  This paper concerns the numerical approximation of low-energy eigenstates of
the linear random Schr\"odinger operator. Under oscillatory high-amplitude
potentials with a sufficient degree of disorder it is known that these
eigenstates localize in the sense of an exponential decay of their moduli. We
propose a reliable numerical scheme which provides localized approximations of
such localized states. The method is based on a preconditioned inverse
iteration including an optimal multigrid solver which spreads information only
locally. The practical performance of the approach is illustrated in various
numerical experiments in two and three space dimensions and also for a
non-linear random Schr\"odinger operator.
