All-at-once solution of time-dependent PDE problems

In this thesis, we examine the solution to a range of time-dependent Partial Differential Equation (PDE) problems. Throughout, we focus on the development of preconditioners for the all-at-once system, which solves for all time-steps in a single coupled computation. The preconditioners developed are used with existing iterative methods and, due to their specific block structure, could be applied in parallel over time. We first develop solvers for the heat equation and the transient convection-diffusion equation. For both of these forward problems, the all-at-once system is non-symmetric. Despite this, in certain cases, we are able to provide rigorous termination bounds for non-symmetric iterative methods, contrary to what is generally possible for non-symmetric systems. The ideas developed for evolutionary PDEs are extended to develop preconditioners for time-dependent optimal control problems. By incorporating the methods designed for the forward problem, we are able to develop block diagonal Schur complement based preconditioners, which also could be implemented in parallel over time. We provide extensive eigenvalue analysis for each preconditioner and demonstrate their effectiveness through numerical computations for a variety of problems. We are able to describe solvers that are robust to various parameters, including the mesh size and number of time-steps.