Bi-fidelity stochastic gradient descent for structural optimization under uncertainty
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ORIGINAL PAPER
Bi-fidelity stochastic gradient descent for structural optimization under uncertainty Subhayan De1 · Kurt Maute1 · Alireza Doostan1 Received: 27 November 2019 / Accepted: 9 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 is used per iteration, resulting in an improved convergence. In the second algorithm, we use gradients from the 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 linear convergence of these two algorithms under appropriate assumptions. Keywords Bi-fidelity method · Optimization under uncertainty · Stochastic gradient descent · Control variate · Stochastic average gradient (SAG) · Stochastic variance reduced gradient (SVRG)
1 Introduction In simulation-based engineering, models, often in the form of discretized (partial) differential equations, are used for purposes such as analysis, design space exploration, uncertainty quantification, and design optimization. In the context of structural optimization, such as shape and topology optimization, the models need to be simulated many times throughout the optimization process [96]. Structures are often subjected to uncertainties in the material properties, geometry, and
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Alireza Doostan [email protected] Subhayan De [email protected] Kurt Maute [email protected]
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Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO 80309, USA
external loads [17,44,65]. Hence, for robust design of these structures, such uncertainties must be accounted for in the optimization process. The most commonly used method to compute the s
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