Some observations on preconditioning for non-self-adjoint and time-dependent problems

Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part Computational Methods for Partial Differential Equations. Here we discuss contrast between symmetric systems equations that arise from self-adjoint problems and non-symmetric non-self-adjoint when iterative methods are employed; such only feasible very large scale computation with PDEs. We then go on to consider all-at-once in approximation time-dependent problems, causality parallel-in-time paradigm, suggesting an approach involves preconditioning initial value time-periodic problems.