  Optimal upper and lower error estimates for strong full-discrete numerical
approximations of the stochastic heat equation driven by space-time white noise
are obtained. In particular, we establish the optimality of strong convergence
rates for full-discrete approximations of stochastic Allen-Cahn equations with
space-time white noise which have recently been obtained in [Becker, S., Gess,
B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicit
space-time discrete numerical approximations of stochastic Allen-Cahn
equations. arXiv:1711.02423 (2017)].
