I am attempting to use an iterative solver to solve $p$ in
$$ Jp = -r $$
where $J$ is an $m\times m$ matrix ($m$ is in the order of $10^5$ and never explicitly stored). $J$ is a dense matrix representing a Jacobian for a certain function, and, in general, is not symmetric.
Unfortunately, this is an ill-posed problem and possibly infinite versions of $p$ exist.
I cannot compute the least-squares version
$$ J^TJp=J^Tr $$
because I only can access the product of $Jp$ and not $J^Tp$ (a limitation of an automatic differentiation approach I am taking).
- Is there a way I can compute $p$ in $Jp = -r$ according to some criterion (e.g. least-squares solution) without transposing $J$?
- If I use an iterative solver (e.g. GMRES, BiCGStab, etc...) with an initial guess of $p=0$, what solution will the solver converge to? The minimum-norm?