  This work deals with the efficient numerical solution of the time-fractional
heat equation discretized on non-uniform temporal meshes. Non-uniform grids are
essential to capture the singularities of "typical" solutions of
time-fractional problems. We propose an efficient space-time multigrid method
based on the waveform relaxation technique, which accounts for the nonlocal
character of the fractional differential operator. To maintain an optimal
complexity, which can be obtained for the case of uniform grids, we approximate
the coefficient matrix corresponding to the temporal discretization by its
hierarchical matrix (${\cal H}$-matrix) representation. In particular, the
proposed method has a computational cost of ${\cal O}(k N M \log(M))$, where
$M$ is the number of time steps, $N$ is the number of spatial grid points, and
$k$ is a parameter which controls the accuracy of the ${\cal H}$-matrix
approximation. The efficiency and the good convergence of the algorithm, which
can be theoretically justified by a semi-algebraic mode analysis, are
demonstrated through numerical experiments in both one- and two-dimensional
spaces.
