  In this contribution, we present a numerical analysis of the continuous
stochastic gradient (CSG) method, including applications from topology
optimization and convergence rates. In contrast to standard stochastic gradient
optimization schemes, CSG does not discard old gradient samples from previous
iterations. Instead, design dependent integration weights are calculated to
form a linear combination as an approximation to the true gradient at the
current design. As the approximation error vanishes in the course of the
iterations, CSG represents a hybrid approach, starting off like a purely
stochastic method and behaving like a full gradient scheme in the limit. In
this work, the efficiency of CSG is demonstrated for practically relevant
applications from topology optimization. These settings are characterized by
both, a large number of optimization variables \textit{and} an objective
function, whose evaluation requires the numerical computation of multiple
integrals concatenated in a nonlinear fashion. Such problems could not be
solved by any existing optimization method before. Lastly, with regards to
convergence rates, first estimates are provided and confirmed with the help of
numerical experiments.
