Block ω-circulant preconditioners for parabolic equations

In this study, a novel class of block omega-circulant preconditioners is developed for the all-at- once linear system that emerges from solving parabolic equations using first and second order discretization schemes for time. We establish a unifying preconditioning framework for omega-circulant preconditioners, extending and modifying the preconditioning approach recently proposed in (Zhang and Xu, 2024 [27]) and integrating some existing results in the literature. The proposed preconditioners leverage fast Fourier transforms for efficient diagonalization, facilitating parallel- in-time execution. Theoretically, these preconditioners ensure that eigenvalue clustering around +/- 1 is achieved, fostering fast convergence under the minimal residual method. Furthermore, when using the generalized minimal residual method, the effectiveness of these preconditioners is supported by the singular value clustering at unity. Numerical experiments validate the performance of the developed preconditioning strategies.