X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/89ef73268fc9f6d2955ed7abe987ea691174d95e..cebfda218ec5cc78ffbd06922f550cf3bae1a46b:/paper.tex

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@@ -607,12 +607,12 @@ GMRES is one of the most widely used Krylov iterative method for solving sparse
 
 In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However in practice the good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.  
 
-Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding methods for multicore processors and multi-GPU machines~\cite{} {\bf MANQUE REF}. 
+Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding techniques for the GMRES method, so-called CA-GMRES, on multicore processors and multi-GPU machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}. 
 
 Compared  to all these  works and  to all  the other  works on  Krylov iterative
 method, the originality of our work is to build a second iteration over a Krylov
 iterative method and to minimize the residuals with a least-squares method after
-a given number of outer iteration.
+a given number of outer iterations.
 
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