Parallel preconditioning for time-dependent PDEs and PDE control

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation.

This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.

The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.

This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)