In this work, we propose an absolute value block α-circulant
preconditioner for the minimal residual (MINRES) method to solve an all-at-once
system arising from the discretization of wave equations. Since the original
block α-circulant preconditioner shown successful by many recently is
non-Hermitian in general, it cannot be directly used as a preconditioner for
MINRES. Motivated by the absolute value block circulant preconditioner proposed
in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput.,
40(2):A1012-A1033, 2018], we propose an absolute value version of the block
α-circulant preconditioner. Our proposed preconditioner is the first
Hermitian positive definite variant of the block α-circulant
preconditioner, which fills the gap between block α-circulant
preconditioning and the field of preconditioned MINRES solver. The
matrix-vector multiplication of the preconditioner can be fast implemented via
fast Fourier transforms. Theoretically, we show that for properly chosen
α the MINRES solver with the proposed preconditioner has a linear
convergence rate independent of the matrix size. To the best of our knowledge,
this is the first attempt to generalize the original absolute value block
circulant preconditioner in the aspects of both theory and performance.
Numerical experiments are given to support the effectiveness of our
preconditioner, showing that the expected optimal convergence can be achieved