Before revision: Solving wave equations in a time-parallel manner is challenging, and the algorithm based on the block \alpha-circulant preconditioning technique has shown its advantage in many existing studies (where \alpha\in(0, 1) is a free parameter). Considerable efforts have been devoted to exploring the  spectral radius of the preconditioned matrix and this leads to many case-by-case studies depending on the used time-integrator.
After revision: Solving wave equations in a time-parallel manner is challenging, and the algorithm based on the block \alpha-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the spectral radius of the preconditioned matrix and this leads to many case-by-case studies depending on the used time-integrator.