inner solver is a Krylov based one. In order to accelerate its convergence, the
outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed.
- At each outer iteration, the sparse linear system $Ax=b$ is partially
- solved using only $m$
- iterations of an iterative method, this latter being initialized with the
- last obtained approximation.
- GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
- inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
- $S$, which is composed by the $s$ last solutions that have been computed during
- the inner iterations phase.
- In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$.
+ At each outer iteration, the sparse linear system $Ax=b$ is partially solved
+ using only $m$ iterations of an iterative method, this latter being initialized
+ with the last obtained approximation. GMRES method~\cite{Saad86}, or any of its
+ variants, can potentially be used as inner solver. The current approximation of
+ the Krylov method is then stored inside a $n \times s$ matrix $S$, which is
+ composed by the $s$ last solutions that have been computed during the inner
+ iterations phase. In the remainder, the $i$-th column vector of $S$ will be
+ denoted by $S_i$.
+$\|r_n\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{n/2} \|r_0\|,$
+In the general case, where A is not positive definite, we have
+
+$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, \,$
+
+
At each $s$ iterations, another kind of minimization step is applied in order to
compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by
the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by
