## Parallel preconditioning for time-dependent PDEs and PDE control

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Andy Wathen**

University of Oxford, UK

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.