\theta_{W_1} \in \R^{n_W - m(m+1)/2}}$. Consequently, for minimizing the $\cH_2$ error, the parameter vector which is subject to optimization reduces to $\theta := \begin{bmatrix}
	\theta_J^\T, \theta_{W_1}^\T, \theta_Q^\T, \theta_{\pG}^\T
\end{bmatrix}^\T.$
\begin{rem}
  Note that $S_0 \geq 0$ always holds since the implicit pH-ODE~\eqref{eq:FOM_decpl} has the same transfer function $\pHtf$ as the original pH-DAE and is therefore positive real, i.\,e., we have that
  \begin{equation*}
    \Phi(\mathsf{i}\omega) := \pHtf(\mathsf{i}\omega) + \pHtf(\mathsf{i}\omega)^{\mathsf{H}} \geq 0,
  \end{equation*}
  for all $\omega \in \R$ and consequently, $\lim_{\omega\rightarrow\infty}\Phi(\mathsf{i}\omega) = 2S_0\geq 0$.
\end{rem}

Now we can formulate the $\cH_2$ optimization problem in the pole-residue framework originally proposed in \cite{Beattie2009a} for unstructured LTI systems and extended to pH-ODE systems in \cite{Moser2020}. Assume that
$(\rJ(\theta)-\rR(\theta))\rQ(\theta)$ is diagonalizable and consider the spectral decomposition
\begin{equation} \label{eq:red_evp}
	(\rJ(\theta)-\rR(\theta))\rQ(\theta) Z(\theta) = Z(\theta)\Lambda(\theta),
\end{equation}
where $\Lambda(\theta) = \text{diag}(\reig_1(\theta),\,...\,,\reig_r(\theta))$ and $Z(\theta)$ contains the right eigenvectors as columns. If the eigenvalues $\reig_i(\theta)~\in~\C$, $i=1,\,\ldots,\,r$ are simple, then the transfer function $\pHtfr(\cdot,\theta)$ may be represented by the partial fraction expansion ${\pHtfr(s,\theta) = \sum_{i=1}^{r} \frac{\rl_i(\theta)\rr_i(\theta)^\T}{s-\reig_i(\theta)}+\rS(\theta)-\rN(\theta)}$, where ${\rl_i(\theta),\,\rr_i(\theta)\in \C^m}$ with
\begin{align*}
	\rl_i(\theta) &= (\rG(\theta)+\rP(\theta))^\T\rQ(\theta) Z(\theta) e_i, \\
	\rr_i(\theta) &= (\rG(\theta)-\rP(\theta))^\T Z(\theta)^{-\T}e_i,
\end{align*}
and where $e_i$ denotes the $i$-th standard basis vector of $\R^r$. Assuming that $\eqref{eq:SN_match}$ holds, we have that
\begin{equation}
	\begin{aligned}
		{\Vert \pHtf-\pHtfr(\cdot,\theta) \Vert}_{\cH_2}^2 = &{\Vert \pHtf_{\rm sp} \Vert}_{\cH_2}^2 \\
				& - 2\sum_{i=1}^{r}\rl_i(\theta)^\T \pHtf_{\rm sp}(-\reig_i(\theta))\rr_i(\theta) \\
					& + \sum_{j,k=1}^{r} \frac{\rl_j(\theta)^\T\rl_k(\theta)\rr_k(\theta)^\T\rr_j(\theta)}{-\reig_j(\theta)-\reig_k(\theta)},
	\end{aligned}
\end{equation}
as shown in \cite[Theorem 2.1]{Beattie2009a}. Since ${\Vert \pHtf_{\rm sp} \Vert}_{\cH_2}^2$ does not depend on $\theta$, it can be neglected in the optimization. Consequently, we define the objective functional
\begin{align*}
\losst(\theta;\pHtf) &:={\Vert \pHtf-\pHtfr(\cdot,\theta) \Vert}_{\cH_2}^2 - {\Vert \pHtf_{\rm sp} \Vert}_{\cH_2}^2 \\
&:= \hat{\mathcal{F}}\Big(\big[\rl_1(\theta)^\T,\ldots ,\rl_r(\theta)^\T,\rr_1(\theta)^\T,\ldots\\ & \quad\quad\quad\quad \quad\rr_r(\theta)^\T,\reig_1(\theta)^\T,\ldots,\reig_r(\theta)^\T\big]^\T\Big) \\
&= (\hat{\mathcal{F}} \circ q)(\theta),
\end{align*}
where
\begin{align*}
	q(\theta) &:= [\rl_1(\theta)^\T,\ldots,\rl_r(\theta)^\T,\rr_1(\theta)^\T,\ldots \\ & \quad\quad\quad\quad\rr_r(\theta)^\T,\reig_1(\theta),\ldots,\reig_r(\theta)]^\T \in \C^{n_q}.
\end{align*}
This functional can be evaluated efficiently because it only requires the solution of the reduced-order eigenvalue problem in \eqref{eq:red_evp} as well as $r$ evaluations of $\pHtf_{\rm sp}$ at $-\reig_i(\theta)$.
The eigenvalues $\reig_i(\theta)$ and rank-one residues $\rl_i(\theta)\rr_i(\theta)^\T$ are functions of the parameter vector $\theta$.
If $\bar{\theta} \in \R^{n_\theta}$ is chosen such that all eigenvalues are simple, then $\mathcal{F}$ is differentiable in a neighborhood of $\bar{\theta}$.
Its derivative is obtained by applying the chain rule, i.\,e., with the differentiation operator $\text{D}$ we obtain
\begin{equation*}
 \text{D}\losst(\bar{\theta}) = \left(\nabla \losst(\bar{\theta})\right)^\T = \text{D}\hat{\mathcal{F}}(q(\bar{\theta}))\cdot \text{D}q(\bar{\theta}),
\end{equation*}
with
\begin{multline*}
	\text{D}\hat{\losst}(q(\bar{\theta})) = \left[\text{D}_{b_1}\hat{\losst}(q(\bar{\theta})),\ldots,\text{D}_{b_r}\hat{\losst}(q(\bar{\theta})), \ldots \right. \\ \left. \text{D}_{c_1}\hat{\losst}(q(\bar{\theta})),\ldots, \text{D}_{c_r}\hat{\losst}(q(\bar{\theta})), \ldots\right. \\ \left.\text{D}_{\lambda_1}\hat{\losst}(q(\bar{\theta})),\ldots,\text{D}_{\lambda_r}\hat{\losst}(q(\bar{\theta}))\right]\in\C^{1\times n_q},
\end{multline*}
and
\begin{align*}
	\text{D}q(\bar{\theta}) &= \left[\text{D}_{\theta_1}{q(\bar{\theta})},\ldots,\text{D}_{\theta_{n_\theta}}{q(\bar{\theta})}\right]\in\C^{n_q\times n_{\theta}}.
\end{align*}
For all $i=1,\,\ldots,\,r$ it holds that
\begin{align*}
    \text{D}_{\rr_i} \hat{\losst}(q(\bar{\theta})) &= 2\rl_i(\bar{\theta})^\T\big(\pHtfr(-\reig_i(\bar{\theta}))-\pHtf(-\reig_i(\bar{\theta}))\big), \\
    \text{D}_{\rl_i} \hat{\losst}(q(\bar{\theta})) &= 2\rr_i(\bar{\theta})^\T\big(\pHtfr(-\reig_i(\bar{\theta}))-\pHtf(-\reig_i(\bar{\theta}))\big)^\T, \\
	\text{D}_{\lambda_i} \hat{\losst}(q(\bar{\theta})) &= -2\rl_i(\bar{\theta})^\T\big(\pHtfr^{\prime}(-\reig_i(\bar{\theta}))-\pHtf^{\prime}(-\reig_i(\bar{\theta}))\big)\rr_i(\bar{\theta}),
\end{align*}
and we refer  to~\cite{Moser2020} for the differentiation of $\text{D} q$.
\begin{rem}
	The partial derivatives in $\mathrm{D}q(\bar{\theta})$ may be computed efficiently with block-wise expressions. For instance, the derivative $\mathrm{D}_{\theta_G}\rl_i(\bar{\theta}) \in \C^{m \times r\cdot m}$ can be computed as
	\begin{align*}
		\text{D}_{\theta_G}\rl_i(\bar{\theta}) &=  \mat{z_i(\bar{\theta})^\T\rQ(\bar{\theta}) & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & z_i(\bar{\theta})^\T\rQ(\bar{\theta})} \\ &= I_m \otimes  z_i(\bar{\theta})^\T\rQ(\bar{\theta}),
	\end{align*}
	where $z_i(\bar{\theta})\in\C^r$ denotes the $i$-th column in $Z(\bar{\theta})$. This is also the case for more complex derivatives that involve the differentiation of the eigenvalue problem in \eqref{eq:red_evp}.
\end{rem}
Here, we highlight some important advantages of the pole-residue framework compared to recently proposed methods that are formulated in the Lyapunov framework (see~\cite{Sato2018,Jiang2019}), in particular for pH-DAEs. These methods require the solution of large-scale Lyapunov equations for the evaluation of $\losst$ and its gradient. Currently no structure-preserving Lyapunov-based methods exist for pH-DAEs; see \cite{SchM22_ppt} for new Lyapunov-based formulations of pH-DAEs. If the strictly proper part of the transfer function can be easily decoupled from the constant polynomial part, then for pH-DAEs with index one, the existing methods for pH-ODEs may be applied to this part.
However, if the splitting into the strictly proper and polynomial part has first to be computed via a factorization method, then the sparsity patterns of the original pH-DAE may be lost
which complicates the repetitive solution of Lyapunov equations for these systems in the large-scale setting. We highlight that the pole-residue framework only requires evaluations of $\pHtf_{\rm sp}$. Since we have that
\begin{equation}
	\pHtf_{\rm sp}(s) = \pHtf(s) - (S_0-N_0),
\end{equation}
for all $s\in\C$, we may work directly with the sparse matrices of the original pH-DAE and do not require the solution of large-scale Lyapunov equations.
\begin{algorithm}[tbh]
	\LinesNumbered
	\SetAlgoLined
	\DontPrintSemicolon
	\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
	\Input{FOM transfer function $\pHtf\in\mathcal{RH}_{\infty}^{\enu \times \enu}$, reduced order $r\in\mathbb{N}$.
	}
	\Output{Reduced pH-ODE of order $r$}
	Compute $S_0,\,N_0$ as in \eqref{eq:S0}--\eqref{eq:N0}.\;
	Initialize $\theta_0$ s.t. $\rS(\theta_0)=S_0$, $\rN(\theta_0)=N_0$.\;
	Solve \vspace{-12pt}\begin{align*}
	    \theta_{\rm fin} &= \argmin\limits_{\substack{\theta\in \R^{n_\theta}}} \losst(\theta; \pHtf) \\
	    &\text{s.t. } \rS(\theta)=S_0,\, \rN(\theta)=N_0.
	\end{align*}\vspace{-18pt} \;
	Construct the ROM with $\theta_{\rm fin}$ as in Lemma \ref{lem:PHParam}.\;
	\caption{PROPT-$\mathcal{H}_2$}
	\label{alg:H2OPT}
\end{algorithm}

Since the $\cH_2$ optimization problem is non-convex, the choice of the initial parameter vector $\theta_0$ will generally impact the fidelity of the final ROM obtained by Algorithm~\ref{alg:H2OPT}. Simple initialization strategies are, for instance, choosing $\theta_0$ randomly or using IRKA-PH (see~\cite{Moser2020, Sato2018}), which generally converges very quickly. Here, we propose another approach that may use \emph{unstructured} ROMs for initialization which is based on the following parameterization.
\begin{lem}\label{lem:pHInit}
	 Let $(\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D})$ be a ROM of state-space dimension $r$ such that $\widetilde{D} = S_0-N_0$ and such that $\widetilde{A}$ has all its eigenvalues in the open left half of the complex plane. Let  $\theta_G \in\R^{r\cdot m}$ and $\theta_K \in\R^{r\cdot p}$ be two parameter vectors and define the matrix-valued functions
	 \begin{align*}
	    \iG(\theta_G) &:= \vtf_m(\theta_G), \\
	 	\rK(\theta_K) &:= \vtf_q(\theta_K).
	 \end{align*}
 	 Let $\iQ(\theta_K)>0$ solve the Lyapunov equation
 	 \begin{equation}
 	 	\widetilde{A}^\T\iQ(\theta_K)+\iQ(\theta_K)\widetilde{A}+\rK(\theta_K)\rK(\theta_K)^\T = 0,
 	 \end{equation}
  	and define
  	\begin{align*}
  		\iJ(\theta_K)&=\frac{1}{2}\left(\widetilde{A} \iQ(\theta_K)^{-1}-\iQ(\theta_K)^{-1}\widetilde{A}^\T\right), \\
  		\iR(\theta_K)&=-\frac{1}{2}\left(\widetilde{A} \iQ(\theta_K)^{-1}+\iQ(\theta_K)^{-1}\widetilde{A}^\T\right).
  	\end{align*}
  	Then the parametric system
  	\begin{align}
  		\label{eq:pHParamInit}
  		\pHsysr(\theta_G,\theta_K):
  		\begin{cases}
  			\!\begin{aligned}
  				\dot x_r(t)= &\big(\iJ(\theta_K)-\iR(\theta_K))\iQ(\theta_K\big)\rx(t) \\ &\quad \quad + \iG(\theta_G) u(t),
  			\end{aligned}\\
  			\!\begin{aligned}
  		 	\ry(t)= &\iG(\theta_G)^\T \iQ(\theta_K) \rx(t)\\ &\quad \quad+ (S_0-N_0) u(t)
  			\end{aligned}\\
  		\end{cases}
  	\end{align}
  	is a pH-ODE system with
  	\begin{equation*}
  		\big(\iJ(\theta_K)-\iR(\theta_K)\big)\iQ(\theta_K) = \widetilde{A}.
  	\end{equation*}
\end{lem}
Let $\widetilde{H}$ denote the transfer function of the (possibly unstructured) ROM $\big(\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D}\big)$ with ${\widetilde{\pHtf}(s)=\sum\limits_{i=1}^{r} \frac{\widetilde{\rl}_i\widetilde{\rr}_i^\T}{s-\widetilde{\reig}_i}+S_0-N_0}$ and ${\widetilde{\rl}_i,\,\widetilde{\rr}_i\in \C^m}$.

Based on the parameterization in Lemma \ref{lem:pHInit}, we can then compute an initial pH model by minimizing the weighted sum of squared errors between the residuals in the Frobenius norm,  i.\,e.,
\begin{equation*}
	\losst_0(\theta_G,\theta_K) := \sum_{i=1}^{r} \frac{1}{|\widetilde{\reig}_i|} \left\Vert \widetilde{\rl}_i\widetilde{\rr}_i^\T - \rl_i(\theta_G,\theta_K)\rr_i(\theta_G,\theta_K)^\T\right\Vert_{\rm F}^2,
\end{equation*}
where
\begin{align*}
	\rl_i(\theta_G,\theta_K) &= \iG(\theta_G)^\T\iQ(\theta_K) \widetilde{Z} e_i, \\
	\rr_i(\theta_G,\theta_K) &= \iG(\theta_G)^\T \widetilde{Z}^{-\T}e_i,
\end{align*}
for $i=1,\,\ldots,\,r$ and $\widetilde{Z}$ is,
again under a diagonalizability assumption, obtained from the spectral decomposition
 \begin{equation*}
	\widetilde{A} \widetilde{Z} = \widetilde{Z}\widetilde{\Lambda},
\end{equation*}
with $\widetilde{\Lambda} = \text{diag}(\widetilde{\reig}_1,\,...\,,\widetilde{\reig}_r)$.

Note that the computation of the gradient of $\losst_0$ is very simple, since it does not involve a differentiation of the eigenvalues or eigenvectors. While the partial gradients of $\rl_i(\cdot)$ and $\rr_i(\cdot)$ with respect to $\theta_G$ are straightforward, the partial gradients of $\iQ(\cdot)$ with respect to the $l$-th entry in $\theta_K$ is the solution of the (reduced-order) Lyapunov equation
\begin{multline*}
	\widetilde{A}^\T \frac{\partial\iQ(\theta_K)}{\partial \theta_{K,l}} + \frac{\partial\iQ(\theta_K)}{\partial \theta_{K,l}}\widetilde{A} \\ + \vtf_q(e_l)\rK(\theta_K)^\T+\rK(\theta_K)\vtf_q(e_l)^\T= 0,
\end{multline*}
where $e_l$ denotes the $l$-th standard basis vector of $\R^{r\cdot p}$. As the number of optimization parameters is reduced to $r(p+m)$, this initialization generally converges very quickly. In combination with Algorithm~\ref{alg:H2OPT}, this enables a \emph{two-step} approach with a more restrictive (yet simpler) pre-optimization of only the residuals and a subsequent (more complex) optimization of all system matrices.
\begin{rem}
	Note that the sample-based SOBMOR method can be tuned to compute a ROM with small $\htwo$ error as well. Instead of using $\loss$ in conjunction with the bisection method outlined in Algorithm~\ref{alg:bisection}, the integral in~\eqref{eq:htwonorm} can be approximated by means of an adaptive quadrature rule (see \cite[Algorithm~1]{Gonnet2010} for a template method). In this way, it is possible to compute the $\htwo$ error and its gradient with respect to the free ROM parameters in terms of the error transfer function at specific sample points. This makes it possible to use the same optimization techniques as in~\cite{Schwerdtner2020}. In particular, \cite[Theorem~3.1]{Schwerdtner2020} for the gradient computation can be reused.

	We denote this method by SOBMOR-$\htwo$. Further details regarding the implementation of the adaptive integration are provided in the Appendix.
\end{rem}

redpajama_set_name: RedPajamaArXiv


text:
My 2017 Birthday Trip To Disneyland Paris! Day One!
On the 12th July, my childhood dreams came true! I don't any child or adult who doesn't dream of heading to Disneyland! This year, along with my best friend Tessa, I finally ventured into the world of Walt Disney to celebrate my 22nd birthday in style at Disneyland Paris.
Throughout the week I will be sharing what happened and what went down each day, starting with the Wednesday we arrived. After booking up to go late 2016, we found a brilliant deal by booking everything separately. Flights were with EasyJet for £80, tickets through the Disneyland Paris website for £132 for three days entrance and our apartment/hotel was the Residence Du Parc Val d'Europe, which I can't remember how much it was exactly, but overall the holiday came to £335 for three days in the park and four nights. Pretty good for a Disney holiday.
Setting off on Wednesday afternoon, we had a super quick flight to Paris CDG, where we met our transfer waiting for us at the other end to head to our hotel. After quickly checking in and dumping our luggage into our room, which was lovely – although it took us a long time to figure out how to open the door – we headed off to do the first important thing of the trip. FIND FOOD.
Is it even a holiday without a plane picture?
Around us, the area was quiet and very much a holiday/home destination. As we walked to the supermarket, we found a place which was created just for dogs to run free in and a Chateau that looked amazing. It was upon this walk to the supermarket where I got my first real glimpse of Disney, as my friend Tessa grabbed my arm and shouted, "Joey I can see the castle." Never in my life has my whole body jumped with so much joy at that little sight. Carrying along the path, I got glimpses of the Indiana Jones ride and Thunder Mountain, both unknown to me at the time.
Food found and another exciting glimpse of the castle to be had, I asked – well more like begged – Tessa to go on an evening walk at around 10:30pm to see if we could get a glimpse of it all at night. Heading out Tessa told me to look up and I got to see the twinkling lights that whirl around the sky to highlight where the park is for those returning to the park to see the illuminations or to enjoy the rides at night.
Walking along and chatting about life, we were both shocked to see fireworks and raced to stand by this river, where I got my first glimpse at the Disney Illuminations. Although they weren't very clear it was a fun game to see what was being reflected onto the castle and to see it in a faraway distance. It was automatic butterflies in the tummy moment. Especially when Star Wars came on, as it was probably the only one I could make out well aha!
By this point, I was so excited and ready to head to the park that I think I raced Tessa home to go to sleep so that we could get up early and enjoy the park. I will let you all know now that on this holiday I may have turned into a little kid again, with excitement, amazement, surprises, some tiredness, a little sprinkle of sass and a tiny amount of hanger all mixing in.
Check in soon to find out what happened on Thursday!
Jennifer Lawrence Stars In The Spooky First Look Trailer For Mother!
Jennifer Lawrence 18:28:00
Whilst at the hospital I received a very interesting email! There was no description and no images within it. Just one word. mother! As well, the email included a link to a trailer and I am so thankful that I clicked on it.
From Darren Afronofsky, the man behind Black Swan, comes mother! A film about a couple's relationship which is tested when unexpected guests move into their home and upset their daily lives.
Starring Jennifer Lawrence, Javier Bardem, Ed Harris and Michelle Pfeiffer, this is a film you are not going to want to miss from the trailer below.
Will you be watching this lovelies? I will be on opening night aha!
Transparent Is Back In This New Season Four Teaser Trailer
Every September I prepare myself for a day of binge watching. And every year without fail I succeed in devouring the latest season of Transparent in one day. This year I am preparing myself to do it all over again, because the season four teaser that was released over the weekend is just amazing!
Honestly, TV has never been or looked this good before…
Don't Miss These Brand New Clips From Pawn Sacrifice and More!
Pawn Sacrifice 18:22:00
And how are we all today? Well, I hope! I am finally here to give you all these exciting new clips, trailers and goodies from the world of film and TV!
From Toby Maguire's new film Pawn Sacrifice to some Netflix newbies, this selection is going to keep you entertained for a long time lovelies…
Life DVD Review
DVD 08:00:00
Life seems like it came to director Daniel Espinosa when he was sat in on a Saturday night watching a film marathon including Alien, Jurassic Park and Gravity. Working to create something fresh but constantly bouncing off the elements featured in these films, Espinosa has made a story about life on Mars with a bromance leading the way.
Starring Ryan Reynolds, Rebecca Ferguson and Jake Gyllenhaal, the film follows scientists on the International Space Station, as they uncover a life form on Mars that could and does cause serious harm to life on Earth. All three actors work well with each other and Reynolds and Gyllenhaal appear to be having the time of their lives when working together and floating around on set.
However, they are let down by the lack lustre script, which doesn't hold many original qualities within it. With six people on board and a rat for company, the film automatically springs to mind the crew members and feline friend that Alien had on board it. And whilst Reynolds does a good job of trying to replace Weaver, it isn't the best space adventure around.
Calvin, the organism/alien that affects the ship comes to life very much like the animals in Jurassic Park. And thankfully so, as it also provides life to the film. First, it is looked at and admired, then it is on a mad murderous rampage. This is the only real character in the film that seems like it knows what it wants on screen and how it is going to get it. It is a hunt or be hunted situation that leads to the thrills you would have hoped.
Ultimately the film is very much a bunch of classics brought together to tell one story that lacks the rhythm and flow of space life that the others capture well. There so I am giving it…
2 ½ Stars
American Gods Series One DVD Review
American Gods 08:00:00
If you don't have an Amazon Prime account, you may have missed out on the fantasy thriller that was American Gods. With many saying it was going to be the new Game of Thrones, the show doesn't hold back from throwing audience members into the mind-blowing world from the get go.
Led by Ricky Whittle, famed for Hollyoaks and The 100, the show based on the book by Neil Gaiman has a standout cast including Emily Browning, Ian McShane, Crispin Glover, Kristin Chenoweth and Gillian Anderson.
All of which who bring to life the world of Shadow Moon, a prisoner who is let out a day earlier than expected due to the death of his wife. Whilst on a flight, he meets the mysterious Mr Wednesday, a con artist and God. After being drawn into a deal with Mr Wednesday, Shadow is faced with some of the weirdest scenarios in his life, which he soon has to learn to deal with whilst also grieving.
From a female god who can suck the life from the men she sleeps with with her vagina to a David Bowie styled God (Very well done by Gillian Anderson), American Gods makes sure to let audiences know that they are not going to be getting the normal fantasy features they would experience.
Every character has a mission. Good or bad. And they will do anything to make sure that they complete their tasks first before something can happen to them Whittle plays the role well, especially with the great McShane alongside him to balance out the character, with his wit and humour. However, some characters should definitely have been explored more, including Browning.
At first, I found the show hard to get into. I watched it originally on Amazon Prime and my mind couldn't handle it, apart from the Irish character who won my heart over automatically. This soon changed though as I watched on. The characters and mystery surrounding the show deepened and this led to intrigue from all sides.
Unlike GOT, you have no idea how this show might go. From the opening five minutes, where things seem to be easy to understand to the last few episodes, there is nothing I can compare it to. It is a comic book which has been brought to life to take us away from our world for an hour. And throw us into a completely random one.
With only eight episodes, this is one show that definitely deserves the binge-watch status it is getting, but take it with care. Whittle may be eye-candy but it is the scenes designs that will draw you in and they can be dangerous to the eye.
Overall, the show is an original piece that highlights how far TV is coming in making gripping adaptations. Whilst a film version of this could have been drawn out too much, the eight hour-long episodes are enough to satisfy your thirst for fantasy once a week. Just don't get to lost in it.
There so I am giving the show...
Dreamscape DVD Review
Dreamscape 07:30:00
The 1980s was a time in which many new film genres were released into the world and film franchises were born. Amongst them was the Dennis Quaid and Kate Capshaw feature, Dreamscape, which has been released on DVD and Blu-ray for the first time today!
Directed by Joseph Ruben, tis sci-fi thriller tells the story of a young psychic called Alex, who is forced into a dream research project. Allowing him to delve into the world of other people's dreams, the film see him fall into a world of unexpected nightmares causing Alex to fear what he may find when the government takes over the project.
Quaid manages to make his characters fear the stand out performance in the film. One that only manages to attract more attention than the plot of the film at times. As the film takes a dark turn it makes you start to think more about what the characters are going through. However, there is a sense that the film is going a little overboard at times.
The film goes a little bit wild when they let the government take over and that is when the charm of it is lost on the audience.
There so I am giving the film...
Electric Dreams DVD Review
I am sure that many of you know the hit song Electric Dreams by Phil Oakley! But did you know that the song came from a hit film called Electric Dreams? Well don't despair if not, because the film is making its Blu-ray debut today and so I thought it would be the perfect time to review it.
In a newly restored format, the film follows Miles, a helpless architect who is struggling to deal with his life, after his computer Edgar decides to ruin it. With bills run up to the full amount, cancelled travel plans and the computer trying to steal his girlfriend, the film captures him at a bad time.
This comedy may seem a bit depressive, but actually the film is a hugely light-hearted treat. The comedy and laughs keep coming throughout and when mixed in with the great soundtrack and killer hit song, the film stands out amongst others.
It is east to see how it earned its cult status, as the small details and hugely witty comments make it a success. The whole film is a mixture of pure delight in fact.
This is why I am giving it...
Future Shock! The Story Of 2000 AD DVD Review
Released back in 1977, Future Shock! The Story Of 2000 AD, was a sci-fi comedy documentary for the upcoming ages. Now for the first time ever, Arrow Videos are releasing it on Blu-ray and DVD to highlight the punk era in Britain in a completely new way.
Based on the best-selling comic 2000 AD, the film is full to the brim of interesting interviews and stories about how it came to life and how important it was to those that read it at the time. It is a celebration of the arts and time put into making magazines and the words within them.
If you have never heard of the comic before then it is nice to find out about it, however, the documentary does dip into boring territory after a while. This would probably not be this way for real fans of the comics, but if not, it does drag on.
The film is one that has been made for readers of 2000 AD. It has been made for them and for their enjoyment, probably not so much for others.
There so I am giving the film…
Happy Birthday To Us! Win A Copy Of Do You Believe On DVD With The Blog To Celebrate!
Blogging 16:03:00
I have a competition for you all now to celebrate the blog's birthday week lovelies! To mark the occasion and the release of Do You Believe on DVD from the 4th September, one of you will be in with the chance of winning the film on DVD.
All you have to do to enter is to head over to the blog's Twitter, follow the blog and retweet the pinned post on the blog's Twitter page. Don't worry if you can find it though lovelies, here is a link for you all to it: https://twitter.com/StartWithThis/status/890950443207208960
You have until 2pm on Friday 4th August to enter lovelies. So head on over to the blog's Twitter and you could be the lucky winners! You can also pre-order the film today: http://amzn.to/2tJmrkI
You Might Want To Grab A Cushion To Watch This New IT Trailer!
IT 11:00:00
Okay so I am about to bring the party down a little bit, but the mood is still going to stay electrifying, as I have the new trailer hot from Comic Con for Warner Bros IT and you might want to grab a cushion.
Bloody clowns! How have they not been banned since this trailer...
Don't Miss These New Clips and Trailers From Amazon Video UK's Comrade Detective and More!
clips 10:39:00
And how are you all today? I wish you all a splendid day because the weekend is finally here and that means that it is officially party time! Today there is going to be a double party for me as I am leaving my old job to start a new one on Tuesday!
So to kick off the celebrations I have a whole bunch of new clips, trailers and more for you to enjoy from Comrade Detective and more lovelies. Let's get the party started...
The Characters Of Kingsman: The Golden Circle Come Alive In These New Posters!
Kingsman: The Golden Circle 16:17:00
After all the excitement on the blog with the birthday celebrations and that amazing Outlander trailer, I don't really know how to explain my joy at these new character posters from Kingsman: The Golden Circle!
Honestly, I swear people want me to burst with film joy aha...
Also if you want to see them in a bit more of an action packed environment, you can watch an exciting video featuring the posters below:
The Blog's Birthday Cake Has Arrived Thanks To The Food Network's Fakery Bakery!
The birthday celebrations are continuing right now lovelies, with a special cake recipe, created by the Food Network, that I think we should all make to mark the amazing celebrations!
To celebrate their new show, Bakers vs. Fakers, which premieres Wednesday 9th August at 9pm on Food Network, the channel found the more and more Brits were buying cakes and claiming them as their own after feeling pressured by TV programmes and baking trends to create their own.
56% of people have even stated that they would look for baking shortcuts to help them when it comes to baking, after being inspired by social media to make their own. However, those days are now behind you with the channel opening The Fakery in London, which will teach you all you need to know about baking the quick and easy way.
Hosting free workshops, The Fakery will see classes being taught by top bakers including Juliet Sear to help enhance the skills you can use on store brought cakes lovelies. Speaking about The Fakery, critically acclaimed baker, author and star faker, Juliet Sear, says: "I'm so excited to be teaming up with Food Network to help launch The Fakery. Research shows that people want to bake more, but don't because they are strapped for time. There are lots of smart shortcuts to help create show-stopping bakes and I am going to be giving away a few of these secrets at The Fakery!"
Whilst, Phillip Luff, Managing Director of Scripps Networks Interactive, UK & EMEA said, "Brits love to get in the kitchen and over the past few years we've seen a huge rise in the demand for baking programmes. When we heard that people want to bake more, but struggle to find the time to bake from scratch, we thought it would be a brilliant idea to open The Fakery, to help our fans find time to create some amazing bakes!"
To celebrate the blog's birthday week, which I hope includes lots of cake, book your free session at The Fakery here lovelies: https://www.eventbrite.co.uk/e/the-fakery-by-food-network-tickets-36306214866
Be honest now lovelies, have you ever faked it? Cake wise I mean aha!
Outlander Season Three Teases Us With This New Trailer
Outlander 15:08:00
Okay so as many of you know, I am obsessed with Outlander and season three is hot on my list to watch this autumn! Anything that is released about it makes me so excited and Comic Con did not help when the cast came together to release the latest trailer for it.
It makes all my dreams come true! Well, Jamie Fraser dreams that is...
Don't Miss These New Clips From Dunkirk and More!
And how are you all today? Well, I hope! Today has been a bit of a hectic day for me in terms of work, as I am about to make a very exciting change and there so I am teaching lots of new people. This has led to me running around like a weirdo through our office aha!
Finally, though I have had a moment to sit back and relax, so I thought I would share with you all these new clips and trailers from Dunkirk and more lovelies. Enjoy...
Time To Celebrate The Blog's Birthday With The Personal Tag
Anniversary 16:23:00
To continue our birthday celebrations on the blog, I thought I would do a fun tag to let you know a few personal things about me and how much the blog means to me! I pass it on to all of you once you have read it lovelies...
1. How Long Have You Been Blogging and What Is Your Best And Worst Thing About It?
This month the blog will be turning five and I have had some of the most amazing things happen to me since I started. The best thing about it is the way I have been able to see myself grow as a person. From meeting people to talking to them about certain shows and films online, it has really helped me come out of my shell. However, the worst thing about it is the way in which bloggers can be taken for granted. I have seen people and companies turn against me because I have said no to something and that can be a real confidence knock. Thankfully as a community we are speaking more and more about it to try and make this stop.
2. How Did You Spend Your Birthday This Year? Any Crazy Traditions?
As many of you probably now know I spent my birthday in Disneyland Paris with one of best friends Tessa and we had a bit of a manic but amazing time whilst there! I do have one crazy tradition, which is that my birthday must end with a screening of a horror film, but we didn't do that this year as we were so tired. We did do the tower of terror though, so I am counting this instead aha!
3. What Is Your Favourite Dessert? All The Gooey Details Please...
Oooo I don't think any birthday is complete without a desert and my favourites are Krispy Kreme donuts with a crème filling and apple crumble with custard! Summer is all about fresh fruit bowls though lovelies.
4. What Is Your Favourite Room In Your House?
My bedroom or our living room! I love my living room and bedroom in winter when the fire is on and we are all cosy. I also love it because it is the place I go when I know that all my work is done for the day and I can finally relax.
5. In Which Ways Do You Entertain Your Nieces And Nephews?
Since they were born I try to be the fun aunt! Whilst I can tell them off, I do try to make them laugh and take them to as many things as I can. I always want them to feel like they can talk to me and if I have to make them do so with a film, art or a play then so be it. As long as they are happy I'm happy!
6. How Do You Store Your Memories?
On my phone, computer and in my emails. I really should back them up, but then I have my mind to do that aha! Only joking, I am actually looking for a good portable hard drive, so do let me know if you have a good one lovelies.
7. Are You A Savoury Or Sweet Type Of Person?
Ooooo it really depends on what mood I am in but right now I am having a savoury moment.
8. If You Could Go Anywhere Would It Be?
I would love to go to Jamaica, South Africa, Rio de Janeiro and India as soon as possible! Please someone make a trip flights airline for these destinations ASAP!
9. Is There Anything You've Ever Regretted?
I regret not taking my mum with me on my first trip to Disneyland Paris! I wished she was there every moment of the day, as I always wanted to experience it with her when I was little. I am now trying to plan a trip though, so hopefully I can get some redemption.
10. What Is The Best Thing You Have Ever Made?
I honestly believe it is my blog! Not only has it helped my writing progress and given me the chance to experience some amazing things, but it has helped me find work and gain confidence to. When I suffered from a mental breakdown last year it really helped me escape the real world and just let all my emotions out.
Look out for more birthday goodness soon lovelies!
Birthday and Christmas Celebrations Collide At The Cadbury Cadvent Event!
Cadbury 10:00:00
Kicking off today's birthday celebrations is well a Christmas post! As many of you will know Christmas is my favourite time of the year and July is one of my favourite months of the year!
So, when you put them together you get the amazing Christmas in July, which has been a trend since I was born for many companies. Yesterday I headed with my friend Chidubem down to the Cadbury Cadvent event to see what goodies and treats my family would be feasting on this year!
Cadbury is a must have in our home during Christmas and this year the company is pulling out all the stops! From snow ball filled crackers that can replace the normal ones on your Christmas table to old school packaging, the company are going back in time and the future to give us everything we need.
Set on three floors, the first magical area was an advent calendar treat fest. Given a number by those by the door, you were made to hunt for the box with your number on it. Once found, you could lift the lid of the box, which had some amazing treats inside. Starting with the normal sized advent and going on up all the bigger sizes, the boxes were filled like Russian dolls with advent gifts that were soon loaded into our sack like totes.
After opening our presents, we headed on up to the next floor where we found our Christmas dinner waiting for us. However, this one was a little different as the table was filled with Snow Balls. Not the alcoholic type but the chocolate and sugar-coated treats that have been reinvented to fit into a cracker that was placed on top of plates to give Cadbury inspired place setting ideas.
Something else that lined the room was a tree with further presents. Presented with a parcel, we found ourselves handed with the mousse chocolate snowman and classic Christmas gifts that Cadbury's have become well known for too little ones. We were also prepped for when we put our own trees up, as we were handed the newest tree decorations from the company, which now feature Oreos in them.
Leaving the room before it became too much, we headed into the final room in the building. Automatically we were transported into a Christmas wonderland, as we sat in old arms chairs and went back in time to when Cadbury began. Something everyone will be doing in December, as the company revealed that the packaging on the bars was going back to how it was when it all began. Vintage has honestly never looked so good!
The final treat of the day was the classic tins. After picking up the modern day looking tubs of Roses and Heroes, a guide showed us an amazing metal tin that was coming out in December. Presented like a soldier's drum, the tin which houses Heroes inside, will donate money to Help the Heroes when purchased. Something important when it comes to Christmas is giving Lovelies and this is the ultimate way to do it.
Heading out of the snow filled building with sacks and hands full, my whole body was excited for December and getting to see others and shops filled with these Cadbury items. I'm officially over summer lovelies and ready for the festivities to begin.
Look out for blog reviews on the items soon!
Don't Miss These New Clips From The Hitman's Bodyguard and More!
How are you all today? The birthday celebrations are about to begin on the blog, but before they do I wanted to share with you all these brand new clips from the world of film and TV lovelies!
Starting us off is The Hitman's Bodyguard, so get ready lovelies...
It Is Birthday Week On The Blog! Time To Celebrate!
This week is birthday week on the blog! For the past five years, you amazing people have been coming to the blog and reading all about what I have been getting up to, finding out the latest film and TV news and ultimately I hope having a good time.
All this week I am hoping to continue to do that lovelies with some exciting posts all about my birthday trip to Disneyland Paris, old school tags, random Christmas inspired items, personal posts and all the film and TV goodness you are used to lovelies.
I will keep you all updated as the week goes on to make it a very special party lovelies because a blog doesn't turn five every day does it?!?
All The Best Bits From The London Valerian Premiere!
Last night London was alight for the premiere of Valerian in London! The stars of the film turned up in the finery to spend the night enjoying the best of the best for the night including Rihanna, Cara Delevingne and Dane DeHaan.
If you missed it lovelies don't worry because I have all the best bits for you to enjoy below...
Don't Miss This New Trailer From The Nut Job 2 and More!
How are you all doing today lovelies? Good, I hope! Although it is July in the UK right now, it feels more like it is the middle of November! The weather has really turned, so I thought I would bring the sunshine to you with these new clips from The Nut Job 2 and more lovelies.
Look out for an exciting announcement soon lovelies!
Dunkirk IMAX Review
Dunkirk 14:19:00
Three years on from Interstellar, Christopher Nolan brings his new film closer to home in a retelling of the WW2 Dunkirk story that continues to shock when told, due to the heroic actions of the soldiers, commanders and British public.
Using familiar faces including Tom Hardy, Cillian Murphy, Mark Rylance, Sir Kenneth Branagh and a mix of new faces such as Fionn Whitehead and Harry Styles, Nolan's film bursts with talented figures who know how to command a screen and give this film the power it deserves. As it is one hell of an emotional ride.
Split into three sections exploring the soldiers on land, in the air and the people from home who came to save them, Dunkirk is a non-linear story that packs a punch. Men trying to find a way off the beach by carrying the injured on to ships and jumping back on them when kicked off and RAF fighters protecting those on the ground from enemy pilots who are bombing their men, the film knows how to make audiences emotions build and build.
When safe areas are believed to be found – for both the characters and the audience members watching - the soldiers face even more heartache, as they try to hide in boats and houses, which end up being disaster zones too. The trust they should have in one another becomes lost in the flowing waves of the beach.
In Dunkirk less is always more for Nolan, who has always been a director who likes to let his characters and audience think rather than speak or hear what others have to say. In Dunkirk, he makes further suspense, as his non-linear story means audiences are never truly sure where they are but stay on the edge of their seats, intent on finding out.
Murphy's shivering soldier is a key example of this, as he is captured in control and out of it within the story, meaning you get to see both sides of the man he plays. Styles' character is another key figure, as he tries to find out who he can trust out of his fellow comrades, leading to the ultimate plot twist. In fact, Whitehead, Styles and Aneurin Barnard, who plays an army private, have made sure that those watching the film don't forget who they are but with very little use of words or powerful speeches to make them remember able.
There aren't many films were silence is key but in Dunkirk it is everything. Silence brings with it the urgency of getting the men off the beach and back home. Silence is the realisation that Rylance's character faces when he sees his country's men drowning and covered in oil. Silence is the thing that causes fear within all of the soldiers. It is a matter of life and death and Nolan has captured this incredibly well.
Especially on IMAX, which is the key to Nolan's Dunkirk success! The 75mm film captures the length and beauty of the screen inch by inch, as it causes your own emotions to be heightened. When scenes show Hardy's character in the cockpit of his Spitfire, the noises make it feel like you are sat beside him. A similar thing is felt when the young actors are trapped in a boat with the water rising in from enemy bullet holes. Your eyes feel like they are doing a work out trying to make sure they are capturing every little moment of Nolan's story.
One thing audience members have to remember though when watching is that this is a real story. While the words, characters and individual stories are all from Nolan, the action that occurs actually happened. The aftermath the soldiers had to live with was real and many did feel guilt. Something they really shouldn't have felt. This is a film that is not only dedicated to them and their time on Dunkirk but it is also a film to highlight to a new generation how much the men who were in WW2 were heroes.
Visually stunning and beautifully told, the film is an educational and cinematic feature that will stand the test of time. Especially as an IMAX masterpiece.
Nolan's Dunkirk has earned this huge...
Sarah Warren Talks To The Blog About New Youtube Web Series LAX!
Interview 13:26:00
Since making her hit film, MLE, in 2015, Sarah Warren has been very busy! From celebrating her work on and off screen, the star has also been working hard on a new web series called LAX, which is on YouTube now.
Speaking to me about the show, we discussed what inspired it, how she found the amazing actors in it and who she would like to join her on a flight to LAX...
Firstly congratulations on the new web series! How did you come up with the idea for LAX?
Thank you very much, and thanks for watching!
"I never take anyone to the airport"
"Because that's a thing you do the first 3 weeks of a relationship. And i don't want anyone asking me, "Why don't you take me to the airport anymore?" - When Harry Met Sally...
I think the moment of arriving to a big airport is so telling: who's picking you up? Does anyone care? What are the smells? Language? Manners? Glottal stopping tones around you? Especially over a few years, going to the same airport, and how does your pick up situation change and does that mean anything? People choosing to spend time waiting, and stop the seeming urgent flow of their lives to stand at an airport and meditate a touch while they wait for someone, is a gift.
I'm pensive about magic. Literal magic yes but also the kind that blossoms stuff; ideas, love - that stomach pouring unstoppable ooo moment which has the power and potential to make anything happen. I am interested in if that can last, if it should, and if so what makes it last. Did you ever land in a city that gave you a little, ooo magic, feeling? Did it last? We are obsessed with promises and extremes as humans: forever, always, never, marriage, religion, diets ... we can be extreme as a species - no kidding I know. I have found it interesting when I watched someone promise me something...not in a cynical way, just truly interested - where does that feeling come from, to promise? Fear? Overwhelming present emotion? Don't know...
I've had many different experiences at airports over the last decade (I've moved continents 4 times), and I thought it would be something (hopefully) relatable and meaningful to explore.
As a cinephile, I was also captivated by the film Locke (which could be a quite unimaginable pitch ... cement pourer man driving in car for hours), and I wanted to make a l'il web series which had that character drama with the honest comedy of Girls. Tall order sure, but let's aim high, no? :)
Airports are a place where filmmakers have captured love, departures and comedy in recent years. What is it about them that makes them so great on screen?
Interesting. I think transitional spaces are honest and freeing - travel, can feel like heightened life (perfect for film) - the goodbyes, hellos, the anticipation, it's all a good cocktail for film which I believe should always be a controlled and pleasurable balance of surprise/suspense and mystery, but mainly; honest. I am looking at the drive home from the airport which is slightly different, because it can be a gradual come down from a high of swirling expectation.
The show very much captures some of the problems that people are facing nowadays but with added comedy. How hard was it to capture these on screen?
I'm glad you think so, I'd like to think it touches people. Would love to know which problems in particular you mean? It's helpful for me to see through your eyes how you interpret the show. Do you mean... humans being flaky? Ha.
It wasn't hard to capture the story conflicts on screen because we had great actors, and sharing a story through complex characters can be cathartic if done not self-indulgently (that's the hope). But it certainly was practically difficult to get it made on zero budget in LA - I mean, we cannot tell you how difficult!
One funny anecdote was that almost every time we showed up at LAX, the exact scene we were about to shoot was there being acted out before us in some duplicate version, by unknowing strangers. It was surreal - good writing I guess? ...Tee..Hee.
The acting in the show is brilliant! How did you find the talent within the show? What was it about these actors that made you pick them?
Thank you kindly for saying so! A few of the roles were written for specific actors I knew. For the others, I set up a casting - many people rush casting but the intention of casting is so crucial that it can't be rushed I believe. A couple actors were stand-up comedians from the Comedy Store I admire. I also studied Improv at UCB and picked my faves from there - actors I feel are excellent listeners, hilarious form the right place in their gut, and good people I dig - that matters to me. Set days are long - why be with impatient pricks when there's so many talented, imaginative, solution oriented, team players out there?
Did anything that happens in the show happen in real? Or have you been close to experiencing?
Let's go for a beer. Kidding, well, we can if you want. Real life? What's that? ... I think writing (for me) can be like dreams in the sense that they are; life adjacent. Lots hasn't happened, some I wish happened, some things happened word for word, some to other people, some professional things that I turned personal for drama sake and vice versa, etc. Some characters are a mix of 4 people in 1. The bottom line is, it can pretty dispiriting and also hysterical what can happen to vulnerable people in a new city - and instead of mopping about it, I've made (hopefully) a relatable comedy that can help people feel less alone and have a giggle - that's the goal.
I find contradiction one of the most compelling things to watch, and this city of cloud-less skies and dreamers holds some beautifully messed up stories - that's not a new discovery, but LAX was my take on some of it.
One thing I've found interesting is LA (or many big cities) is not usually a place people come to see or to love, they often come to be seen or be loved. That imbalance can make good stories.
Can you tell us anything about what you are working on next?
I have two features and a series in development right now - one of which will be shot in Egypt. I've also just been cast in a film that I'm excited about ... this is a bit secret and intentionally vague - apologies! I'm also working on a book for actors on camera technique. Mainly, I'd love humans to watch LAX right now...watch it and hire me - thanks! :)
Finally, one of the characters in the show says, "You should never come to Hollywood alone." Who would you most like to share a flight to LAX with? And why?
Other than some of the lovely humans in my immediate life ... 3 hypothetical situation people right now come to mind (sorry I'm cheating and using 3). I realize in advance that I will be talking about these incredibly accomplished humans like I'm their equal, but you know, a gal can dream...
Sue Perkins - to laugh and bond about dogs, food, life, comedy - human observations. Jill Soloway - to just praise her really, then explain to her over tomato juice (which I only drink on planes) the ridiculously parallels (I like to think) in our life/taste/art experience.
Ira Glass - to see one another ... I cannot tell you how much I admire that man... his heart, his brain, his efficient organization - I sense he might be one of the only people I wouldn't feel alone sitting next to on a plane, because I think I would see him and he would see me, instead of any other BS and isn't that nice? He just gets humans, and asks the right questions - he's king of story.
Is my adoration of these 3 people creepy enough? You asked! :)
Oh oh wait, and Louis C.K., and Mike Mills, ooo ooo and the Duplass Brothers and Sarah Silverman - how many seats am I allowed??
As many as you want with this hit show Sarah! You can watch the entire series here lovelies: https://www.youtube.com/channel/UC4pFxdO1_TVvU529UHmhahg
Your Monday Isn't As Bad As Those In These New Annabelle: Creation Clips!
Annabelle Creation 11:39:00
And how are you all? Well, I hope! We have started a new week today and I know Monday's are a right nightmare for many of us, so I thought I would be ironic and give you all some spooky clips to start the day!
Fresh from Annabelle: Creation, these clips will make this Monday look like a breeze...
And now to make your Monday all cute again with this new trailer from The Jungle Bunch and more lovelies...
The Lost City Of Z DVD Review
Charlie Hunnam is going in a very different direction in all his latest roles. And whilst King Arthur may not have been his finest, his turn in The Lost City of Z makes up for it. Playing British explorer Percy Fawcett, Hunnam joins Robert Pattinson, Tom Holland and Sienna Miller to bring to life his character's story so well, you will think you are in a history lesson.
Based on the best-selling book, the film focuses on Fawcett's life and his journeys into the jungle. Heading into places many would fear to go during the 1900s, Fawcett shared with his wife (Miller), son (Holland) and advisor (Pattinson) the worlds that he uncovered, even when those around him ridicule him for the things he states upon returning.
Director James Gray has used his camera to make it seem like every moment is intimate. Like you are discovering the new worlds at the same time as the characters on screen are. His trust in his actors pays off throughout and he knows how to bring the book to life, with detailed shots of jungle life.
Hunnam and Pattinson are the real stands out in the film. Together they share a passion for getting the adaptation just right. They fit the characters well and make for easy watching (not just because they are both good looking), as they take their skills and apply them well. Holland and Miller, however, are a bit let down within the story.
Whilst Miller brings strength to her character, the focus on Hunnam makes audiences forget about her until she returns on screen. As for Holland, it would have been incredible to see him even more within the story, to allow for a deeper relationship to be shown between his character and Hunnam's.
When placed all together though and under the eye of Gray, the film does something that many would properly not have expected it to have done. It works and it works superbly well. The cast clicks through their determination and the books detail has not been forgotten throughout creation.
Resident Evil: Vendetta DVD Review
Resident Evil has been one of the most successful film franchises of all time and now, after recently ending, it has had a bit of a makeover. Out on DVD and Blu-ray as of now, Resident Evil: Vendetta is an animated version of the film, which hooks you into the world of the fame a little bit more than the live-action films do.
With animations that reflect those of the world-famous game, viewers are treated to the story of returning characters Chris Redfield and Leon S.Kennedy, as well as new face Rebecca Chambers, as they work to stop a deadline virus ruining New York.
Director Takanori Tsujimoto's feature animation has a story of vengeance but also one of love for the gamers who made this story possible. The creation of such a plot from the game could have only come from the mind of someone who knows what is expected by other fans. Information they would have only known by being fans of the games and franchise itself.
Whilst the story isn't so brutal, it is full of action and beautiful sequences that some could say was lost in the live-action stories towards the end. The amount of care for detail is key here. However, it is easy at times to let the characters run a little wild because of this.
Filled to the brim with added special features, the filmmakers make sure that you miss no part in the process of bringing the story from gaming screen to the big screen. Talking through early sketches to the way in which motion capture was used in the film, the creators have allowed audiences to see more of the thoughts that go into the filmmaking process than the films before did. It makes the world of the game and the place this story came to life seem much more valuable.
Marvel Present New Comic Con Thor: Ragnarok Starring A Lot Of Hulk!
As many of you will now know I am a huge fan of Thor! Well maybe Chris Hemsworth, but for the sake of this post let's just say Thor aha! After taking to the panel yesterday at Comic Con, the cast and crew presented the first full trailer for Thor: Ragnarok and let's just say my body is ready!
Loki, Hulk, Thor and a whole host of fresh faces have made up this Marvel masterpiece…
Plus as an extra treat lovelies, check out this new poster for Black Panther which was also unveiled at the event...

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Educating Greater Manchester, review: an intriguing primer for a looming career crisis
A standard issue entry in this always enjoyable series, but the juicy stuff is still to come...
By Gabriel Tate 3 November 2020 • 10:16pm
Channel 4 returned to Salford's Harrop Fold School where Drew Povey was head teacher
"Harrop Fold School is now under new leadership." The real story behind Educating Greater Manchester (Channel 4) came at the end of its opening episode.
Returning for retakes might have suggested a lack of imagination, but this second visit to the Salford secondary reaped unexpected rewards in late 2018 when it was put into special measures and its head, Drew Povey, was suspended for allegedly falsifying attendance records.
Povey subsequently resigned, claiming to have been the victim of a "personal vendetta". All this lies in the future; in the meantime, I couldn't help scrutinising footage for evidence of a school on the precipice. Largely to no avail, it has to be said.
This was business as usual for a series which has long juggled gritty reality with heart-wrenching personal stories and knockabout playground silliness, somehow tying it all up into a neatly inspirational package by the end of the hour.
We saw the challenging behaviour of Jacob from Year 7 – munching sweets in class, swearing in Spanish – put in the context of his dyslexia, and his attitude transformed by a Povey pep talk where the latter revealed he, too, was dyslexic.
"[I feel] better, knowing someone with dyslexia has gone all the way to the top," Jacob remarked, adding with killer timing, "Well, sort of."
Harrop Fold School pupil Jacob
Katelyn, familiar from the previous series, was still struggling to focus on her studies with newly diagnosed ADHD, but Povey stepped in (again) to inspire her to pursue studying to be a midwife by taking responsibility for her actions.
A knife scare was traced to one of the school's leading lights who had brought in a trick knife and (on Povey's recommendation) redeemed himself by posting a cautionary YouTube video. The only Povey-free zone was the crackdown on sales of sweets, where kids raking in £25 a day could have taught The Wire's corner king Bodie a thing or two; the staff, meanwhile, drew tactics straight from the Jimmy McNulty school of intel gathering. It was great fun.
On this evidence, Povey's pastoral methods were hard to fault, some excruciating banter about Hitler and diarrhoea aside. So what went wrong? Did something get lost in the edit?
His journey from here to "influential leadership authority with a unique multi-sector viewpoint on creating innovative and sustainable change" (according to the profile on his website for his current role as a coach and mentor) will be an intriguing one.
Channel 4,
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Image credit: NASA The invasion of Mars continues. Next up: NASA's Mars Explorer Rover-A which has been cleared to launch Sunday, June 8 at 1805 GMT (2:05pm EDT). Rover-A and its twin carry a suite of geological instruments to examine rocks and soil searching for evidence of past water on Mars. Scientists back on Earth … Continue reading "NASA Ready for Twin Rover Launch"