\label{sec03:eq04}
\end{equation}
where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a
-solution of the global linear system.%The advantage such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis.
-The advantage of such a Krylov subspace is that we need neither an
-orthogonal basis nor synchronizations between the different clusters
-to generate this basis.
+solution of the global linear system. The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations between the different clusters to generate this basis.
The multisplitting method is periodically restarted every $s$
iterations with a new initial guess $\tilde{x}=S\alpha$ which
\For {$k=1,2,3,\ldots$ until the global convergence}
\State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do}
\State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l$
+\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l^k$
\State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters
-\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^j=\sum^L_{i=1}A_{li}X_i^j$
+\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$
\State\textbf{end for}
-\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$}
+\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
\State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$
\State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters
\EndFor
\Function {InnerSolver}{$x^0$, $j$}
\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
-\State Solving the local splitting $A_{ll}X_l^j=Y_l$ with the parallel GMRES method, such that $X_l^0$ is the initial guess.
+\State Solving the local splitting $A_{ll}X_l^j=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the initial guess
\State \Return $X_l^j$
\EndFunction
\Statex
\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
-\State Solving the normal equations $R^TR\alpha=R^Tb$ in parallel by $L$ clusters using the parallel CGNR method
-\State Compute the local minimizer: $\tilde{X}_l^k=S_l\alpha$
+\State Solving the normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using the parallel CGNR method
+\State Compute the local minimizer: $\tilde{X}_l^k=S_l^k\alpha^k$
\State \Return $\tilde{X}_l^k$
\EndFunction
\end{algorithmic}
