  Topology optimization under uncertainty (TOuU) often defines objectives and
constraints by statistical moments of geometric and physical quantities of
interest. Most traditional TOuU methods use gradient-based optimization
algorithms and rely on accurate estimates of the statistical moments and their
gradients, e.g., via adjoint calculations. When the number of uncertain inputs
is large or the quantities of interest exhibit large variability, a large
number of adjoint (and/or forward) solves may be required to ensure the
accuracy of these gradients. The optimization procedure itself often requires a
large number of iterations, which may render TOuU computationally expensive, if
not infeasible. To tackle this difficulty, we here propose an optimization
approach that generates a stochastic approximation of the objective,
constraints, and their gradients via a small number of adjoint (and/or forward)
solves, per iteration. A statistically independent (stochastic) approximation
of these quantities is generated at each optimization iteration. The total cost
of this approach is only a small factor larger than that of the corresponding
deterministic TO problem. We incorporate the stochastic approximation of
objective, constraints and their design sensitivities into two classes of
optimization algorithms. First, we investigate the stochastic gradient descent
(SGD) method and a number of its variants, which have been successfully applied
to large-scale optimization problems for machine learning. Second, we study the
use of the proposed stochastic approximation approach within conventional
nonlinear programming methods, focusing on the Globally Convergent Method of
Moving Asymptotes (GCMMA). The performance of these algorithms is investigated
with structural design optimization problems utilizing a Solid Isotropic
Material with Penalization (SIMP), as well as an explicit level set method.
