  We introduce a time-implicit, finite-element based space-time discretization
scheme for the backward stochastic heat equation, and for the forward-backward
stochastic heat equation from stochastic optimal control, and prove strong
rates of convergence. The fully discrete version of the forward-backward
stochastic heat equation is then used within a gradient descent algorithm to
approximately solve the linear-quadratic control problem for the stochastic
heat equation driven by additive noise.
