  A fast two-level linearized scheme with unequal time-steps is constructed and
analyzed for an initial-boundary-value problem of semilinear subdiffusion
equations. The two-level fast L1 formula of the Caputo derivative is derived
based on the sum-of-exponentials technique. The resulting fast algorithm is
computationally efficient in long-time simulations because it significantly
reduces the computational cost $O(MN^2)$ and storage $O(MN)$ for the standard
L1 formula to $O(MN\log N)$ and $O(M\log N)$, respectively, for $M$ grid points
in space and $N$ levels in time. The nonuniform time mesh would be graded to
handle the typical singularity of the solution near the time $t=0$, and Newton
linearization is used to approximate the nonlinearity term. Our analysis relies
on three tools: a new discrete fractional Gr\"{o}nwall inequality, a global
consistency analysis and a discrete $H^2$ energy method. A sharp error estimate
reflecting the regularity of solution is established without any restriction on
the relative diameters of the temporal and spatial mesh sizes. Numerical
examples are provided to demonstrate the effectiveness of our approach and the
sharpness of error analysis.
