  In this work, we propose an absolute value block $\alpha$-circulant
preconditioner for the minimal residual (MINRES) method to solve an all-at-once
system arising from the discretization of wave equations. 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 $\alpha$-circulant
preconditioner. Since the original block $\alpha$-circulant preconditioner is
non-Hermitian in general, it cannot be directly used as a preconditioner for
MINRES. Our proposed preconditioner is the first Hermitian positive definite
variant of the block $\alpha$-circulant preconditioner for the concerned wave
equations, which fills the gap between block $\alpha$-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 a properly chosen $\alpha$ the MINRES solver
with the proposed preconditioner achieves 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 the concerned problem. Numerical
experiments are given to support the effectiveness of our preconditioner,
showing that the expected optimal convergence can be achieved.
