X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7d1a785e0f85113eb0bc24d6b4421a95d200d9ce..dfdee1f9086e9574b81e615edd998441ed3b82ae:/paper.tex diff --git a/paper.tex b/paper.tex index 693edae..c8305c5 100644 --- a/paper.tex +++ b/paper.tex @@ -716,7 +716,7 @@ $error$, which is defined by $||Ax_k-b||_2$. Let us summarize the most important parameters of TSIRM: \begin{itemize} -\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method; +\item $\epsilon_{tsirm}$: the threshold that stops the TSIRM method; \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method; \item $s$: the number of outer iterations before applying the minimization step; \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method; @@ -727,9 +727,9 @@ Let us summarize the most important parameters of TSIRM: The parallelization of TSIRM relies on the parallelization of all its parts. More precisely, except the least-squares step, all the other parts are obvious to achieve out in parallel. In order to develop a parallel version of -our code, we have chosen to use PETSc~\cite{petsc-web-page}. For -line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and -efficient since the matrix $A$ is sparse and since the matrix $S$ contains few +our code, we have chosen to use PETSc~\cite{petsc-web-page}. In +line~\ref{algo:matrix_mul}, the matrix-matrix multiplication is implemented and +efficient since the matrix $A$ is sparse and the matrix $S$ contains few columns in practice. As explained previously, at least two methods seem to be interesting to solve the least-squares minimization, CGLS and LSQR. @@ -764,7 +764,7 @@ the parallelization of CGLS which is similar to LSQR. In each iteration of CGLS, there is two matrix-vector multiplications and some -classical operations: dot product, norm, multiplication and addition on +classical operations: dot product, norm, multiplication, and addition on vectors. All these operations are easy to implement in PETSc or similar environment. It should be noticed that LSQR follows the same principle, it is a little bit longer but it performs more or less the same operations. @@ -846,8 +846,11 @@ $\begin{array}{ll} which concludes the induction and the proof. \end{proof} -%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower -%than the one of the GMRES method. +Remark that a similar proposition can be formulated at each time +the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$, +with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases +regarding $A$, +that the proposed TSIRM converges while the GMRES($m$) does not. %%%********************************************************* %%%********************************************************* @@ -881,13 +884,14 @@ torso3 & 2D/3D problem & 259,156 & 4,429,042 \\ \label{tab:01} \end{center} \end{table} -Chosen parameters are detailed below. As by default the restart of GMRES is -performed every 30 iterations, we have chosen to stop the GMRES every 30 -iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen -to minimize the least-squares problem with the following parameters: +Chosen parameters are detailed below. +We have stopped the GMRES every 30 +iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default +setting of GMRES. $s$, for its part, has been set to 8. CGLS + minimizes the least-squares problem with parameters $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to -$\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R) -Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc. +$\epsilon_{tsirm}=1e-10$. These experiments have been performed on an Intel(R) +Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version of PETSc. In Table~\ref{tab:02}, some experiments comparing the solving of the linear @@ -1174,11 +1178,13 @@ experiments up to 16,394 cores have been led to verify that TSIRM runs For future work, the authors' intention is to investigate other kinds of -matrices, problems, and inner solvers. The influence of all parameters must be +matrices, problems, and inner solvers. In particular, the possibility +to obtain a convergence of TSIRM in situations where the GMRES is divergent will be +investigated. The influence of all parameters must be tested too, while other methods to minimize the residuals must be regarded. The number of outer iterations to minimize should become adaptative to improve the overall performances of the proposal. Finally, this solver will be implemented -inside PETSc. This would be very interesting because it would allow us to test +inside PETSc, which would be of interest as it would allow us to test all the non-linear examples and compare our algorithm with the other algorithm implemented in PETSc.