I have a matrix of the form:
$M:=\begin{pmatrix} S_1 & & & \\\ Q_1 & S_2 & & \\\ & ... & ... & \\\ & & Q_n & S_n\end{pmatrix}$
where the blocks are square. The $S_i$ are asymmetric, the $Q_i$ are symmetric and positive definite. Would you have a reference/hints for efficient solution algorithms for systems of the type $Mx=b$?
What further properties are in case needed on the $Q_i$ blocks? We can assume that the resulting matrix $M$ is non singular.
I encountered such structure while solving parabolic PDEs on time dependent domains.
The suggested method should be faster than solving $S_ix_i-Q_{i-1}x_{i-1}=b_i$ recursively.
Moreover, we can assume that the number of blocks is large (e.g. $1e4$) and that the size of each block is also large, e.g. $1e3$.
Parallel strategies are very welcome.