  The dielectric response function and its inverse are crucial physical
quantities in materials science. We propose an accurate and efficient strategy
to invert the dielectric function matrix. The GW approximation, a powerful
approach to accurately describe many-body excited states, is taken as an
application to demonstrate accuracy and efficiency. We incorporate the
interpolative separable density fitting (ISDF) algorithm with
Sherman--Morrison--Woodbury (SMW) formula to accelerate the inversion process
by exploiting low-rank properties of dielectric function in plane-wave GW
calculations. Our ISDF--SMW strategy produces accurate quasiparticle energies
with $O(N_{\mathrm{r}}N_{\mathrm{e}}^2)$ computational cost $(N_{\mathrm{e}}$
is the number of electrons and $N_{\mathrm{r}}=100$--$1000N_{\mathrm{e}}$ is
the number of grid points) with negligible small error of $0.03$ eV for both
complex molecules and solids. This new strategy for inverting the dielectric
matrix can be \(50\times\) faster than the current state-of-the-art
implementation in BerkeleyGW, resulting in two orders of magnitude speedup for
total GW calculations.
