  A coarse grid correction (CGC) approach is proposed to enhance the efficiency
of the matrix exponential and $\varphi$ matrix function evaluations. The
approach is intended for iterative methods computing the matrix-vector products
with these functions. It is based on splitting the vector by which the matrix
function is multiplied into a smooth part and a remaining part. The smooth part
is then handled on a coarser grid, whereas the computations on the original
grid are carried out with a relaxed stopping criterion tolerance. Estimates on
the error are derived for the two-grid and multigrid variants of the proposed
CGC algorithm. Numerical experiments demonstrate the efficiency of the
algorithm, when employed in combination with Krylov subspace and Chebyshev
polynomial expansion methods.
