  The presence of uncertainty in material properties and geometry of a
structure is ubiquitous. The design of robust engineering structures,
therefore, needs to incorporate uncertainty in the optimization process.
Stochastic gradient descent (SGD) method can alleviate the cost of optimization
under uncertainty, which includes statistical moments of quantities of interest
in the objective and constraints. However, the design may change considerably
during the initial iterations of the optimization process which impedes the
convergence of the traditional SGD method and its variants. In this paper, we
present two SGD based algorithms, where the computational cost is reduced by
employing a low-fidelity model in the optimization process. In the first
algorithm, most of the stochastic gradient calculations are performed on the
low-fidelity model and only a handful of gradients from the high-fidelity model
are used per iteration, resulting in an improved convergence. In the second
algorithm, we use gradients from low-fidelity models to be used as control
variate, a variance reduction technique, to reduce the variance in the search
direction. These two bi-fidelity algorithms are illustrated first with a
conceptual example. Then, the convergence of the proposed bi-fidelity
algorithms is studied with two numerical examples of shape and topology
optimization and compared to popular variants of the SGD method that do not use
low-fidelity models. The results show that the proposed use of a bi-fidelity
approach for the SGD method can improve the convergence. Two analytical proofs
are also provided that show the linear convergence of these two algorithms
under appropriate assumptions.
