A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations

The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind all-at-once linear systems, which are solved via parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] Lin Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that BDF ($p\geq 2$) is not selfstarting, when they exploited to solve time-dependent PDEs. In this note, we focus on 2-step often superior trapezoidal rule Riesz fractional diffusion equations, but its resultant discretized block triangular Toeplitz with low-rank perturbation. Meanwhile, first give an estimation condition number systems then adapt previous work construct two circulant (BC) preconditioners. Both invertibility BC preconditioners eigenvalue distributions matrices discussed details. efficient implementation also presented especially handling computation dense structured Jacobi matrices. Finally, numerical experiments involving both one- two-dimensional equations reported support our theoretical findings.