13-01-2014 V1

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 64b36e0ab5b35811a5ee564bacbabfa5a48e8348..ea6cfe92e3e4d0dc013ccbc80055215e27b7ea5c 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -6,6 +6,12 @@
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 

+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+

 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
 
 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
 


@@ -223,10 +229,7 @@ S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 \label{sec03:eq04}
 \end{equation}
 where   for  $j\in\{1,\ldots,s\}$,  $x^j=[X_1^j,\ldots,X_L^j]$   is  a
 \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
 
 The  multisplitting   method  is  periodically   restarted  every  $s$
 iterations  with   a  new  initial   guess  $\tilde{x}=S\alpha$  which


@@ -258,16 +261,18 @@ gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 \begin{algorithmic}[1]
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 \begin{algorithmic}[1]

-\State Load $A_l$, $B_l$, initial guess $x^0$
+\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$

 \State Initialize the minimizer $\tilde{x}^0=x^0$
 \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 Initialize the minimizer $\tilde{x}^0=x^0$
 \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} 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\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
 \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


@@ -276,15 +281,15 @@ gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
 
 \Function {InnerSolver}{$x^0$, $j$}
 \State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
 
 \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 \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}
 \State \Return $\tilde{X}_l^k$
 \EndFunction
 \end{algorithmic}