# Applying the inverse of the Schur complement without matrix-matrix products

If one attempts to solve a (block matrix) saddle point problem such as $$\begin{bmatrix} A & -B^T \\ B & 0 \end{bmatrix} \begin{bmatrix} u \\ p \end{bmatrix} = \begin{bmatrix} f \\ 0 \end{bmatrix}$$ using a Schur complement structure, then one needs to form the inverse of the Schur complement, $$S = BA^{-1}B^T$$. We are assuming that $$A$$ is nonsingular but nothing else. It may be indefinite and nonsymmetric. Additionally, $$B$$ may be rectangular. Herein lies my issue. Generally, we want to avoid matrix-matrix products. And so, if we have to solve something like $$CDx=b$$ for invertible matrices $$C$$ and $$D$$, we might do a two step process:

1. Solve $$Cy=b$$.
2. Solve $$Dx=y$$.

This is done all the time when using a Krylov solver with preconditioning (if I'm off base please correct me), for example.

However, to apply $$S^{-1}$$ to some vector $$b$$, we need to solve $$Sx=b$$ or equivalently $$BA^{-1}B^Tx=b$$ also without forming matrix-matrix products. There has to be a standard way of doing this, but I have not been able to track it down or come up with one on my own, since $$B$$ need not be nonsingular or even square. Thank you for your suggestions.

• Thank you for the comment. I don't think I fully understand your comment. I don't know how I would implement your suggestion. What I really want is $S^{-1}b$. As is usually the case, I will compute the action of the inverse by solving $Sx=b$. Therefore, x is unknown and $B^Tx$ cannot be computed directly (at least not in the straight forward way as a matrix-vector product). Feb 18, 2022 at 19:16
• I misread your question, never mind Feb 18, 2022 at 19:24
• Ah ok. Thank you for the interest in my question anyway. Your comment is helpful on some level at least since a Krylov solver such as GMRES only needs access to the action of $S$ and not its inverse on various vectors. That might be the best that can be done. Feb 18, 2022 at 19:30
• @Chessnerd321 Are you able to supply a reference re GMRES and $S$? I too am trying to invert $S$ in order precondition a system. Dec 11, 2023 at 2:41
• @Olumide This review paper is the best reference I could find. Depending on your application, it likely has what you need. In short, what I said is basically correct. People find a problem-dependent preconditioner for $S$ and then use GMRES since it does not need access to $S$, just it's action. Dec 11, 2023 at 12:17