text:
\section{Introduction}

Solving the Helmholtz equation numerically is a difficult task, especially when dealing with high-frequency regimes, heterogeneous media or reflecting boundary conditions. Over the last decades a lot of effort and progress has been made in developing efficient algorithms to solve the ill-conditioned linear system resulting from the Helmholtz operator's discretization. Domain decomposition methods (DDM) try to overcome these difficulties. They are hybrid methods that combine direct solvers in subdomains and iterative matching of the solutions across the subdomains. The original domain decomposition method introduced by Schwarz~\cite{Schwarz:1870:UGA} only works for overlapping domain decomposition. P. L. Lions~\cite{Lions:1988:SAM} introduced a new variant of this algorithm where the Dirichlet interface conditions are replaced by Robin interface conditions, his method can be applied to both overlapping and nonoverlapping subdomains. He showed convergence for the elliptic case and the proof was extended by Despr\'es~\cite{Despres:1991:DDM} to the Helmholtz equation and later on to the time-harmonic Maxwell equations~\cite{Despres:1992:ADD}. It is known that the presence of overlaps helps to speed up the convergence, however nonoverlapping based methods are often used to avoid to deal with the construction of the normal derivative of the solution. More recently, sweeping-type domain decomposition methods  have been made popular due to their capability to achieve nearly-linear asymptotic complexity. A sweeping algorithm was first proposed and analyzed  in~\cite{Nataf:1997:CRO} for convection-diffusion operators. Sweeping approaches for Helmholtz problems have recently seen its interest renewed as a preconditionner to speed up the convergence of the solver: the double sweep preconditionner of Vion and Geuzaine for non overlapping decomposition with high order interface conditions  \cite{vion_double_2014,vion_parallel_2016}, the PML-based sweep method of Stolk \cite{c_stolk_rapidly_2013}, and the polarized traces method of Zepeda-N\'u\~nez and Demanet \cite{zepeda-nunez_method_2016}. There also exists sweeping-type methods that are not domain decomposition based methods, such as the sweeping PML preconditionner of Engquist and Ying \cite{engquist_sweeping_2011,engquist_sweeping_2011-1}, the source transfer method~\cite{Chen:2013:source} , see~\cite{gander:2019:class} for a complete panorama and relations between these methods.\\

For a decomposition of the domain into layers and when equipping the local subproblems with exact absorbing boundary conditions (ABC), the total number of iterations is equal to the number of subdomains, this is due to the nilpotency of the iteration operator valued matrix. In practice, the exact ABC (which are also the optimal interface conditions, see \cite{Nataf:1994:OIC}) procedure is tedious to implement and computationally expensive. On the other hand, for the non-exact ABC such as Robin condition, the nilpotency property is lost and convergence deteriorates. More precisely, these boundary conditions at the interfaces produce spurious reflected waves that significantly increase the number of iterations to converge, in particular for heterogeneous media and high frequency regimes. \\

We propose to precondition the discrete Helmholtz system by an overlapping splitting double-sweep algorithm that allows for overlapping subdomains and prevents spurious interface reflections from hindering the convergence. Enabling overlapping subdomains enables to leverage its beneficial effect on the damping of high frequency modes~(see e.g.~\cite{Dolean:2015:IDDSiam}, \S~2.2.) of the error whereas the splitting prevents its adversary effect on the convergence of propagative modes. This is useful since in the non-overlapping approach~\cite{vion_double_2014,vion_parallel_2016}, the quality of the ABC is nearly the only way of impacting the convergence of the algorithm, and when dealing with more complex problem such as Maxwell equations high order ABCs are harder to handle.\\

Note that in double sweep algorithms only one or two subdomain solves are done concurrently. In order to avoid having idle processes, in \cite{vion_parallel_2016} it is proposed to pipeline the algorithm with respect to multiple right hand sides. This is useful in seismic inversion  problems or in far-field pattern computations.\\

We will first state the problem in \S~\ref{sec:problem_setting}. Then we explain in \S~\ref{sec:sds} how we substructure the Helmholtz problem and formulate a modified version of the double sweep algorithms introduced in \cite{Nataf:1997:CRO,vion_double_2014,vion_parallel_2016}, then how we use this overlapping splitting double sweep algorithm to build a preconditionner that efficiently speed up the convergence of the solver. In~\S~\ref{sec:convergence_rates}, the convergence of the algorithm is studied when it is applied to a decomposition of the plane into vertical strips. In \S~\ref{sec:resnum}, we present numerical results for the original problem, i.e. the Helmholtz problem.\\

\section{Statement of the problem and some algorithms}
\label{sec:problem_setting}

We consider the Helmholtz equation in a bounded domain $\Omega \subset \mathbb{R}^{2}$ with frequency $\omega$, velocity $c$ and wavenumber $k$ defined by $k^{2} = \omega^{2}/c^{2}$:
\begin{equation}
\label{helmholtz_equation}
\begin{array}{l}
\left(- k^2 - \Delta\right) u = f \text{ in }\Omega\\
	+ \text{ appropriate boundary conditions on }\partial\Omega\,.
\end{array}
\end{equation}
We consider a layered decomposition of $\Omega$ into $N$ slices $(\Omega_i)_{1\le i \le N}$  without overlap, see Figure~\ref{fig:decomposition}. More precisely, for each $1\le i \le N$, $\Omega\setminus\Omega_i$ is written as the disjoint union of two open subsets $\Omega_{i,l}$ and $\Omega_{i,r}$ where $\Omega_{i,l}$ is on the left of $\Omega_i$ and $\Omega_{i,r}$ on its right.
The boundary $\partial\Omega_i \setminus \partial\Omega$ is written as the disjoint union
of $\Gamma_{i,l}$ and $\Gamma_{i,r}$ where $\Gamma_{i,l}$ is on the
left of $\Omega_i$ and $\Gamma_{i,r}$ is on its right
($\Omega_{1,l}=\emptyset$ and $\Omega_{N,r}=\emptyset$) (see Fig.~\ref{fig:subdomain}).
The outward normal from $\Omega_i$ on $\Gamma_{i,l}$
(resp. $\Gamma_{i,r}$) is denoted by $\vec n_{i,l}$ (resp.
$\vec n_{i,r}$).
\begin{figure}[ht!]
  \centering
  \includegraphics[width=0.6\textwidth]{./figure_subdomains.png}
  \caption{Decomposition into vertical strips}
  \label{fig:decomposition}
\end{figure}
The problem~\eqref{helmholtz_equation} can be solved iteratively using a domain decomposition method where we solve locally on each subdomain $\Omega_i$ the equation \eqref{helmholtz_equation} with appropriate boundary conditions on the physical boundaries and interfaces~\cite{Despres:1991:DDM}. The method writes:\\
 Solve in parallel:
\begin{equation}
\label{jacobi_algo_helmholtz}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) u_{i}^{n+1} &= f\ \text{ in } \Omega_{i},\ 1\le i\le N \\
\mathcal{B}_{i,l}\left(u_{i}^{n+1}\right) &= \mathcal{B}_{i,l}\left(u_{i-1}^{n}\right) \ \text{on} \ \Gamma_{i,l},\ 2\le i\le N \\
\mathcal{B}_{i,r}\left(u_{i}^{n+1}\right) &= \mathcal{B}_{i,r}\left(u_{i+1}^{n}\right) \ \text{on} \ \Gamma_{i,r},\ 1\le i\le N-1 \\
&+ \text{ appropriate boundary conditions on }\partial\Omega\cap \partial\Omega_i\,.
\end{aligned}
\right.
\end{equation}
where $\mathcal{B}_{i,l}$ and $ \mathcal{B}_{i,r}$ are the interface conditions. For sake of simplicity, we consider first-order ABC as interface conditions:
\begin{equation}
	\label{eq:IBC0}
\left\{
\begin{aligned}
\mathcal{B}_{i,l} &= \partial_{\vec{n}_{i,l}} + Ik  \\
\mathcal{B}_{i,r} &= \partial_{\vec{n}_{i,r}} + Ik
\end{aligned}
\right.
\end{equation}
where $I^2=-1$ and $\vec{n}_{i,r}$ (resp. $\vec{n}_{i,l}$) is the outward normal to domain $\Omega_i$ on $\Gamma_{i,r}$ (resp. $\Gamma_{i,l}$). It is known that higher-order ABC lead to significative improve of the convergence speed, see e.g. ~\cite{Gander:2001:OSH,Antoine:12:QON}.\\

 A more efficient variant of algorithm~\ref{jacobi_algo_helmholtz} was introduced in~\cite{Nataf:1997:CRO}. It consists in double sweeps over the subdomains:\\
{\bf Left to right sweep:}
\begin{equation}
\label{leftsweep_algo_helmholtz}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) u_{i}^{n+1/2} &= f\ \text{ in } \Omega_{i},\ 1\le i\le N \\
\mathcal{B}_{i,l}\left(u_{i}^{n+1/2}\right) &= \mathcal{B}_{i,l}\left(u_{i-1}^{n+1/2}\right) \ \text{on} \ \Gamma_{i,l},\ 2\le i\le N \\
\mathcal{B}_{i,r}\left(u_{i}^{n+1/2}\right) &= \mathcal{B}_{i,r}\left(u_{i+1}^{n}\right) \ \text{on} \ \Gamma_{i,r},\ 1\le i\le N-1 \\
&+ \text{ appropriate boundary conditions on }\partial\Omega\cap \partial\Omega_i\,.
\end{aligned}
\right.
\end{equation}
{\bf Right to left sweep:}
\begin{equation}
\label{rightsweep_algo_helmholtz}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) u_{i}^{n+1} &= f\ \text{ in } \Omega_{i},\ 1\le i\le N \\
\mathcal{B}_{i,l}\left(u_{i}^{n+1}\right) &= \mathcal{B}_{i,l}\left(u_{i-1}^{n+1/2}\right) \ \text{on} \ \Gamma_{i,l},\ 2\le i\le N \\
\mathcal{B}_{i,r}\left(u_{i}^{n+1}\right) &= \mathcal{B}_{i,r}\left(u_{i+1}^{n+1}\right) \ \text{on} \ \Gamma_{i,r},\ 1\le i\le N-1 \\
&+ \text{ appropriate boundary conditions on }\partial\Omega\cap \partial\Omega_i\,.
\end{aligned}
\right.
\end{equation}

\section{Overlapping Splitting double sweep}
	\label{sec:sds}

In this section, we define a variant of algorithm~\eqref{leftsweep_algo_helmholtz}-\eqref{rightsweep_algo_helmholtz} which has a superior convergence. Numerical results will show that it benefits better from the overlap and have a better parallelism. This algorithm is written in terms of the substructured problem that we define first.

\subsection{Substructuring}
\label{sub:substructuring}

Substructuring the algorithm~\eqref{jacobi_algo_helmholtz}, the iterative method can be reformulated considering only surfacic unknowns on the interfaces:
\begin{equation}
\label{hl_hr_reformulation}
\left\{
\begin{aligned}
h^{n}_{i,l} &\vcentcolon = \mathcal{B}_{i,l}\left(u^{n}_{i}\right), \ \text{ on }\ \Gamma_{i,l} \ \ \text{ for } \ 2\le i\le N \\
h^{n}_{i,r} &\vcentcolon =  \mathcal{B}_{i,r}\left(u^{n}_{i}\right), \ \text{ on }\ \Gamma_{i,r} \ \ \text{ for } \ 1\le i\le N-1\,.
\end{aligned}
\right.
\end{equation}
Considering the global vector $h^n$ containing the local unknowns $(h_{i,l}^{n})_{2\le i \le N}$ and $(h_{i,r}^{n})_{1\le i \le N-1}$, we can reformulate the additive Schwarz method~\eqref{jacobi_algo_helmholtz} as a Jacobi algorithm on $h^n$:
\begin{equation}
\label{global_jacobi}
h^{n+1} \vcentcolon= \mathscr{T}(h^{n}) + G
\end{equation}
where the iteration operator $\mathscr{T}$ can be written in the form of an operator valued matrix and $G$ refers to the contribution of the right-hand side $f$, see~\cite{Nataf:1994:OIC}. Therefore, we look for a vector $h$ such that,
\begin{equation}
\label{global_substructuring}
\left( Id - \mathscr{T}\right)(h) = G\,.
\end{equation}
In order to define more precisely the operator $\mathscr{T}$,
 we introduce for each subdomain an operator $S_{i}$ which takes three arguments, two surfacic functions $h_{l}$ and $ h_{r}$ and a volume function $f$:
 \begin{equation}
	 \label{eq:Si}
 S_i(h_{l}, h_{r}, f) \vcentcolon= v
 \end{equation}
 where $v:\,\Omega_i \mapsto \mathbb{C}$ satisfies:
\begin{equation}
\label{operator_S_helmholtz}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) v &= f \ \ \text{in} \ \Omega_{i} \\
\mathcal{B}_{i,l}(v) &= h_{l} \ \text{on} \ \Gamma_{i,l} \quad \left(2\le i \le N\right)\\
\mathcal{B}_{i,r}(v) &= h_{r} \ \text{on} \ \Gamma_{i,r} \quad \left(1\le i \le N-1\right)\\
&+ \text{ appropriate boundary conditions on }\partial\Omega\cap \partial\Omega_i\,,
\end{aligned}
\right.
\end{equation}
for  $1 < i < N$. For $i=1$, the definition of $S_1$ is similar except that it takes only the two arguments $(h_r,f)$ since domain $\Omega_1$ has no left interface and similarly operator $S_N$ takes only the two arguments $(h_l,f)$ since domain $\Omega_N$ has no right interface. {\bf As of now, for sake of simplicity and by abuse of notation, $S_1(h_l,\ h_r,\ f )$ $(\text{resp. } S_N(h_l,\ h_r,\ f))$  will refer to $S_1(h_r,f)$ $(\text{resp. } S_N(h_l,\ f))$. }

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{./Local_problem_figure.pdf}
\caption{Local problem on the subdomain $\Omega_{i}
     }
\label{fig:subdomain}
\end{figure}
Next, we introduce the surfacic right hand-side $G$ by
\begin{equation}
	\label{eq:Gglobal}
	\begin{array}{rcl}
G_{i+1,l} &\vcentcolon =& \mathcal{B}_{i+1, l}(S_i(0 , 0 , \ f)),\ \ 1\le i \le N-1 \\
G_{i-1,r}  &\vcentcolon =&  \mathcal{B}_{i-1, r}(S_i(0 , 0 , \ f)),\ \ 2\le i \le N\,.
	\end{array}
\end{equation}
and the substructured operator $\mathscr{T}$ by:
\begin{equation}
	\begin{array}{rcl}
\mathscr{T}(h)_{i+1,l} &\vcentcolon =& \mathcal{B}_{i+1, l}(S_i(h_{i,l},\ h_{i,r}, 0)),\ \ 1\le i \le N-1 \\
\mathscr{T}(h)_{i-1,r}  &\vcentcolon =&  \mathcal{B}_{i-1, r}(S_i(h_{i,l},\ h_{i,r}, 0)),\ \ 2\le i \le N\,.
	\end{array}
\end{equation}

Using this framework, we write the substructured form of the {\bf double sweep algorithm} as follows:\\
{\bf Forward sweep}\\
\begin{equation}
	\label{eq:DSalgosubstructuredForward}
	\begin{array}{rcl}
h_{i+1,l}^{n+1/2} &\vcentcolon =& \mathcal{B}_{i+1, l}(S_i(h_{i,l}^{n+1/2},\ h_{i,r}^n, f))\,,\\
h_{i-1,r}^{n+1/2}  &\vcentcolon =&  \mathcal{B}_{i-1, r}(S_i(h_{i,l}^{n+1/2},\ h_{i,r}^n, f))\,,
	\end{array}
\end{equation}
{\bf followed by a Backward sweep}\\
\begin{equation}
	\label{eq:DSalgosubstructuredBackward}
	\begin{array}{rcl}
h_{i+1,l}^{n+1} &\vcentcolon =& \mathcal{B}_{i+1, l}(S_i(h_{i,l}^{n+1/2},\ h_{i,r}^{n+1}, f))\,,\\
h_{i-1,r}^{n+1}  &\vcentcolon =&  \mathcal{B}_{i-1, r}(S_i(h_{i,l}^{n+1/2},\ h_{i,r}^{n+1}, f))\,.
	\end{array}
\end{equation}

\subsection{Overlapping Splitting Double Sweep preconditioner (OSDS)}
\label{sub:splitted_double_sweep_preconditioner}

We explain now the rationale behind the overlapping splitting double sweep preconditioner that we define in this section. Note first that by linearity of the operators $(S_i)_{1\le i\le N}$, the contribution of each subdomain can be split into two contributions, one for each of its two interfaces:
\begin{equation}
	\begin{array}{rcl}
\mathscr{T}(h)_{i+1,l} &=& \mathcal{B}_{i+1, l}(S_i(h_{i,l},\ 0 ,\ 0))\ \ +\ \mathcal{B}_{i+1, l}(S_i( 0 , \ h_{i,r} ,\ 0 )),\ \ 1\le i \le N-1 \\
\mathscr{T}(h)_{i-1,r}  & =& \mathcal{B}_{i-1, r}(S_i( 0 ,\ h_{i,r},\ 0))\ +\ \mathcal{B}_{i-1, r}(S_i(h_{i,l}, \ 0 ,\ 0)),\ \ 2\le i \le N
	\end{array}
\end{equation}
Had we used exact absorbing boundary conditions (EABC) $\mathcal{B}^{EABC}$ instead of the zero-th order ones~\eqref{eq:IBC0}  in equations~\eqref{eq:Si}-\eqref{operator_S_helmholtz}, we would have with obvious abuse of notation:
\begin{equation}
	\begin{array}{rcl}
 \mathcal{B}^{EABC}_{i+1, l}(S^{EABC}_i(0,\ h_{i,r}, 0)) &=& 0,\ \ 1\le i \le N-1 \\
\mathcal{B}^{EABC}_{i-1, r}(S^{EABC}_i(h_{i,l},0, 0)) &=& 0,\ \ 2\le i \le N\,.
	\end{array}
\end{equation}
Then, the operator would have the following form
\begin{equation}
	\begin{array}{rcl}
{\mathscr{T}^{EABC}}(h)_{i+1,l} & =& \mathcal{B}^{EABC}_{i+1, l}(S^{EABC}_i(h_{i,l} , 0 , 0)),\ \ 1\le i \le N-1\,, \\
{\mathscr{T}^{EABC}}(h)_{i-1,r}  & =&  \mathcal{B}^{EABC}_{i-1, r}(S^{EABC}_i( 0 ,\ h_{i,r}, 0)),\ \ 2\le i \le N\,.
	\end{array}
\end{equation}
The operator $\mathscr{T}^{EABC}$ is a nilpotent operator of order $N-1$ so that the inverse of $I-\mathscr{T}^{EABC}$ is easy to compute as a finite Neumann series:
\[
  (I-\mathscr{T}^{EABC})^{-1} = \sum_{i=0}^{N-2} (\mathscr{T}^{EABC})^i\,.
\]
This formula induces a convergence in $N-1$ steps of a domain decomposition method with exact absorbing boundary conditions. In practice, the absorbing boundary conditions are non exact, therefore we have
\begin{equation}
	\begin{array}{rcl}
 \mathcal{B}_{i+1, l}(S_i(0,\ h_{i,r}, 0)) &\neq& 0,\ \ 1\le i \le N-1\,, \\
\mathcal{B}_{i-1, r}(S_i(h_{i,l},0, 0)) &\neq& 0,\ \ 2\le i \le N\,,
	\end{array}
\end{equation}
and we loose the nilpotency property of $\mathscr{T}$. This led us to define a new
 operator ${\mathscr{T}_{OSDS}}$
\begin{equation}
	\label{eq:TsdsDefinition}
	\begin{array}{rcl}
{\mathscr{T}_{OSDS}}(h)_{i+1,l} &\vcentcolon =& \mathcal{B}_{i+1, l}(S_i(h_{i,l} , 0 , 0)),\ \ 1\le i \le N-1 \\
{\mathscr{T}_{OSDS}}(h)_{i-1,r}  &\vcentcolon =&  \mathcal{B}_{i-1, r}(S_i( 0 ,\ h_{i,r}, 0)),\ \ 2\le i \le N
	\end{array}
\end{equation}
which by definition is a nilpotent operator of order $N-1$. We propose to use this newly defined operator to build a preconditioner for \eqref{global_substructuring}. The left-preconditioned system reads:
\begin{equation}
\label{precondglobal_substructuring}
( Id - {\mathscr{T}_{OSDS}}\, )^{-1} \left( Id - \mathscr{T}\right)(h) = ( Id - {\mathscr{T}_{OSDS}}\, )^{-1}G
\end{equation}
More intuitively, the key idea is to cancel out the reverse contribution at the interfaces that should not happen for the exact ABC case in order to prevent spurious interface reflections from hindering the convergence. In fact, these boundary conditions at the interfaces produce spurious reflected waves that significantly increase the number of iterations to converge, in particular for heterogeneous media and high frequency regimes. Note that for a non overlapping domain decomposition, the OSDS algorithm is similar to the double sweep method of~\cite{vion_double_2014,vion_parallel_2016}. Our approach addresses the case of overlapping subdomains that benefits the convergence rate.

 \section{Convergence rates}
 \label{sec:convergence_rates}

We first recall the convergence rates proved in~\cite{Nataf:1997:CRO} for the fixed point method related to the Jacobi and double sweep methods for the convection-diffusion equation and sketch how a similar estimate is derived for the OSDS algorithm. Then, we consider the Helmholtz equation.

It was proved in Theorem~2.4 of~\cite{Nataf:1997:CRO} that under some geometric assumptions the fixed point algorithms achieve geometric convergence with the following estimates:\\

Jacobi method~\eqref{hl_hr_reformulation}:

\begin{equation}
	\label{eq:cvEstimateGlobalJacobi}
	||u_i^n-u||_{H^2_{st}(\Omega_i)}\le C \rho^{[{n\over 2(N-1)}]}
	\sup_j ||u_j^0-u||_{H^2_{st}(\Omega_j)} \ \ n\ge 2N+1 \,,
\end{equation}

Double sweep algoritm~\eqref{eq:DSalgosubstructuredForward}-\eqref{eq:DSalgosubstructuredBackward}:

\begin{equation}
	\label{eq:cvEstimateGlobalDS}
||u_i^n-u||_{H^2_{st}(\Omega_i)}\le C \rho^n
\sup_j ||u_j^0-u||_{H^2_{st}(\Omega_j)}\ \  n\ge 3 \,,
\end{equation}
 (writing $[.]$ for the integer part)  where $C$ and $\rho$
are constants, $C>0$, $0<\rho<1$ and $||\,\cdot\, ||_{H^2_{st}}$ denotes some Sobolev norm. \\

Under the same assumptions, it can be checked that for $n$ even, we have
\begin{equation}
	\label{eq:cvEstimateGlobalOSDS}
||u_i^n-u||_{H^2_{st}(\Omega_i)}\le 2\, {\sqrt{\rho}}^{\,n}
\sup_j ||u_j^0-u||_{H^2_{st}(\Omega_j)}\ \   n\ge 2\,,
\end{equation}
 for the fixed point method related to the overlapping splitting double sweep defined as follows:
 \begin{equation}
 \label{eq:fixedpointOSDS}
 ( Id - {\mathscr{T}}_{OSDS}\, )(h^{n+1}) = (\mathscr{T}-\mathscr{T}_{OSDS})(h^n)+G\,.
 \end{equation}

Note that the expression for $\rho$ (not shown here) is the same for all these estimates. The differences lie in the exponent of $\rho$ and in the constant. The constant $C$ in eq.~\eqref{eq:cvEstimateGlobalDS} may be large (see estimate~\eqref{eq:C} below) whereas in eq.~\eqref{eq:cvEstimateGlobalOSDS} its value is simply $2$. A seemingly advantage of \eqref{eq:cvEstimateGlobalDS} with respect to \eqref{eq:cvEstimateGlobalOSDS} is the square root of $\rho$ ratio in the convergence rate. In fact this difference is nullified when you realize that an iteration of DS takes twice as much time as an iteration of OSDS. Indeed, during one iteration the DS algorithm, only one subdomain is active at a time in the order $1\rightarrow 2 \rightarrow \ldots \rightarrow N$ and then in the reverse order $N \rightarrow N-1 \rightarrow \ldots \rightarrow 1$. Whereas during one iteration of OSDS, two subdomains are active at a time in the order $1\rightarrow 2 \rightarrow \ldots \rightarrow N$ for one subdomain and concurrently in the reverse order $N \rightarrow N-1 \rightarrow \ldots \rightarrow 1$ for the other one. Thus, the elapsed time of one iteration of DS is thus twice as much time as one iteration of OSDS using two cores. Note that using two cores for the DS algorithm brings no advantage in terms of elapsed time. So overall, these estimates give an edge to OSDS over DS in terms of the constants in formulas~\eqref{eq:cvEstimateGlobalDS} and \eqref{eq:cvEstimateGlobalOSDS}.\\

We give a few hints on the proof of~\eqref{eq:cvEstimateGlobalOSDS}. The convergence of OSDS depends on the spectral norm of
\begin{equation}
\label{eq:TfixedpointOSDS}
R_{OSDS} := ( Id - {\mathscr{T}}_{OSDS}\, )^{-1}(\mathscr{T}-\mathscr{T}_{OSDS})\,.
\end{equation}
In order to estimate it, we first recall the notations and results from~\cite{Nataf:1997:CRO}. The operator ${\mathscr T}$ can be written in the form of an
operator  valued matrix
$$ {\mathscr T}(H)=
\left[ \begin{array}{cc}
\begin{array}{cccc}
   0&\ &\ &0\\
   \times&\ddots&\ &\ \\
   \ &\ddots&\ddots&\ \\
   0&\ &\times&0
   \end{array}&
\begin{array}{cccc}
  \times&\ &\ &0\\
   \ &\ddots&\ &\ \\
   \ &\ &\ddots&\ \\
   0&\ &\ &\times
   \end{array}\\
\begin{array}{cccc}
  \times&\ &\ &0\\
   \ &\ddots&\ &\ \\
   \ &\ &\ddots&\ \\
   0&\ &\ &\times
   \end{array} &  \begin{array}{cccc}
   0&\times&\ &0\\
   \ &\ddots&\ddots&\ \\
   \ &\ &\ddots&\times\\
   0&\ &\ &0
  \end{array}
\end{array}\right]\
\left[\begin{array}{c} h_{2,l}\\ \vdots \\ \vdots \\ h_{N,l}\\
h_{1,r}\\ \vdots\\ \vdots \\ h_{N-1,r}
\end{array}\right]$$ where the crosses correspond to non zero
operators.

 In order
to write them in a compact form, we  introduce four $(2N-2)\times
(2N-2)$ operator valued matrices:
$$ {\mathscr M}_l=({\mathscr M}_{l,mn})_{1\le m,\,n\le 2N-2}\hbox{ with } {\mathscr
M}_{l,mn}=\left\{\begin{array}{cl}
                {\mathscr T}_{mn}&1\le m,\,n\le N-1\\
                0&\hbox{otherwise}
               \end{array}\right.\hfill
$$
$$ {\mathscr A}_l=({\mathscr A}_{l,mn})_{1\le m,\,n\le 2N-2}\hbox{ with } {\mathscr
A}_{l,mn}=\left\{\begin{array}{cl}
                {\mathscr T}_{mn}&1\le m\le N-1,\ N\le n\le 2N-2\\
                0&\hbox{otherwise}
               \end{array}\right.\hfill
$$
$$ {\mathscr A}_r=({\mathscr A}_{r,mn})_{1\le m,\,n\le 2N-2}\hbox{ with } {\mathscr
A}_{r,mn}=\left\{\begin{array}{cl}
                {\mathscr T}_{mn}&N\le m\le 2N-2,\ 1\le n\le N-1\\
                0&\hbox{otherwise}
               \end{array}\right.\hfill
$$
$$ {\mathscr M}_r=({\mathscr M}_{r,mn})_{1\le m,\,n\le 2N-2}\hbox{ with } {\mathscr
M}r_{mn}=\left\{\begin{array}{cl}
                {\mathscr T}_{mn}&N\le m,\,n\le 2N-2\\
                0&\hbox{otherwise}
               \end{array}\right.\hfill
$$ so that we have ${\mathscr T}={\mathscr M}_l+{\mathscr A}_l+{\mathscr M}_r+{\mathscr A}_r$. We recall relations~(23) page~369 in~\cite{Nataf:1997:CRO}:
\begin{equation}
\begin{array}{lll}
{\mathscr M}_r^{N-1}={\mathscr M}_l^{N-1}=0;&{\mathscr M}_l\,{\mathscr M}_r={\mathscr M}_r\,{\mathscr M}_l=0;&{\mathscr A}_l^2={\mathscr A}_r^2=0\\
{\mathscr A}_l\,{\mathscr M}_l={\mathscr A}_r\,{\mathscr M}_r=0;&{\mathscr M}_l\,{\mathscr A}_r={\mathscr M}_r\,{\mathscr A}_l=0&\ \ \ \\
\end{array}\label{eq:tabref}
\end{equation}
It is worth noticing that these relations come from
the structure of the matrices and do not depend on the value of the
coefficients.\\

Using these notations and the nilpotency of ${\mathscr M}_l$ and ${\mathscr M}_r$, it can be checked that $R_{OSDS}$ has the following expression:
\[
R_{OSDS} = \sum_{i=0}^{N-2} {\mathscr M}_r^i\,{\mathscr A}_r + \sum_{i=0}^{N-2} {\mathscr M}_l^i\,{\mathscr A}_l\,.
\]
Then using cancellation relations~\eqref{eq:tabref}, it can be checked that for $n$ even:
\[
R_{OSDS}^n = \left( \sum_{i=0}^{N-2} {\mathscr M}_r^i\,{\mathscr A}_r  \sum_{i=0}^{N-2} {\mathscr M}_l^i\,{\mathscr A}_l\right)^{n/2}   + \left( \sum_{i=0}^{N-2} {\mathscr M}_l^i\,{\mathscr A}_l  \sum_{i=0}^{N-2} {\mathscr M}_r^i\,{\mathscr A}_r\right)^{n/2} \,.
\]
The estimate follows from the formula for $\rho$ given in Theorem~5.1. of~\cite{Nataf:1997:CRO}.

\subsection{Application to Helmholtz equation}
\label{sub:application_to_helmholtz_equation}

The above formula are obtained by algebraic means and remain thus valid if the equation to be solved is the Helmholtz equation in place of the convection-diffusion equation.

In this section, as in~\cite{Nataf:1997:CRO}, the space $\mathbb{R}^2$ is decomposed into $N$ vertical strips in the following manner: Let $\Omega_i=]l_i,\,L_i[\times \mathbb{R}$, $1\le i\le N$ with $-\infty=l_1<l_2\le L_1<\ldots<L_{i-2}<l_i\le L_{i-1} <\ldots<L_N=+\infty$, we have $\mathbb{R}^2=\cup_{i=1}^N\bar\Omega_i$. This simple decomposition into straight strips makes it possible to use the Fourier transform along the interfaces (the dual variable of $y$ is denoted by $\xi$). It is then possible to compute the Fourier symbols of the operators of the previous section. They will be expressed in function of the symbols of the Dirichlet-to-Neumann map $\lambda(\xi)$ of a half space:\\
for propagative modes ($|\xi|<|k|$):
\[
 \lambda(\xi) := I k \sqrt{1 - \xi^2/k^2}
\]
and for vanishing modes ($|\xi|>|k|$):
\[
 \lambda(\xi) := \sqrt{\xi^2 - k^2}\,,
\]
They will also be expressed in terms of $\rho_j(\xi)$ (the convergence rate for a decomposition of $\mathbb{R}^2$ into two non overlapping half spaces), an interface condition whose Fourier symbol is of the form $B:=\partial_{\vec n}+\lambda_j(\xi)$ (in this work we have simply taken $\lambda_j(\xi)=I k$):
\[
	\rho_j(\xi) := \frac{\lambda(\xi) - \lambda_j(\xi)}{\lambda(\xi) + \lambda_j(\xi)}\,,
\]
and of the width $\delta$ of the overlap between subdomains taken constant for the sake of simplicity. More specifically, we obtain the following formulas for the Fourier symbols of the different operators:
 \begin{eqnarray}
 (A_r)_{n+N-1,n}(\xi)
 &=\displaystyle
 {\rho_j(e^{-\lambda \delta}+e^{-\lambda(L_{n+1}-l_{n+1})})\over
 1-{\rho_j}^2e^{-2\lambda(L_{n+1}-l_{n+1})}},\ \ 1\le n\le
N-1\,,\nonumber\\
 (A_l)_{n,n+N-1}(\xi) &=\displaystyle
 {\rho_j(e^{-\lambda \delta}+e^{-\lambda(L_{n}-l_{n})})\over
 1-{\rho_j}^2e^{-2\lambda(L_{n}-l_{n})}},\ \ 1\le n\le
N-1\,,\nonumber\\
 (M_l)_{n+1,n}(\xi)
 &=\displaystyle
 {e^{-\lambda(L_{n+1}-l_{n+1}-\delta)}(1-{\rho_j}^2)\over
 1-{\rho_j}^2e^{-2\lambda(L_{n+1}-l_{n+1})}},\ \ 1\le n\le N-2\,,
 \label{eq:schurreform}\\
 (M_r)_{n,n+1}(\xi)
 &=\displaystyle
 {e^{-\lambda(L_{n-N+2}-l_{n-N+2}-\delta)}(1-{\rho_j}^2)\over
 1-{\rho_j}^2e^{-2\lambda(L_{n-N+2}-l_{n-N+2})}},\ \ N\le n\le 2N-3\,.
\nonumber\end{eqnarray}

The convergence factor $\rho$ is given by:
\[
	\rho(\xi) := \| A_r(\xi)\|\,\| A_l(\xi)\|\,(\sum_{i=0}^{N-2}\|M_l(\xi)\|^i)\,(\sum_{i=0}^{N-2}\|M_r(\xi)\|^i)\,,
\]
and the constant $C$ in the convergence estimates of Jacobi and DS methods, equations~\eqref{eq:cvEstimateGlobalJacobi} and \eqref{eq:cvEstimateGlobalDS} has the following form:
\[
	C(\xi) := \left(1+\frac{\rho(\xi)}{\|A_l(\xi)\|}\right)\left(1+\frac{\rho(\xi)}{\|A_r(\xi)\|}+\frac{\rho(\xi)}{\|A_r(\xi)\|\,\|A_l(\xi)\|}\right)   \,.
\]
These formula are identical to formula~(22) of~\cite{Nataf:1997:CRO} except for the formula of $\lambda$ and $\rho_j$.\\

We now give some estimates valid for the Helmholtz equation and our approximation $\lambda_j=Ik$ of the symbol $\lambda(\xi)$ of the Dirichlet to Neumann map. We differentiate between the propagative and vanishing modes starting with the former one. In order to simplify the expressions, we assume that all subdomains have a width at least equal to some number $H>0$ and the overlap is smaller than $H/2$.

\subsubsection{Propagative modes $|\xi|<|k|$}
\label{sub:propagative_modes}
The expression of the two-subdomain convergence factor $\rho_j$ is then:
\[
	\rho_j(\xi) = \frac{\sqrt{1-\xi^2/k^2}-1}{\sqrt{1+\xi^2/k^2}+1}\ \text{ and }\ |\rho_j(\xi)| < 1\,,
\]
and $\lambda(\xi)$ is a purely imaginary number so that the exponentials in formula~\eqref{eq:schurreform} are of modulus exactly $1$, so that we have the following estimates:
\[
	|A_r(\xi)_{n+N-1,n}|\, , \, |A_l(\xi)_{n,n+N-1}| \le |\rho_j(\xi)| \frac{2}{1-\rho_j^2(\xi)}\,,
\]
and
\[
	|(M_r(\xi))_{n,n+1}|\, , \, |(M_l(\xi))_{n+1,n}| \le 1\,.
\]

\vspace*{1em}

It is then easy to prove the following estimates:
\begin{equation}
	\label{eq:rhoestimateprop}
	\rho(\xi) \le \rho_j(\xi)^2 \frac{4}{(1-\rho_j^2(\xi))^2}\times (N-1)^2\,,
\end{equation}
It is interesting that in these expressions, the overlap is of no help so that convergence of the fixed point method can only be proved for small enough $|\xi/k|\le \eta_N$ where the threshold $\eta_N$ depends on the number of subdomains.

Note also that
\begin{equation}
	\label{eq:C}
	C(0) \ge (N-1)^2\,
\end{equation}
has a quadratic growth with the number of subdomains.

\subsubsection{Vanishing modes $|\xi|>|k|$}
\label{sub:vanishing_modes_xi_k}
The expression of the convergence factor $\rho$ is then:
\[
	\rho_j(\xi) = \frac{\sqrt{\xi^2-k^2}-Ik}{\sqrt{\xi^2-k^2}+Ik}
\]
whose modulus is exactly one and $\lambda(\xi)$ is a real positive number so that the following estimates hold:
\[
	|A_r(\xi)_{n+N-1,n}|\, , \, |A_l(\xi)_{n,n+N-1}| \le 2  \frac{e^{-\lambda(\xi)\,\delta}}{1-e^{-2\lambda(\xi)H}}\,,
\]
and
\[
	|(M_r(\xi))_{n,n+1}|\, , \, |(M_l(\xi))_{n+1,n}| \le 2\, \frac{e^{-\lambda(\xi)H/2}}{1-e^{-2\lambda(\xi)H}} \,.
\]
For all $\eta > 1$ and $|\xi/k|\ge\eta$, it can be proved as in~\cite{Nataf:1997:CRO} Lemma~6.1 that for a sufficiently large overlap $\delta$ we have
\begin{equation}
	\label{eq:rhoestimateevan}
|\rho(\xi)|< e^{-2\delta\,\lambda(\xi)} \,.
\end{equation}

These results show convergence for a Fourier number $\xi$ far enough from the cut-off frequency $k$ which makes sense since for $|\xi|=|k|$ even in the two-subdomain case the convergence factor is equal to one. \\

Now instead of a domain $\Omega$ equals to the plane $\mathbb{R}^2$, we take an infinite waveguide $\Omega=\mathbb{R}\times (0,L)$ for some $L>0$ with Dirichlet boundary conditions on top and bottom. Then, the above computation can be made provided that we replace the Fourier transform by Fourier series of the type
\[
	u(x,u) = \sum_{\xi \in \mathbb{N}^*} u_\xi \sin(\xi \pi \frac{y}{L})\,,
\]
For that it is sufficient to substitute, in the previous formulas, $\xi$ by $\xi\pi/L$ for $\xi \in \mathbb{N}^*$. The symbol $\lambda$ of the Dirichlet to Neumann map of the half waveguide has the following expression:
\[
	\lambda(\xi) = \sqrt{-k^2 + (\xi\pi/L)^2}\,.
\]
Thus if $kL/\pi$ is an integer (resonant waveguide) we  have $\lambda=0$ for $\xi=kL/\pi$ whereas if $kL$ is not an integer we never encounter the difficulty of having $\lambda=0$. \\

\vspace{0.5em}

The above estimates \eqref{eq:rhoestimateprop}-\eqref{eq:rhoestimateevan} on $\rho(\xi)$ and \eqref{eq:C} on $C(\xi)$ apply to the estimates of convergence for the various methods (Jacobi~\eqref{eq:cvEstimateGlobalJacobi}, Double Sweep~\eqref{eq:cvEstimateGlobalDS} and Overlapping Splitting Double Sweep~\eqref{eq:cvEstimateGlobalOSDS}) up to a change of the $H^2$ norm for a norm on the Fourier transform of the unknowns for a fixed Fourier number $\xi$.

\section{Numerical results}
	\label{sec:resnum}

In this section, we present numerical results when solving the substructured equation~\eqref{global_substructuring} with the GMRES algorithm right preconditioned by $Id$ (Jacobi method), $( Id - {\mathscr{T}_{DS}}\, )^{-1}$ (Double sweep algorithm) and $( Id - {\mathscr{T}_{OSDS}}\, )^{-1}$ (Overlapping Splitting Double sweep algorithm). The Helmholtz equation is discretized with a P1 finite element using the FreeFem++ domain specific language~\cite{Hecht:2012:NDF}. Note that we use a careful variational discretisation of the normal derivative ensuring that the solution obtained converges to the solution of the problem without domain decompositions.

\subsection{Homogeneous waveguide}

First, we consider the homogeneous waveguide test case with a layered decomposition into $N$ subdomains. More specifically, we consider a rectangular geometry $\left( \Omega = [0, N] \times [0, 1] \right)$ made of a homogeneous medium. On the upper and lower sides of the waveguide, we impose homogeneous Dirichlet conditions (cf. black lines in Figure~\ref{fig:HomogeneousSolution}). In addition, we perform a multimode excitation on the left side and we impose an absorbing boundary conditions on the right side. The global problem is written as
\begin{equation}
\label{homogeneous_problem}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) u &= f\ \text{ in } \Omega\\
\left(\partial_{\vec{n}} + Ik \right) u &= 0 \ \text{ on }  \{x=N\}\times [0, 1] \\
\left(\partial_{\vec{n}} + Ik \right) u &= u_{g} \ \text{ on }  \{x=0\}\times [0, 1]\\
u &= 0 \ \text{ on } [0, N] \times \{y=0, y=1\}
\end{aligned}
\right.
\end{equation}
where $u_g =  e^{-120(y-0.5)^2)}sin(\pi y)$.

\begin{figure}[ht!]
  \centering
  \includegraphics[width=0.8\textwidth]{./u_homog_20pi_BLACK.png}
  \caption{Homogeneous waveguide ($k=20\pi$)}
  \label{fig:HomogeneousSolution}
\end{figure}

\begin{figure

  \includegraphics[width=1.\textwidth]{./Homogeneous_waveguide_error_curves.png}
  \caption{Homogeneous waveguide residual curves (tol$=10^{-6}$), Jacobi in blue, DS in green and OSDS in orange}
  \label{fig:HomogeneousResidualCurves}
\end{figure}

We considered two values for the wave number: $k =20$ and $k = 20\pi$. The width of the overlap is 4 mesh size. In Tables~\ref{tab:homowaveguidenpplo24_20} and \ref{tab:homowaveguidenpplo24_20pi} we show iteration counts. We use two stopping criteria for the residual: $10^{-6}$ and $10^{-3}$ the latter is written in brackets. As expected, the Jacobi method needs many iterations to converge. For both Jacobi and double sweep algorithms, the iteration counts vary linearly w.r.t. the number of subdomains. The iteration counts for the Overlapping Splitting Double Sweep algorithm are significantly lower and display a sublinear behavior w.r.t. the number of subdomains.

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{3cm} p{3cm} p{3cm}|  }

 \hline
 N & Jacobi & DS & OSDS \\
 \hline
 5   & 28 (11)      &  18 (7) & 9 (3)     \\
 10  & 65 (27)    &  37 (15)  & 12 (4)      \\
 20  & 165 (51)    &  74 (36) & 23 (7)   \\
 40  &  356 (86)  &  151 (72) & 36 (10)    \\
 80  &  781 (155) & 296 (144)  &  67 (18)  \\
 \hline
\end{tabular}
\end{center}
\caption{Homogeneous waveguide, $k=20$, $\delta = 4h$, TOL=$10^{-6} (10^{-3})$, nppwl = 24 , P1 }
\label{tab:homowaveguidenpplo24_20}
\end{table}

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{3cm} p{3cm} p{3cm}|  }

 \hline
 N & Jacobi  & DS & OSDS\\
 \hline
 5   & 25   & 16 & 9   \\
 10  & 62  & 33 & 20   \\
 20  & 130   & 76 & 33 \\
 40  & 280  & 170 & 60   \\

 \hline
\end{tabular}
\end{center}
\caption{Homogeneous waveguide, $k=20\pi$, $\delta = 4h$, TOL=$10^{-6}$, nppwl = 24 , P1 }
\label{tab:homowaveguidenpplo24_20pi}
\end{table}

\subsection{Influence of the overlap}

We have also tested the effect of the width of the overlap on the convergence. We considered two test cases: the homogeneous waveguide (see Table~\ref{tab:homowaveguideovrl}) and the wedge (see Table~\ref{tab:wedgeovrl}) that is defined in more detail in \S~\ref{sec:wedgetest}. We observe that for the waveguide solved by the Overlapping Splitting Double Sweep method, the iteration count  decreases significantly with the overlap. This monotonical decrease in the iteration count contrasts with the behaviour of the other two methods. We see that for the Jacobi and double sweep (DS) methods, the overlap has a very little effect. For the Jacobi method it improves slightly the iteration counts whereas for the DS method, it might deteriorate the iteration count. For the wedge test case, all methods benefit monotonically from the size of the overlap but once again the reduction in the iteration count is more pronounced for the Overlapping Splitting Double Sweep method where the iteration count is reduced by a factor $1.83$ when the overlap is increased from $2h$ to $16h$.

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }

 \hline
 $\delta$ & Jacobi & DS & OSDS\\
 \hline
 2   &  159  &  69  & 27 \\
 4   &  165  &  74  & 23 \\
 8   &  160  &  76  & 20 \\
 16  &  143  &  73  & 18 \\
 \hline
\end{tabular}
\end{center}
\caption{Homogeneous waveguide, $k=20$, $\delta$ varies, TOL=$10^{-6}$, nppwl = 24 , P1 }
\label{tab:homowaveguideovrl}
\end{table}

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }

 \hline
 $\delta$ & Jacobi & DS & OSDS\\
 \hline
 2   &  259  &   127 &  97 \\
 4   &  245  &   117 &  83 \\
 8   &  221  &   105 &  69 \\
 16  &  202  &    91 &  53 \\
 \hline
\end{tabular}
\end{center}
\caption{Wedge, $\omega=40\pi$, $\delta$ varies, TOL=$10^{-6}$, nppwl = 24 , P1 }
\label{tab:wedgeovrl}
\end{table}

\subsection{Open cavity test}

Same as before, the domain is rectangular with an homogeneous medium and its length increases with the number of subdomains. The open cavity test is challenging due to the homogeneous Dirichlet conditions imposed on three sides (cf. black lines in Figure~\ref{fig:CavitySolution}). In addition, we perform an excitation on the left side. The Dirichlet conditions create rebounds leading to an increase in the number of reflections, this phenomenon is exacerbated for high-frequency regimes. The global problem can be written as
\begin{equation}
\left\{
\begin{aligned}
\left(- k^2 - \Delta\right) u &= f\ \text{ in } \Omega\\
\left(\partial_{\vec{n}} + Ik \right) u &= g \ \text{ on }  \Gamma \\
u &= 0 \ \text{ on } \partial\Omega \backslash \Gamma
\end{aligned}
\right.
\end{equation}
where $\Gamma\vcentcolon = \{x=0\}\times [0,1]$ and $g =  exp^{-ik\left(x cos(\theta) + y sin(\theta)\right)},\ \theta = \frac{\pi}{8}$. It corresponds to an incident plane wave propagating at an angle $\frac{\pi}{8}$ w.r.t. to the horizontal direction, see Fig.~\ref{fig:CavitySolution}. This creates numerous reflections on the lateral boundaries of the open cavity.

\begin{figure}[ht!]
  \centering
  \includegraphics[width=0.8\textwidth]{./u_global_cavity_20pi_BLACK.png}
  \caption{Open cavity solution ($k=20\pi$)}
  \label{fig:CavitySolution}
\end{figure}

Iteration counts are given in Tables~\ref{tab:cavityomega20npplo24} and \ref{tab:cavityomega20pinpplo24} for two wave numbers. One can notice that we took an overlap of $8h$ instead of $4h$ for the waveguide because we want to take advantage of the fact that the overlap contributes to accelerate convergence. Once again, for a given number of subdomains, the Overlapping Splitting DS method is systematically better than the other methods. As for the scalability, we observe that in terms of the number of subdomains, the iteration count varies surlinearly for the Jacobi method, linearly for the DS method and sublinearly for the Overlapping Splitting DS method. \\

\vspace{1em}

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }

 \hline
 N & Jacobi & DS & OSDS\\
 \hline
 5   & 54   &  29  & 15 \\
 10  & 124  &  56 & 19\\
 20  & 279  &  109 & 29 \\
 40  & 551   &  202 & 45 \\
 80  & +1200 &  416  & 78  \\
 \hline
\end{tabular}
\end{center}
\caption{Open cavity, $k=20$, $\delta=8h$, TOL=$10^{-6}$, $nppwl = 24$ , P1 }
\label{tab:cavityomega20npplo24}
\end{table}

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }

 \hline
 N & Jacobi  & DS & OSDS\\
 \hline
 5   & 72   & 47 & 31 \\
 10  & 154 & 94 & 47 \\
 20  & 331 & 223 & 82 \\
 40  & 754 &  453 & 149\\
 \hline
\end{tabular}
\end{center}
\caption{Open cavity, $k=20\pi$, $\delta=8h$, TOL=$10^{-6}$, $nppwl = 24$ , P1 }
\label{tab:cavityomega20pinpplo24}
\end{table}

\begin{figure}[h!]

  \includegraphics[width=1\textwidth]{./Cavity_error_curves.png}
  \caption{Open cavity residual curves (tol$=10^{-6}$), Jacobi in blue, DS in green and OSDS in orange}
  \label{fig:CavityResidualCurves}
\end{figure}

\subsection{Wedge test}
\label{sec:wedgetest}

We consider the classical test case of the wedge, see e.g. \cite{vion_double_2014}, a rectangular domain $[0, 600] \times [0, 1000]$ with three different velocities in regions separated by non-parallel boundaries (Fig.~\ref{fig:wedgesolution} left). Starting from the top, the velocities are $c=2000$, $c=1500$ and $c=3000$. Sommerfeld conditions are imposed on the bottom, right and left boundaries. The abrupt variations of the wavenumber produce internal reflections in different directions. A typical solution is shown on Figure~\ref{fig:wedgesolution} right. The global problem can be written as

\begin{equation}
\left\{
\begin{aligned}
\left(- k(x,y)^2 - \Delta\right) u &= f\ \text{ in } \Omega\\
\left(\partial_{\vec{n}} + Ik \right) u &= g \ \text{ on }  \Gamma_R\\
\left(\partial_{\vec{n}} + Ik \right) u &= 0 \ \text{ on }  \partial \Omega \backslash \Gamma_R
\end{aligned}
\right.
\end{equation}
where $\Gamma_R := [0, 600] \times \{y=1000\}$ and $g =  e^{-120(y-0.5)^2}sin(\pi y)$. \\

\begin{figure}[ht!]
\centering
\includegraphics[width=0.3\textwidth]{./Velocity_wedge.png}
\hspace{4em}
\includegraphics[width=0.2955\textwidth]{./u_global_wedge_160pi.png} \\
\caption{Heterogeneous media: Wedge (\textbf{Left}: Velocity model, \textbf{Right}: Solution (real) for $\omega=160\pi$)}
\label{fig:wedgesolution}
\end{figure}

Convergence curves are plotted in Fig.~\ref{fig:wedgeErrorCurves} and iteration counts are given in Tables~\ref{tab:wedgenpplo24_40pi} and \ref{tab:wedgenpplo24_60pi}. The OSDS method is clearly superior to the Jacobi and DS methods. When increasing the number of subdomains, the ratio in favor of the OSDS method compared to the DS method increases up to reaching a value of nearly $4$ for a domain decomposition into $40$ strips. Interestingly, we see that for a low tolerance on the residual (TOL=$10^{-3}$), the OSDS iteration counts are almost independent of the number of subdomains. Note also that compared to the DS method, the convergence curves of the OSDS method does not suffer of any stagnation. \\

\begin{figure}[ht!]

   \includegraphics[width=1\textwidth]{./Wedge_error_curves_RobinH.png}
   \caption{Wedge residual curves (tol$=10^{-6}$), Jacobi in blue, DS in green and OSDS in orange}
   \label{fig:wedgeErrorCurves}
\end{figure}

\begin{table}[htp]
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }
 \hline
 N & Jacobi & DS  & OSDS\\
 \hline
 5   &    28 (17)   &  19  (11) &   13 (6)  \\
 10  &  55 (31)     &  31  (16) &   14 (7)   \\
 20  &  110  (55)  & 58 (29)   &  18  (8)  \\
 40  &   203 (88)  &   103 (47) &   27 (9)     \\
 \hline
\end{tabular}
\end{center}
\caption{Wedge, $\omega=40\pi$, $\delta=16h$, TOL=$10^{-6} (10^{-3})$, $nppwl = 24$ , P1 }
\label{tab:wedgenpplo24_40pi}
\end{table}

\begin{table}[ht!
\begin{center}
\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}| }

 \hline
 N & Jacobi  & DS & OSDS\\
 \hline
 5   &  28 (15)    &  18 (10)  &  12 (5)     \\
 10  &  56 (30)   &   31 (15) &  14  (6)   \\
 20  &  111 (53)  & 57  (28)  &  18 (7)    \\
 40  &  206 (85)  &  111 (55) &   30 (9)  \\
 \hline
\end{tabular}
\end{center}
\caption{Wedge, $\omega=60\pi$, $\delta=16h$, TOL=$10^{-6} (10^{-3})$, $nppwl = 24$ , P1 }
\label{tab:wedgenpplo24_60pi}
\end{table}

\section{Prospects}
\label{sec:conclusion}

We have introduced an overlapping splitting double sweep algorithm which yields improved convergence for various problems: the waveguide, the open cavity and wedge test cases. It is analyzed for a plane decomposed into vertical strips. Many aspects deserve further investigations, in this section we only list some of them. As previously mentioned, for a first study of the new algorithm we only use first-order absorbing boundary conditions at the interfaces. It is known that higher-order ABC lead to drastically better stability properties and to a quicker convergence. This should reflect in better iteration counts for all methods. Also, an intrinsic problem with both double sweep and overlapping splitting double sweep methods is that due to the sequentiality of the algorithm, subdomains are idle most of the time. To be more precise, in a double sweep, only one process is active at a time whereas in an overlapping splitting double sweep, two processes are active at a time (one for the left sweep and one for the right sweep performed concurrently). A solution to overcome this is to introduce a pipelining technique that can be applied to  multiple right-hand sides problems to improve parallelism and achieve significant speed-ups, see~\cite{vion_parallel_2016}.
As a last remark, note that in a volumic version of the algorithm the local solve induced by the definition of the overlapping splitting double sweep operator ${\mathscr{T}_{OSDS}}$, eq.~\eqref{eq:TsdsDefinition} could be replaced by a cheaper approximate solve such as incomplete factorizations~\cite{engquist_sweeping_2011}.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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On this Monday, August 8th, 2016 of the Infowars Nightly News, Donald Trump details his plan to make America great again and for starters that means no more Obama Care. Meanwhile, the mainstream media continues to pretend that there's nothing wrong with Hillary Clinton's health. Plus Iran wants more money from Obama as they urge him to use his remaining time in office to return 2 billion dollars in frozen Iranian assets. - Watch now
Monday: The Alex Jones Show. Amazon Wants Control Over Your House/ Pollings Want Control Over Your Vote
On this Monday, August 8th, 2016 edition of the Alex Jones Show, we look into Hillary's health problems after he had trouble climbing a simple flight of stairs. Doctors and psychologists suggest she lacks the health - both physically and mentally - to serve as president. Paul Ryan's challenger, Paul Nehlen, exposes the TPP and Ryan's love for amnesty to illegals. Also, German investigative journalist Max Bachmann reports on the migrant crisis in Europe and his project to stop World War 3. And journalist Michael Snyder lists off the greatest threats humanity is facing. We also expose the secret group that's running U.S. elections and how they'll attempt to steal the presidency from Donald Trump. - Watch now
Sunday: The Alex Jones Show. Breaking: Secret Group Running U.S. Elections Discovered
On this Sunday, August 7th, 2016 transmissions of the Alex Jones Show, we give an extensive history of voter fraud in the United States. We also look at the blatant dishonesty of the media as it devises new scandals for Donald Trump. Meanwhile, a new Reuters/Ipsos poll puts Trump within 3 points striking distance. And Hillary gets hit hard after comments that she "short circuited" when she said FBI Director Comey characterized her testimony as "truthful." - Watch now
Friday: The Alex Jones Show. Clinton caught trying to steal election / Obama attacks Infowars
On this Friday, August 5th, 2016 edition of the Alex Jones Show, we discuss the latest in the presidential campaign and how the Obama administration is covering up our economic downfall. NSA whistleblower Bill Binney reveals how the NSA has Hillary's missing emails and he also discusses his new Kickstarter project. We also analyze how the State Dept. openly laughed at the idea of transparency in government. - Watch now

redpajama_set_name: RedPajamaCommonCrawl


Kurt Jensen Lars M. Kristensen 1 Coloured Petri Nets Department of Computer Science Kurt Jensen ADA-220 Coloured Petri Nets Modelling.
5 Kurt Jensen Lars M. Kristensen 5 Coloured Petri Nets Department of Computer Science Lectures  Q3: Coloured Petri Nets 1.  Monday 10-12 in Shannon-159 (60-105 min).  Thursday 10-12 in Shannon-159 (60-105 min).  Q4: Coloured Petri Nets 2.  Only 1-2 meetings (status presentations).  Dates and time will be decided later.  It is possible to follow only Coloured Petri Nets 1.  Preparing for lectures:  You are expected to read the material prior to the lectures.  I may skip the easier parts in the material and concentrate on the more difficult parts.
7 Kurt Jensen Lars M. Kristensen 7 Coloured Petri Nets Department of Computer Science Mandatory projects  Q3: Coloured Petri Nets 1.  Project 1: CPN Modelling (2 weeks; parallel to lectures).  Project 2: State Space Analysis (2 weeks; parallel to lectures).  Project 3: Performance Analysis (1,5 weeks; parallel to lectures).  Each documented in a 10-15 pages project report.  Q4: Coloured Petri Nets 2.  Larger project on CPN modelling and validation (full time).  You choose the project.  1-2 progress presentations.  Documented in a 15-20 pages project report.  Conducted in groups of 2-3 persons.  Start forming the groups now!
8 Kurt Jensen Lars M. Kristensen 8 Coloured Petri Nets Department of Computer Science Exam(s)  Coloured Petri Nets 1:  Individual oral examination. (20 minutes, no advance preparation).  Based on topics from the textbook material and the two projects.  Everyone is required to take the CPN1 exam.  Coloured Petri Nets 2:  Individual oral examination. (20 minutes, no advance preparation).  Project and associated report will serve as a basis.
9 Kurt Jensen Lars M. Kristensen 9 Coloured Petri Nets Department of Computer Science Practical information www.cs.au.dk/~cpnbook/courses/CPN OBS!
Download ppt "Kurt Jensen Lars M. Kristensen 1 Coloured Petri Nets Department of Computer Science Kurt Jensen ADA-220 Coloured Petri Nets Modelling."
Issues in Drug Abuse: Advanced Topics in Addiction (Psychology 486) Course Objectives, Organization, & Requirements Michael A. Bozarth, Ph.D. Department.
CG152 Introduction: slide 1 CS223 The Project: slide 1 Today's Lecture is in 3 Parts Part – 1: The Project Doron Part – 2: Going through the skipped over.
Kurt Jensen Lars M. Kristensen 1 Coloured Petri Nets Department of Computer Science Coloured Petri Nets Modelling and Validation of Concurrent Systems.
½ hour Chapter 2 Java in 30 minutes 1. 2 Rationale ½ hour Teaching a computer language like a logical system is possible. But not necessarily helpful.
Introduction to Financial Management. Overview of Financial Management Introduction Keys to Success Stay up in class (complete assignments on time each.
Computers and Society Lecture 1: administrative details and an introduction to the class Professor: Evan Korth New York University.
Introduction to Financial Management. Overview of Financial Management Introduction Keys to Success Recitations Class Structure - Syllabus Text – Financial.
Project Workshops Schedule. 2 Important Points to Note Project Workshops will be Thursday at 1315 in LC 50 These workshops are compulsory for all project.
ICS 324 Database Systems Second Semester (072) Information & Computer Science Department.
IS 421 Information Systems Management James Nowotarski 16 September 2002.
Overview of Financial Management  Introduction  Keys to Success  Recitations  Class Structure - Syllabus  Text – 2nd Preliminary Draft of Fin. Mgmt.
BAIM 503 ISSUES IN INTERNATIONAL POLICY Ilan Vertinsky Monday 18:30 – 22:00.
Modelling with Coloured Petri Nets Søren Christensen Department of Computer Science University of Aarhus.
Lecture 1 Concepts of Programming Languages Arne Kutzner Hanyang University / Seoul Korea.
King Saud UniversityCSC112 - First Semester 2009- 2010 1 CSC 112 Java Programming I Introduction.


text:
Reduce Waste: Lose the Cafeteria Trays
Making sustainability sexy
Lloyd Alter lloydalter
Put down that tray, Snow White; studies at universities have shown that diners think twice about loading up on food when they cannot carry it so easily. This reduces waste, helps the campus bottom line and the students' waistline. According to Inside Higher Ed, "More and more campuses are looking at this from both an environmental perspective and an economic one," said Varun Avasthi, director of dining services at Colby College, which experiments with trayless policies. "If you're not wasting as much food you're not buying as much food."
At Alfred University, food and beverage waste dropped between 30 and 50 percent, totalling 1,000 pounds of solid waste and 1123 gallons of liquid waste every week.
Students aren't happy, arguing that it is none of the college's business how much food they take.
"Unfortunately, there are some habits that need to be broken," said Avasthi, who is also Maine's district manager for food service provider Sodexho. "There are logistical issues, sure. Students don't want to go back up and get another glass. That's where the pushback typically occurs. It's a convenience factor. We're so used to having everything all the time whenever we want it." ::Inside Higher Ed via http://blogs.wsj.com/informedreader/2008/01/30/a-novel-way-to-cut-waste-lose-the-cafeteria-trays/?mod=WSJBlog
Put down that tray, Snow White; studies at universities have shown that diners think twice about loading up on food when they cannot carry it so easily. This reduces waste, helps the campus bottom line and the students' waistline. According
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Steam cleaning is considered to be one of the most eco-friendly alternatives to cleaning practices. This process can be applied to disinfect or sanitise nearly all surfaces - including flooring, kitchen platforms - except for some delicate fabrics used in upholstery and curtains, reason being the basic characteristic of using only water to produce steam and clean.
What makes steam cleaning so important is its ability to dissolve and remove any type of dirt without the use of strong detergents. In fact, steam can also dissolve hard-to-remove substances like ground-in wax, glue and chewing gum. Likewise, through steam cleaning techniques polluting effects on the environment are avoided, leading to a safer place to work and stay. "This is the most efficient sanitising method wherein dust mites, pests, allergens are removed without the need of harmful chemicals, in respect of human healthcare," says Gabriella Bianco, Communications and PR Manager, IP Cleaning Srl.
IPC has enriched their steam products range with four different high-performing steam generators, conceived for different field of application. From the smallest to the biggest, these can be provided with equipment specifically designed to perform and ease all steam cleaning activities. The range includes hoses, extension tubes, multi-purpose brushes of various forms (triangular or rectangular), brush polyester bristles, professional nozzles and steam lances. These accessories can be used for everyday cleaning operations and also to execute periodical treatments of the surfaces.
Continental Trading has recently introduced the STEAM BOX that has an automatic water refilling system with electromechanical water level control inside the boiler, pressure gauge, hour meter and a stainless steel body. "We have also introduced the CAR HYGIENE PLUS range equipped with full range of car cleaning and hygiene," adds Adnan Naziri, Continental Trading.
Elsea's product range includes various models designed for steam cleaning that includes the Steam Wave, Steam Plus and Steam Poseidon: each model has its own features and accessories kit. "In particular, the Steam Poseidon is the latest launch among the ELSEA steam generators with a more powerful suction of dust and liquids and with a strong and industrial structure," says D'Ago stino.
On a commercial level, steam cleaning can be an effective solution with versatile cleaning pressures and high-temperature, significantly reducing both water consumption and waste disposal costs. D'Agostino says particular industries such as food, healthcare, mechanical, HORECA and car wash benefit greatly from using steam machines. IPC Group, too, believes in the appeal of steam cleaning in the food industry – processing, storing, transport and food retail – assures a safe environment since all types of allergens and dust mites are killed without any chemicals. "Steam is highly recommended for HORECA environments in particular for the cleaning of kitchens, common services, pools, fitness centers and finally guest rooms, during both daily maintenance and extraordinary cleaning activities," comments Bianco.
There have been a lot of technological advancements in the steam cleaning process. For instance all the steam cleaning machines produced by ELSEA work as dry steam cleaners. "This guarantees the top results in sanitising and degreasing the surfaces and spaces. Dry saturated steam is defined as steam, free from micro drops of water. Besides, ELSEA's most recent innovations in the steam cleaning sector include the automatic selection of dry/wet steam delivery and the design of a new ozone-sanitising treatment machine," D'Agostino contributes. Furthermore, steam generation and diffusion are those elements recently optimised in IPC's steam products range. With the growing concern towards conserving water and the environment steam cleaning can be an effective alternative to cleaning practices.
An important challenge in the food industry nowadays is the contamination that could be caused by the decomposition of the food ingredients and other deposits hidden in the edges and between the wall junctions inside the food production machines and in the upper parts of the facilities. By using steam cleaning, all kinds of audit requirements according to the local food industry laws are easy to meet.
Dry steam cleaning is perfect to meet the highest local hygiene standards of the food industry in each country, including the most spread Hazard Analysis and Critical Control Points (HACCP). Dry steam is used for the work surface areas, kitchen extractor hoods, ventilation ducts, machinery and equipment used to cook or serve the food, including scales, measuring instruments, metal grills etc.
Steam machines are the perfect fit for the healthcare industry, with a focus on the sanitisation and the prevention of infectious diseases. Before it was put in practice by changing the detergents often in order to fight the bacteria that are neutralised by the chemical agents. In fact, all germs are immunised by steam at 160°. It is widely used in hospitals and clinics, operating rooms and other environments where the instruments must be kept sterile including ambulances and veterinary assistance. More precisely, the use of dry steam cleaning is indicated, but not limited to the sanitization of pillows and blankets, mattresses and wheelchairs, waiting rooms and corridors.
Hotels, restaurants and cafes, independent on their dimensions, have very high standards of cleaning and sanitary needs, given to the frequent guests' change and their high expectations of this criteria. Also, here steam cleaning is the right solution to clean and sanitise floors, walls, work surfaces and bathrooms, as well as to remove stains, including gums from carpets, fabrics, furniture and mattresses.
The evident advantages of steam cleaning in the hospitality industry are the reduced use of chemical detergents, the rapidity and the improvement of the air quality. Besides, it is very versatile, since it can be used for tile and majolica surfaces, parquet and listels, polishing chromatic surfaces and tubes, cleaning lifts and ventilation ducts and so on.
Steam cleaning can be performed on vehicles of all types and sizes, even those not suitable for the standard car washes. Since, the steam is also much more delicate compared to the water jet or brush cleaning, it is suitable to clean vintage cars, luxury limousines, motorcycles, bicycles and scooters without fear of damaging their paint jobs, etc. In any case, steam is perfect to clean and sanitize vehicles' glass, dashboards, mechanical, electrical and electronic parts. Being very efficient, any kind of stain on the seats and carpets are quickly eliminated. If cleaned by steam, the internal upholstery dries very fast, so the car can be immediately delivered to the client.


text:
Cotentin. In concert on Saturday, these three fishermen will sing in a nice restaurant
Through Writing La Presse de la Manche
The Channel Press
The Sons & Ships trio will be in concert at Saint-Pierre-Eglise (Manche). (©DP)
After several months without welcoming musiciansGuillaume and Martha, from restaurant Le P'tit Bistro at Saint-Pierre-Eglise (Channel), reconnect with musical evenings. For this premiere of 2023programmed on the counter, place at Fils & Ships with Jean-Pierre, Romain and Titouan, all from the neighboring town of Fermanville.
One of them, Titouan, is a mechanic on board a fishing vessel with tides to be maintained for several weeks. So, as he is there at the moment, we take the opportunity to invite them and open this new year in music.
Guillaume Deloye of the P'tit Bistro
Fils & Ships, it's 100% Fermanville with at the start father and son duo, quickly joined by Romain. All are involved in the local maritime life shared between the maritime high schoolthe SNSMdeep-sea fishing, trawlers locals and the daily life of the fishermen of Barfleur.
In the wake of "Our Sailors"
Les Fils & Ships, a trio that is warming up and setting off quietly, above all to share, in the wake of their friends from " Our Sailors This group is made up of renowned musicians and singers, who have gained a foothold in the world of local fishing for two years now following a shared residence between Fermanville and Omonville-la-Rogue.
Les Fils & Ships is guitar, violin, harmonica, a duo of voices who interpret their own texts imbued with the daily experiences of the life of fishermen-sailors.
From our correspondent Nathalie BONNEMAINS
Convenient. Saturday 21 January, from 7 p.m., concert by Fils & Ships. Information at the P'tit Bistrot on 02 33 43 65 93.
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Carlow Nationalist — Intel to cut another 30 jobs through redundancy programme
Johnny de Mol's return gives HLF8 half more viewers


In irritable bowel syndrome, there is no damage or visible sign of disease in the large intestine, but it does not work properly. An accurate understanding of symptoms is one of the most important aspects of diagnosing IBS. It is equally important to exclude other possible causes of gastrointestinal (GI) symptoms.
IBS is usually diagnosed by a gastroenterologist or a primary care physician.
There is no single test for irritable bowel syndrome. Instead, the doctor will consider your medical history and may perform a variety of diagnostic tests.
The doctor will take a thorough history, asking detailed questions about bowel symptoms, how long you have had them, and how often they occur. They may also ask questions about your psychological health and any history of trauma – many people with IBS have depression or anxiety and a history of abuse. The physician may ask about family history of GI problems and other conditions. A clear picture may emerge from the medical history to help the doctor assess risk factors that strengthen the suspicion of IBS or rule out other conditions.
There are two sets of criteria doctors may consider when evaluating your symptoms for the possibility of IBS.
If your symptoms are caused by irritable bowel syndrome, the results of all tests should be normal. If any tests reveal abnormalities, your doctor may suspect that your GI symptoms are caused by another condition.
If you have any warning signs of another bowel disease, the doctor may order additional tests.
Imaging tests can include X-rays, computed tomography (CT) scans, ultrasounds, and magnetic resonance imaging (MRI). You may need to swallow a contrast solution such as barium before some imaging scans to provide a clearer image. A barium enema may be necessary before imaging of the colon. Imaging scans can detect signs of inflammatory bowel disease such as scarring, intestinal blockages, abscesses, and fistulas (abnormal passages between the bowel and other organs or tissues).
Endoscopy and biopsy are performed to check for signs of damage in your bowel. IBS does not cause damage, so the presence of lesions or changes to the bowel may indicate that symptoms are caused by inflammatory bowel disease, celiac disease, microscopic colitis, or another GI condition.
Dozens of other conditions can produce IBS-like symptoms, and these must be ruled out in order to confirm irritable bowel syndrome. The process of ruling out similar conditions is referred to as differential diagnosis. To list just a few, conditions that may resemble IBS include Crohn's disease, ulcerative colitis, microscopic colitis, fibromyalgia, parasites such as Giardia, celiac disease, and lactose intolerance. The side effects of many medications may include symptoms similar to those seen in IBS.
Your doctor may be able to rule out many of these conditions quickly based on your medical and family history or simple blood tests. Other disorders may require time and repeated tests before they can be confirmed or ruled out. The presence of other diseases in addition to IBS may complicate the differential diagnosis and eventually result in multiple diagnoses.
What is the prognosis for IBS?
Irritable bowel syndrome is not fatal. People with IBS live a normal average lifespan. In some people, IBS goes into remission – causes no symptoms – for many years. However, IBS can reappear. There is no cure for IBS.
At what age does IBS develop?
Most people develop IBS before age 50.
My blessings to slam the gi dr. Mine needs it tol.


text:
Solution home Marketing Using the Reports Outputs for Communications
Using the Report Outputs for Communications Print
Using the Report Outputs for Communications
Building a query and generating results is of little value unless you can easily do something with the results. PatronBase Marketing lets you produce reports in the form of Mailing Labels for several popular label sheets. You can generate a Contact List that will provide a list of phone numbers for a tele-marketing campaign. Mail Merge assists in producing letters customised for each patron. You can send an e-Mail Message to those patrons who have provided their email addresses. SMS Message allows sending an SMS txt message to patrons' cellphones as simply as sending a report to a printer. When you execute the query, an SMS Editor will open allowing you to compose the message to the patrons identified by the query.
Messages can be sent to patrons by SMS over the cellular network using iTagg integration. This feature does require a subscription to the iTagg service and custom configuration, so speak to your PatronBase agent if you are interested.
Address to the attention of... will be especially useful when addressing organisations. Choose the appropriate Relationship from the list of options to identify who the letter will be addressed to.
The Privacy box encourages respect for the specified wishes of a patron regarding their preferred form of contact. There may be times when it is necessary to ignore these privacy settings. For instance, if a performance of a show has been cancelled at short notice because a key performer has taken ill, it may be necessary to email and/or phone every known patron to inform them of the cancellation or postponement of the show and to communicate their options. Patrons would appreciate this considerate contact, even if they had specifically stated a wish for posted communications. Therefore, the Privacy settings allows for queries to override the privacy settings chosen by patrons, but this should be done with great discretion.
Customise the Channels and Privacy Levels in Administration See Setting Up Patrons for more information.


A quick one just to show you the limited edition lacquers released alongside the Chanel Rouge Allure Velvet collection and the effect you get from using the Mat Top Coat.
I can't wait to try one of these babies on myself!
Rose Caché, Rose Exubérant, Pirate, and the Mat Top Coat retail for $25 each. All are limited edition. Available at major department stores and chanel.com.
I love that matte top coat, but I hear they're discontinuing it!!
So cool! Wonder how it wears?
I've never been a huge fan of matte nails but that velvet over red looks amazing!


Guests resolving to save some green as part of their New Year's resolution can opt for Red Robin's $6.99 everyday value-focused Tavern Double burgers with Bottomless Steak Fries. The Taco Tavern Double™ features two fire-grilled beef patties topped with fresh guac, crunchy tortilla strips, Pepper Jack, lettuce and zesty salsa. Another crazy-delicious combo sure to please guests' appetites and budgets is the Sir Acha Tavern Double™ with two fire-grilled beef patties, sriracha onion straws, American cheese, pickles, lettuce, tomatoes and spicy mayo.


text:
Springsteen: "I want my country back"
Supporting Obama, the rock legend takes a swing-state tour, once again.
By Joan Walsh
Published October 6, 2008 7:07PM (EDT)
I wrote this week on Open Salon about how, after 2000 and 2004, I no longer trust my electoral instincts. I shut down optimism quickly. As I put it, I've taught myself to view the electorate through Rove-colored glasses; most recently, I wasn't sure the Obama-Biden ticket had won the debates these last two weeks until I saw polls confirming that they did.
Conversely, in 2004 I was sure Bruce Springsteen's decision to publicly back the Democratic ticket and tour for John Kerry meant that his fans, even the conservative ones, would move to the Democrats. When he said, "the country that we carry in our hearts is waiting," I was sure Kerry would win. Of course, I was wrong; the country is still waiting. So I won't get too excited about Springsteen's weekend swing-state tour, and what it means. But I sure did like this speech, and you can watch his 42-minute set at the Philadelphia rally here.
"Hello Philly,
"I am glad to be here today for this voter registration drive and for Barack Obama, the next President of the United States.
"I've spent 35 years writing about America, its people, and the meaning of the American Promise. The Promise that was handed down to us, right here in this city from our founding fathers, with one instruction: Do your best to make these things real. Opportunity, equality, social and economic justice, a fair shake for all of our citizens, the American idea, as a positive influence, around the world for a more just and peaceful existence. These are the things that give our lives hope, shape, and meaning. They are the ties that bind us together and give us faith in our contract with one another.
"I've spent most of my creative life measuring the distance between that American promise and American reality. For many Americans, who are today losing their jobs, their homes, seeing their retirement funds disappear, who have no healthcare, or who have been abandoned in our inner cities. The distance between that promise and that reality has never been greater or more painful.
"I believe Senator Obama has taken the measure of that distance in his own life and in his work. I believe he understands, in his heart, the cost of that distance, in blood and suffering, in the lives of everyday Americans. I believe as president, he would work to restore that promise to so many of our fellow citizens who have justifiably lost faith in its meaning. After the disastrous administration of the past 8 years, we need someone to lead us in an American reclamation project. In my job, I travel the world, and occasionally play big stadiums, just like Senator Obama. I've continued to find, wherever I go, America remains a repository of people's hopes, possibilities, and desires, and that despite the terrible erosion to our standing around the world, accomplished by our recent administration, we remain, for many, a house of dreams. One thousand George Bushes and one thousand Dick Cheneys will never be able to tear that house down.
"They will, however, be leaving office, dropping the national tragedies of Katrina, Iraq, and our financial crisis in our laps. Our sacred house of dreams has been abused, looted, and left in a terrible state of disrepair. It needs care; it needs saving, it needs defending against those who would sell it down the river for power or a quick buck. It needs strong arms, hearts, and minds. It needs someone with Senator Obama's understanding, temperateness, deliberativeness, maturity, compassion, toughness, and faith, to help us rebuild our house once again. But most importantly, it needs us. You and me. To build that house with the generosity that is at the heart of the American spirit. A house that is truer and big enough to contain the hopes and dreams of all of our fellow citizens. That is where our future lies. We will rise or fall as a people by our ability to accomplish this task. Now I don't know about you, but I want that dream back, I want my America back, I want my country back.
"So now is the time to stand with Barack Obama and Joe Biden, roll up our sleeves, and come on up for the rising."
Joan Walsh is the author of "What's the Matter With White People: Finding Our Way in the Next America."
MORE FROM Joan Walsh


text:
Paisley 2021
Candidate City
About the bid
About Paisley
Paisley stories
Paisley 2021 story map
Paisley YMCA launch digital entrepreneur sessions for young people
Renfrewshire's young entrepreneurs of the future are being supported through the launch of Paisley YMCA's iEnterprise digital youth incubator programme.
The iEnterprise programme, which received funding from the Big Lottery Fund's Young Start Fund, is a dedicated entrepreneurial space which aims to help young people furthest away from the employment market to find work or start up their own business.
The first session which recently took place saw 23 young people aged from 11 to 18 years come along to participate and learn more digital skills.
The principle behind iEnterprise is for young people to identify a need within their community, develop an idea to address that need through entrepreneurial thinking using digital technology to help develop and implement their ideas.
Over the next 40 weeks participants will be given access to software and hardware as well as training and certification to develop their ideas. They will also be encouraged to take on project management roles, set budgets, solve problems, set goals and work as a team to achieve their objective.
Darran Gillan, Youth and Community Development Manager, Paisley YMCA said: "Our aim is to support and develop young people in order to help them reach their full potential through digitally based cultural and educational projects and programmes that develop an entrepreneurial spirit.
"The £50,000 Young Start fund from the Big Lottery Scotland will support that transition for young people into the world of work over the next two years and enable them to be more attractive candidates to employers."
In addition to equipping young people with valuable transferable skills the programme aims to improve young peoples' digital literacy skills, further develop their problem solving and cognitive thinking abilities, gain increased confidence using digital technology in intuitive ways that makes sense to them and developing their knowledge to use technology as a problem-solving tool.
Using digital technology to help transform people's lives by creating opportunities is a key aspect of Paisley's bid to be UK City of Culture 2021.
Paisley 2021 Bid Director, Jean Cameron, said: "The iEnterprise programme is an excellent project which will help young people explore their creativity through digital channels. One of the main ambitions of the Paisley 2021 bid is to increase participation, raise aspirations and improve future prospects through culture and this programme looks to achieve all of these outcomes.
"The young people taking part in these sessions can use the skills they learn in a range of creative and interesting ways helping to broaden their horizons and open them up to the range of career opportunities. Projects like iEnterprise help demonstrate that culture is digital."
During our pilot programme which took place in 2016 local young people developed wrestling, graphic design and music events businesses.
iEnterprise will be open at Paisley YMCA on Monday, Wednesday and Friday evenings from 5pm - 8pm for the next 40 weeks.
For more information on iEnterprise contact darran@paisleyymca.org
© Paisley for UK City of Culture 2021


text:
Q: Listview not working for Android Studio I'm trying to create a listView for lecturers in my faculty. There's something wrong, the app don't work, and the preview shows the default listView.
my xml code is:
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:paddingBottom="@dimen/activity_vertical_margin"
android:paddingLeft="@dimen/activity_horizontal_margin"
android:paddingRight="@dimen/activity_horizontal_margin"
android:paddingTop="@dimen/activity_vertical_margin"
>

<ListView
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:id="@+id/listView">

</ListView>
</RelativeLayout>

And my java code is:
import android.os.Bundle;
import android.support.annotation.Nullable;
import android.support.v7.app.AppCompatActivity;
import android.widget.ArrayAdapter;
import android.widget.ListView;

import java.util.ArrayList;

class lectures extends AppCompatActivity {
ListView ListView2;
ArrayAdapter<String> adapter;
public static String[] names = {"Dr.Aiad","Mr.Hamed","Mr.Qahtan","Dr.reza","Mr.Osamah",};

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_main);
    ListView2 = (ListView)findViewById(R.id.listview);
    adapter = new ArrayAdapter<String>(this, android.R.layout.activity_list_item, names);
    ListView2.setAdapter(adapter);

}
}

A: Replace your code with this :)
import android.os.Bundle;
import android.support.annotation.Nullable;
import android.support.v7.app.AppCompatActivity;
import android.widget.ArrayAdapter;
import android.widget.ListView;

import java.util.ArrayList;

class lectures extends AppCompatActivity {
ListView listView2;
ArrayAdapter<String> adapter;
 String[] names = {"Dr.Aiad","Mr.Hamed","Mr.Qahtan","Dr.reza","Mr.Osamah"};

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_main);
    listView2 = (ListView)findViewById(R.id.listview);
    adapter = new ArrayAdapter<String>(this, R.layout.activity_list_item, names);
    listView2.setAdapter(adapter);

 }
}

Hope it helps you!! :)

redpajama_set_name: RedPajamaStackExchange


text:
KeNIA's latest beneficiary, The Sixth Sense eyeing 10,000 Visually Impaired users in 2021
KeNIA's latest beneficiary, The Sixth...
Funding and Investment
Lawrence Otieno
2019/2020 Leaders in Innovation Fellowship (LIF) beneficiary, The Hope Tech Plus Limited with its signature product, the Sixth Sense is optimistic of gaining 10,000 users by end year,2021.
The startup which currently has a user base of about 1,600, targets the visually impaired with their hand-held mobility assistive device dubbed The Sixth Sense – a small, handheld walking companion to the visually impaired which imitates echolocation and sonar techniques by using
ultrasound sensors to detect objects. The device vibrates in different ways to warn the user when they're close to an object, with the frequency of vibrations increasing as user gets closer to an object.
This unique mobility aid is a brainchild of Brian Mwenda, an assistive tech enthusiast and Electrical and Electronic Engineer whose passion for innovation was triggered by his close association with the physically challenged.
"When I joined Secondary School, We used to learn together with friends with physical disabilities who we had to assist in one way or another. When I joined the University of Nairobi to study Electrical and Electronic engineering, I again met friends with Physical challenges and I had an urge to develop a product to assist them," he says. Mwenda adds that the field of Assistive technology has been neglected for a long time, but the technology and resources currently available in the world should be a reason to reverse the trend and innovate even more around the assistive technology.
It is very difficult for people with disabilities to fit in and compete fairly in most societies. People with visual impairment, for instance find moving around challenging as they lack adequate assistive devices to guarantee safe and unaided movement. While most rely on a white cane, they still require help from a sighted person to navigate obstacles, a dependency that Mwenda's team seeks to alleviate.
Currently incubated at the Meru University's Innovation, Incubation and Entrepreneurship Centre (IIEC), the Sixth Sense completely replaces white canes and gives sufficient information about the physical environment of a user for them to navigate safely.
The Startup recently received a commercialization grant from the Kenya National Innovation Agency (KeNIA) to further their innovation and improve their technology, and Mwenda says that most of the funds will go towards the development of the hardware, user education and trials over the next one year.
"We've just started the partnership with KeNIA and for the next one year we're looking at more collaboration. KeNIA has committed to connect us with stakeholders in the industry to help us scale up our product with a view of reaching as many people as possible. We are looking at a product which is socially acceptable by the community. Even though it has never been in the market, it has the potential to significantly transform someone's life."
The youthful innovator however says that although the startup journey is quite challenging, the training accorded to them by KeNIA and its partner institution, the Royal Academy of Engineers created a sort of a community around where they can talk, support each other, meet up and collaborate. The agency has also supported them in terms of publicity and branding.
The Sixth Sense has won numerous awards since its inception three years ago; notable among them is the 2019 National Diversity & Inclusion Awards & Recognition, where it won the Diversity and Inclusion Award for Inclusive and Innovative Deployment of Technology category.
The team is planning to scale up to the global market, noting that mobility challenges for the visually impaired is not only for the Kenyan populace but is a global challenge, and are urging the young and upcoming innovators to always think about inclusivity and universal accessibility in whatever innovative product they are working on.
At the same time, Mwenda says the team is working round the clock to create a world with zero barriers to participation for over 285 million visually impaired people globally by providing true freedom of independent and unaided movement which in turn opens up a world of opportunities in education, entrepreneurship, social and economic activities.


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Every Time Hillary Clinton Mentioned People of Color During Her DNC Speech
"So let's put ourselves in the shoes of young Black and Latino men and women who face the effects of systemic racism and are made to feel like their lives are disposable."
Kenrya Rankin Jul 28, 2016 10:57PM ET
Democratic presidential candidate Hillary Clinton acknowledges the crowd as she arrives on stage during the fourth day of the Democratic National Convention at the Wells Fargo Center, July 28, 2016, in Philadelphia, Pennsylvania. Photo Credit: Alex Wong/Getty Images
At 10:47 p.m. ET on July 28, Hillary Clinton made history when she accepted the Democratic party's nomination to represent the party as it seeks to hold on to the White House in this fall's general election. It is the first time in America's history that a woman has earned the nomination for a major party.
She used her time on stage at the Democratic National Convention to lay out her plan for America and rally the millions of party members watching to unite under the same banner, sending a special shoutout to Bernie Sanders' diehard supporters. "I want you to know. I heard you," she said to Bernie Sanders' supporters. "Your cause is our cause. Our country needs your ideas, energy and passion. That is the only way we can turn our progressive platform into real change for America. We wrote it together, now let's go out and make it happen together."
But how many times did she explicitly address people of color and the issues that uniquely impact them? Here is every instance we counted:
"Isn't [Donald Trump] forgetting…mothers who lost children to violence and are building a movement to keep other kids safe?"
"So, it's true. I sweat the details of policy, whether we're talking about the exact level of lead in the drinking water in Flint, Michigan, the number of mental health facilities in Iowa or the cost of your prescription drugs. Because it's not just a detail if it's your kid. If it's your famly, it's a big deal. And it should be a big deal to your president, too."
"My primary mission as president will be to create more opportunities and more good jobs with rising wages right here in the United States. From my first day in office, to my last, especially in places that for too long have been left out and left behind. From our inner cities to our small towns. From Indian country to coal country. From comunities ravaged by addiction to regions hollowed out by plant closures."
"I believe that when we have millions of hardworking immigrants contributing to our economy, it would be self defeating and inhumane to try to kick them out. Comprehensive immigration reform will grow our economy and keep families together, and it's the right thing to do."
"We have to heal the divides in our country. Not just on guns. But on race, immigration and more. And that starts with listening. Listening to each other, trying as best we can to walk in each others shoes. So let's put ourselves in the shoes of young Black and Latino men and women who face the effects of systemic racism and are made to feel like their lives are disposable. Let's put ourselves in the shoes of police officers kissing their kids and spouses goodbye every day, heading off to do a dangerous and necessary job. We will reform our criminal justice system from end to end, and rebuild trust between law enforcement and the communities they serve. And we will defend, we will defend all our rights. Civil rights, human rights and voting rights. Women's rights and workers' rights. LGBT rights and the rights of people with disabilities."
"And we will stand up against mean and divisive rhetoric wherever it comes from. You know, for the past year many people made the mistake of laughing off Donald Trump's comments, excusing him as an entertainer just putting on a show. They thought he couldn't possibly mean all the horrible things he says, like when he called women pigs or said that an American judge couldn't be fair because of his Mexican heritage. Or when he mocks and mimics a reporter with a disability, or insults prisoners of war like John McCain, a hero and a patriot who deserves our respect. Now at first, I admit, I couldn't believe it either. It was just too hard to fathom that someone who wants to lead our nation could say those things, could be like that. But here's the sad truth: There is no other Donald Trump. This is it. And in the end, it comes down to what Donald Trump doesn't get: America is great because America is good. So enough with the bigotry and bombast."
Did we miss anything? Refresh our memory in the comments.
#DemsInPhilly
Mothers of the Movement
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Rhinelander: Monday, April 9, 2018
ANNOUNCING THE RIBBON CUTTING OF OUR BREWERY AND TOUR CENTRE ON FRIDAY APRIL 13 AND SATURDAY APRIL 14 IN DOWNTOWN RHINELANDER WI
"RHINELANDER BEERS--REFRESHING AS THE NORTHWOODS"
Ms. Jyoti Auluck, Founder, President and CEO of Rhinelander Brewing Company is pleased to announce the Grand Opening of our Brewery and Tour Center. The newly renovated Brewery, Taproom and Gift Shop are located at 43 South Brown Street in historic downtown Rhinelander, WI.
Ribbon cutting will take place at 5 PM on Friday, April 13. Live music, 2 hour LIVE RADIO REMOTE and reports from COYOTE FM. The celebration continues on Saturday, April 14 – 1 to 9 PM. Ten of our finest beers will be on tap for you to enjoy. There will be specials in our Gift Shop on branded apparels, accessories and our packaged beers. The first 500 people who mention Promo Code: "Hodag 1882" starting on Friday at 4 PM and continuing on Saturday will get a Free PINT of beer. Mentioning the Promo code will also allow them to get a Free Rhinelander/Hodag branded Growler with every fill of the selected beers on tap. Jyoti Auluck and Al Ewan, head brewer and assistant brewers Brad Schmidt and Brian Carpenter along with the Taproom Manager Brenda O'Rourke will be present to enjoy a pint with all the people that show up.
The new Taproom has been fully renovated over the last 6 months. The historic building has new signage in front and the entire renovation was done with an authentic Northwoods theme, under the leadership of our Rhinelander based staff. The Rhinelander area contractors and suppliers who were used exclusively for the renovation work did quality work and completed the work in a timely fashion.
The new Taproom offers Hodag Beers/Rhinelander Brewery branded Growlers at the most competitive prices and of course, ten of our tastiest craft, lager and light beers. The beers offered include Rhinelander Lager, Uptown Girl Light, Thirsty Miner American IPA, Double IPA and Chocolate Stout, Over The Barrel Hard Soda and 4 HODAG beers made right here in Rhinlander. Our Gift Shop and packaged beer selection has been greatly expanded.
A number of new items are being added to the Rhinelander Brewery Museum, including dozens of old Shorty Cartoon posters. Ms. Auluck is urging local residents to donate additional items to the brewery any historic Rhinelander Brewery memorabilia. Donated items are being displayed for enjoyment by locals and visitors alike. The donators' names will be acknowledged in the museum with the display of the artifacts.
Al Ewan and the team has already produced 4 Hodag Ales: Irish Style Red Ale, Blonde Ale, Belgian Double and a Single Hop Session IPA. More unique brews are already in the tank, an English Bitters, Wheat Ale and a Rye Kolsh will be ready in coming weeks. Demand has been exceptional and the brewers are hustling to keep up.
BACKGROUND ON RHINELANDER BREWERY:
As people in the Midwest will recall fondly, the historic Rhinelander Brewery produced the finest beers at its brewery from 1882 to 1967, owned by Otto Hilgermann and Henry Danner located at Ocala Street in Rhinelander, next to the Pelican River. The former brewery location is now the Eye Clinic of Wisconsin at 2 E Ocala St. It was closed just like hundreds of other breweries closed with the advent (and aggressive tactics) of multi-national mega breweries. Interestingly, Mr. Hilgermann and his family lived in the house across the street from the brewery at 919 Arbutus St. complete with a 5 foot tunnel under the street and a 2 inch beer pipeline to his house. For many decades, Rhinelander Shorty was the most successful regional brand sold in the Midwest.
During the period from 1882 to 1967, Rhinelander Brewery put Rhinelander on the map. The returnable Rhinelander Export Lager and of course, the Rhinelander 7 oz Shorty is still etched in the memory of millions of Midwesterners. The historic Joseph Huber Brewery in Monroe, WI kept on producing Rhinelander beers in the 12 oz returnables from 1970s for another 35 years. The famous Rhinelander returnables lost their lustre in the early 2000s when the customers gradually stopped bringing back their Rhinelander returnable bottles. Joseph Huber Brewery switched Rhinelander beer to cans and soon the Northwoods lost one of its most famous beers.
In June 2011, the Canadian born Ms. Auluck re-launched Rhinelander Shorty with great success. It took her 2 years, requiring a sizeable capital investment and acquiring the original recipe and taste of the original Rhinelander Shorty beer.
The response throughout Wisconsin and the Midwest was nothing short of amazing. Almost every retail store and bar started selling the Rhinelander Shorty throughout US and parts of Canada.
Keeping the "best quality" that the original Rhinelander Brewery stuck to for 85 years of its existence, Ms. Auluck expanded the Rhinelander family of beers to Rhinelander Lager and Rhinelander Light 12oz, Thirsty Miner Craft Beers, as well as seasonal Rhinelander Oktoberfest. Also in early 2016, she launched her Rhinelander Hard Root Beer and her company now sells over 1 million pints of beer per year throughout the country making it one of the largest breweries in Wisconsin.
Rhinelander Brewing Company's new location on Brown Street sells a lot of its beers and other alcoholic products but it does NOT sell its own products that are carried and marketed by other Rhinelander based retailers. This is Ms. Auluck's way of supporting local businesses and a way of thanking Trig's Cellar 70, Stein's, and other local retailers who have supported her immensely over the last 7 years, thereby contributing greatly to her success.
Ms. Jyoti Auluck, President & CEO of Rhinelander Brewing Co is available for interviews. E-mail her at President@RhinelanderBrewery.com or by telephone at (715) 550-BEER.


Follow the winding brick pathways of the Village throughout the 42 acres of lush landscaping and charming 18th century-style architecture for a unique shopping and dining experience. Peddler's Village has something for everyone, from 70 quality specialty shops and six restaurants to the 70-room Golden Plough Inn and Giggleberry Fair family entertainment center. Enjoy free seasonal festivals and year-round family events. For more information, please visit www.peddlersvillage.com or call 215/794-4000.


Issue #1 of 2! Want to see the comic book live up to its ultimate potential? Then come along with groundbreaking writer/artist team T.P. Louise and Ashley Wood ( Popbot ) as they presents another opus. Join Les Morts 13 and a rotating cast of characters and overlapping, intersecting stories for what promises to be one of the most unique tales ever told in comics. Take your comic book expectations and throw 'em out the window for this Aventure of a lifetime.
Issue #2 of 2! Want to see the comic book live up to its ultimate potential? Then come along with groundbreaking writer/artist team T.P. Louise and Ashley Wood ( Popbot ) as they presents another opus. Join Les Morts 13 and a rotating cast of characters and overlapping, intersecting stories for what promises to be one of the most unique tales ever told in comics. Take your comic book expectations and throw 'em out the window for this Aventure of a lifetime.
With another storm brewing and the city's deathtoll rising, Coast Guard veteran Virgil Lafleur is forced to adopt more gruesome methods in order to protect his fellow citizens from the horrors on the horizon... From screenwriter Mark Landry and artist extraordinaire, Ashley Marie Witter (Squarriors, Interview with the Vampire), comes this visceral revenge thriller that merges the real-world hopes and horrors of a post-disaster community with an engaging thread of political corruption, class divide and blood-curdling terror!