Density functional theory (DFT) calculations in CP2K employ the Gaussian and plane waves (GPW) method. In GPW, the description of the total density on realspace grids is typically the computationally most expensive part. By introducing a local resolution-of-the-identity (LRI) approach, the linear scaling of the GPW approach can be retained, while reducing the prefactor for the grid operations. The combined approach, LRIGPW, is comprehensively described in J. Chem. Theory Comput., 2017.
In LRIGPW, the atomic pair densities $\rho_{\mathrm{AB}}$ are approximated by an expansion in a set of fit functions centered at atom A $\{f_i^{\mathrm{A}}(\mathbf{r})\}$ and atom B $\{f_j^{\mathrm{B}}(\mathbf{r})\}$,

\begin{equation} \rho_{\mathrm{AB}}\approx \sum_i{a_i^{\mathrm{A},(\mathrm{AB})}f_i^{\mathrm{A}}}(\mathbf{r}) + \sum_j{a_j^{\mathrm{B},(\mathrm{AB})}f_j^{\mathrm{B}}}(\mathbf{r}). \end{equation} The fit functions are also Gaussian-type functions and provided as auxiliary basis set.

LRIGPW is specified in the QS section by setting METHOD LRIGPW.

&QS
  METHOD LRIGPW
  &LRIGPW
     LRI_OVERLAP_MATRIX INVERSE
     SHG_LRI_INTEGRALS
  &END
&END QS

Further specifications can be given in the LRIGPW subsection. LRIGPW requires additionally an auxiliary basis set as input.

 &DFT
    BASIS_SET_FILE_NAME BASIS_LRIGPW_AUXMOLOPT
    BASIS_SET_FILE_NAME BASIS_MOLOPT
    ...
 &END DFT
 &SUBSYS
    &KIND O
      BASIS_SET DZVP-MOLOPT-GTH
      POTENTIAL GTH-PBE-q6
      LRI_BASIS_SET LRI-DZVP-MOLOPT-GTH-MEDIUM
    &END KIND
    ...
 &END SUBSYS

Auxiliary basis sets are available for the MOLOPT basis sets. All auxiliary basis sets have been generated by simple geometric progression without any need for further optimization. These basis sets are available in different sizes: MEDIUM and LARGE. Using the large auxiliary basis sets, the accuracy is improved, but the computational overhead increases.
The auxiliary basis for LRI are generally quite large leading to a potentially ill-conditioned overlap matrix, Equation (10) in

• Ice XV: ice_lrigpw.tar.gz