[Krylov_multi.git] / krylov_multi.tex

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



\begin{document}
\author{Raphaël Couturier \and Lilia Ziane Khodja}

\maketitle


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\begin{abstract}
In  this paper  we  revisit  the Krylov  multisplitting  algorithm presented  in
\cite{huang1993krylov}  which  uses  a  sequential  method to  minimize  the  Krylov
iterations computed by a multisplitting algorithm. Our new algorithm is based on
a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
parallel GMRES method inside each block and on a parallel Krylov minimization in
order to improve the convergence. Some large scale experiments with a 3D Poisson
problem  are  presented  with  up   to  8,192  cores.   They  show  the  obtained
improvements compared to a classical GMRES both in terms of number of iterations
and execution times.
\end{abstract}

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\section{Introduction}
Iterative methods are used to solve  large sparse linear systems of equations of
the form  $Ax=b$ because they are  easier to parallelize than  direct ones. Many
iterative  methods have  been proposed  and  adapted by  many researchers.   For
example, the GMRES method and the  Conjugate Gradient method are very well known
and  used by  many researchers~\cite{S96}. Both methods  are based  on the
Krylov subspace which consists in forming  a basis of a sequence of successive
matrix powers times the initial residual.

When  solving large  linear systems  with  many cores,  iterative methods  often
suffer  from scalability problems.   This is  due to  their need  for collective
communications  to  perform  matrix-vector  products and  reduction  operations.
Preconditioners can be  used in order to increase  the convergence of iterative
solvers.   However, most  of the  good preconditioners  are not  scalable when
thousands of cores are used.

Traditional iterative  solvers have  global synchronizations that  penalize the
scalability.   Two  possible solutions  consists  either  in using  asynchronous
iterative  methods~\cite{ref18} or  to  use multisplitting  algorithms. In  this
paper, we will  reconsider the use of a multisplitting  method. In opposition to
traditional  multisplitting  method  that  suffer  from  slow  convergence,  as
proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
drastically improve the convergence.


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\section{Related works and presention of the multisplitting method}
A general framework  for studying parallel multisplitting has  been presented in~\cite{o1985multi}
by O'Leary and White. Convergence conditions are given for the
most general case.  Many authors improved multisplitting algorithms by proposing
for  example  an  asynchronous  version~\cite{bru1995parallel}  and  convergence
conditions~\cite{bai1999block,bahi2000asynchronous}   in  this  case   or  other
two-stage algorithms~\cite{frommer1992h,bru1995parallel}.

In~\cite{huang1993krylov},  the  authors  proposed  a  parallel  multisplitting
algorithm in which all the tasks except  one are devoted to solve a sub-block of
the splitting  and to send their  local solutions to  the first task which  is in
charge to  combine the vectors at  each iteration.  These vectors  form a Krylov
basis for  which the first task minimizes  the error function over  the basis to
increase the convergence, then the other tasks receive the updated solution until
convergence of the global system. 

In~\cite{couturier2008gremlins}, the  authors proposed practical implementations
of multisplitting algorithms to solve  large scale linear systems. Inner solvers
could be  based on sequential direct method  with the LU method  or sequential iterative
one with GMRES.

In~\cite{prace-multi},  the  authors have  proposed a  parallel  multisplitting
algorithm in which large blocks are solved using a GMRES solver. The authors have
performed large scale experiments up-to  32,768 cores and they conclude that
asynchronous  multisplitting algorithm  could be more  efficient  than traditional
solvers on an exascale architecture with hundreds of thousands of cores.

So compared to these works, we propose in this paper a practical multisplitting method based on parallel iterative blocks and gives better results than classical GMRES method for the 3D Poisson problem we considered.
\\

The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways 
\begin{equation}
A = M_\ell - N_\ell,
\label{eq01}
\end{equation}
where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by iteration based on the obtained splittings as follows
\begin{equation}
x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
\label{eq02}
\end{equation}
where $E_\ell$ are non-negative and diagonal weighting matrices and their sum is an identity matrix $I$. The convergence of such a method is dependent on the condition
\begin{equation}
\rho(\displaystyle\sum^L_{\ell=1}E_\ell M^{-1}_\ell N_\ell)<1.
\label{eq03}
\end{equation}
where $\rho$ is the spectral radius of the square matrix.

The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems
\begin{equation}
v_\ell^k=M^{-1}_\ell N_\ell x_\ell^{k-1} + M^{-1}_\ell b,~\ell\in\{1,\ldots,L\},
\label{eq04}
\end{equation}
to be solved independently by a direct or an iterative method, where $v_\ell$ is the solution of the local sub-system. Thus the computations of $\{v_\ell\}_{1\leq \ell\leq L}$ may be performed in parallel by a set of processors. A multisplitting method using an iterative method as an inner solver is called an inner-outer iterative method or a two-stage method. The results $v_\ell$ obtained from the different splittings~(\ref{eq04}) are combined to compute solution $x$ of the linear system by using the diagonal weighting matrices
\begin{equation}
x^k = \displaystyle\sum^L_{\ell=1} E_\ell v_\ell^k,
\label{eq05}
\end{equation}    
In the case where the diagonal weighting matrices $E_\ell$ have only zero and one factors (i.e. $v_\ell$ are disjoint vectors), the multisplitting method is non-overlapping and corresponds to the block Jacobi method.

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\section{A two-stage method with a minimization}
Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. The multisplitting of this linear system is defined as follows
\begin{equation}
\left\{
\begin{array}{lll}
A & = & [A_{1}, \ldots, A_{L}]\\
x & = & [X_{1}, \ldots, X_{L}]\\
b & = & [B_{1}, \ldots, B_{L}]
\end{array}
\right.
\label{sec03:eq01}
\end{equation}  
where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. In this work, we use a row-by-row splitting without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. So, the multisplitting format of the linear system is defined as follows
\begin{equation}
\forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell, 
\label{sec03:eq02}
\end{equation} 
where $A_{\ell m}$ is a sub-block of size $n_\ell\times  n_m$ of the rectangular matrix $A_\ell$, $X_m\neq  X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.

Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system
\begin{equation}
\left\{
\begin{array}{l}
A_{\ell \ell}X_\ell = Y_\ell \mbox{,~such that}\\
Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_m,
\end{array}
\right.
\label{sec03:eq03}
\end{equation}
is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.  

It should  be noted that the convergence  of the inner iterative  solver for the
different  sub-systems~(\ref{sec03:eq03})  does   not  necessarily  involve  the
convergence of the multisplitting method.  It strongly depends on the properties
of       the       global      sparse       linear       system      to       be
solved~\cite{o1985multi,ref18}. Furthermore, the  splitting of the linear system
among  several clusters  of  processors  increases the  spectral  radius of  the
iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the
number of  splitting is, the larger the  spectral radius is.  In  this paper, we
based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
convergence and improve the scalability of the multisplitting methods.

In order to accelerate the convergence, we implemented the outer iteration of the multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
\begin{equation}
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 solution of the global linear system. The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations between clusters to generate this basis.

The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
\begin{equation}
R\alpha=b,
\label{sec03:eq05}
\end{equation}
where $R=AS$ is a dense rectangular matrix of size $n\times s$ and $s\ll n$. This leads us to solve a system of normal equations
\begin{equation}
R^TR\alpha=R^Tb,
\label{sec03:eq06}
\end{equation}
which is associated with the least squares problem
\begin{equation}
\text{minimize}~\|b-R\alpha\|_2,
\label{sec03:eq07}
\end{equation}  
where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate 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]
\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
\Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
\State Load $A_\ell$, $B_\ell$
\State Set the initial guess $x^0$
\State Set the minimizer $\tilde{x}^0=x^0$
\For {$k=1,2,3,\ldots$ until the global convergence}
\State Restart with $x^0=\tilde{x}^{k-1}$:
\For {$j=1,2,\ldots,s$}
\State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
\State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
\State Exchange local values of $X_\ell^j$ with the neighboring clusters
\State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$ 
\EndFor 
\State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
\State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
\EndFor

\Statex

\Function {InnerSolver}{$x^0$, $j$}
\State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
\State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
\State \Return $X_\ell^j$
\EndFunction

\Statex

\Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
\State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
\State \Return $\tilde{X}_\ell^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}

The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $\ell$ represent the local data for cluster $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and CGNR method executed in parallel by all clusters to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. The first one is performed at line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.

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\section{Experiments}
In order to illustrate  the interest  of our algorithm. We have  compared our
algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
situations.  We have chosen to focus on only one problem which is very simple to
implement: a 3 dimension Poisson problem.

\begin{equation}
\left\{
                \begin{array}{ll}
                  \nabla u&=f \mbox{~in~} \omega\\
                  u &=0 \mbox{~on~}  \Gamma=\partial \omega
                \end{array}
              \right.
\end{equation}

After discretization, with a finite  difference scheme, a seven point stencil is
used. It  is well-known that the  spectral radius of  matrices representing such
problems are very close to 1.  Moreover, the larger the number of discretization
points is,  the closer to 1  the spectral radius  is.  Hence, to solve  a matrix
obtained for  a 3D Poisson  problem, the number  of iterations is high.  Using a
preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
preconditioners are not scalable when using many cores.

%Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
In the following we present some experiments we could achieved out on the Hector
architecture,  a UK's  high-end computing  resource, funded  by the  UK Research
Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
16-core AMD  Opteron 2.3 Ghz  and 32 GB  of memory. Machines  are interconnected
with a 3D torus.

Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows
the  size of  the  3D Poisson  problem.  The size  is chosen  in  order to  have
approximately  50,000 components  per core.   The second  column  represents the
number of  cores used. In parenthesis,  there is the decomposition  used for the
Krylov multisplitting. The  third column and the sixth  column respectively show
the execution time for the GMRES  and the Krylov multisplitting codes. The fourth
and  the   seventh  column  describes   the  number  of  iterations.    For  the
multisplitting  code, the  total number  of inner  iterations is  represented in
parenthesis. For  the GMRES code (alone  and in the  multisplitting version) the
restart parameter is fixed to 16. The precision of the GMRES version is fixed to
1e-6. For  the multisplitting,  there are two  precisions, one for  the external
solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which
is fixed to 1e-10. It should be noted  that a high precision is used but we also
fixed  a maximum number of  iterations for each  internal step. In  practice, we
limit the  number of iterations in the internal step to  10. So an internal  iteration is finished
when the precision is reached or  when the maximum internal number of iterations
is reached. The precision and the maximum number of iterations of CGNR method are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c||c|c|c||c|c|c||c|} 
\hline
\multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} &  \multicolumn{3}{c||}{GMRES} &  \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
 \cline{3-8}
           &                   &  Time (s) & nb Iter. & $\Delta$  &   Time (s)& nb Iter. & $\Delta$ & \\
\hline
$468^3$ & 2,048 (2x1,024)        &  299.7    & 41,028    & 5.02e-8  &  48.4    & 691(6,146) & 8.24e-08  & 6.19   \\
\hline
$590^3$ & 4,096 (2x2,048)        &  433.1    & 55,494    & 4.92e-7  &  74.1    & 1,101(8,211) & 6.62e-08  & 5.84   \\
\hline
$743^3$ & 8,192 (2x4,096)        & 704.4     & 87,822    & 4.80e-07 &  151.2   & 3,061(14,914) & 5.87e-08 & 4.65    \\
\hline
$743^3$ & 8,192 (4x2,048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   & 1,531(12,721) & 1.47e-07& 6.39  \\
\hline

\end{tabular}
\caption{Results}
\label{tab1}
\end{center}
\end{table}


From these  experiments, it can be  observed that the  multisplitting version is
always  faster   than  the  GMRES   version.   The  acceleration  gain   of  the
multisplitting version is between 4 and 6.  It can be noticed that the number of
iterations is drastically reduced with the multisplitting version even it is not
neglectable. Moreover, with 8,192 cores, we  can see that using 4 clusters gives
better performance than simply using 2 clusters. In fact, we can remark that the
precision with 2 clusters is slightly  better but in both cases the precision is
under the specified threshold.

\section{Conclusion and perspectives}
We  have implemented  a  Krylov  multisplitting method  to  solve sparse  linear
systems  on large-scale computing  platforms.  We  have developed  a synchronous
two-stage  method based  on the  block Jacobi  multisaplitting which  uses GMRES
iterative  method as  an inner  iteration.  Our  contribution in  this  paper is
twofold. First we provide a multi cluster decomposition that allows us to choose
the  appropriate size  of  the clusters  according  to the  architecures of  the
supercomputer.  Second,   we  have  implemented  the  outer   iteration  of  the
multisplitting method  as a  Krylov subspace method  which minimizes  some error
function.  This  increases the convergence  and improves the scalability  of the
multisplitting method.

We  have tested  our multisplitting  method to  solve the  sparse  linear system
issued from  the discretization of  a 3D Poisson  problem. We have  compared its
performances to the  classical GMRES method on a  supercomputer composed of 2,048
to 8,192 cores. The experimental results showed that the multisplitting method is
about 4  to 6  times faster  than the GMRES  method for  different sizes  of the
problem split into  2 or 4 blocks when using  multisplitting method. Indeed, the
GMRES  method  has  difficulties to  scale  with  many  cores while  the  Krylov
multisplitting  method  allows to  hide  latency  and  reduce the  inter-cluster
communications.

In future  works, we plan to conduct  experiments on larger number  of cores and
test  the  scalability  of  our   Krylov  multisplitting  method.  It  would  be
interesting  to validate its  performances to  solve other  linear/nonlinear and
symmetric/nonsymmetric problems.  Moreover, we intend  to develop multisplitting
methods based  on asynchronous iteration in which  communications are overlapped
by computations.  These methods would  be interesting for platforms  composed of
distant  clusters interconnected  by  a high-latency  network.  In addition,  we
intend  to investigate  the  convergence  improvements of  our  method by  using
preconditioning  techniques  for  Krylov  iterative methods  and  multisplitting
methods with overlapping blocks.

\section{Acknowledgement}
The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.

%Other applications (=> other matrices)\\
%Larger experiments\\
%Async\\
%Overlapping\\
%preconditioning


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