STRUCTURAL & MULTIDISCIPLINARY OPTIMIZATION GROUP

Surrogate-based design optimization

Design optimization requires a large number of expensive simulations/experiments. The growth of computational capabilities has allowed the consideration of phenomena of constantly increasing level of complexity. The gain in realism makes the use of numerical simulators extremely challenging, since the complexity of the simulators makes their evaluation computationally expensive, preventing systematic call during optimization. To reduce the computational cost, cheap-to-evaluate surrogate models, also known as meta-models, are often used in place of the actual simulation models. It consists of replacing the expensive model by a simpler mathematical model (or surrogate) fitted to a few chosen simulations at a set of points called a design of experiments (DOE). Twenty years ago, polynomial response surfaces were almost exclusively used as surrogates for engineering design. With increased computational power, more sophisticated surrogates gained popularity: neural networks, kriging models, support vector regression, and even the simultaneous use of multiple surrogates. Figure 1 illustrates the capabilities of surrogate models through a kriging model of a commonly used benchmark function.

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(a) Actual response.	(b) Kriging model with 12 observations

Figure 1: Surrogate modeling in action.

A surrogate-based design optimization (SBDO) cycle consists of choosing points in design space (design of experiments), conducting simulations at these points and fitting surrogates to expensive responses. If the fitted surrogate satisfies measures of accuracy, we use it to conduct design optimization and then verify the optimum we obtain by exact simulation. Then, if it appears that further improvements in the design can be made by improving the surrogate, we zoom on regions of interest and conduct another cycle. This process is illustrated by Figure 2.

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Figure 2: Surrogate based design optimization.

In surrogate-based design optimization, the research of the Structural and Multidisciplinary Optimization Group involves:

· Uncertainty quantification: besides prediction, surrogates also provide uncertainty estimates. These estimates are used to select points to be sampled in the next optimization cycle and also to stop the optimization task. Our research focus on (i) using the information given by multiple surrogates to improve or provide uncertainty estimates; and (ii) using the uncertainty estimates to improve the robustness of the optimization results.

· Uncertainty minimization: an appropriate choice of the design of experiments allows to minimize the uncertainty (and the error) of the surrogate. Our research focus on (i) adapted design of experiments for constrained optimization (when a surrogate is used to approximate a constraint function), and (ii) efficient allocation of resources for reliability based optimization.

· Assessing the value of another cycle in surrogate-based optimization. Our research focus on providing accurate estimates of the probability of achieving a target level of improvement in the next cycle.

· Cross-validation and bootstrap: with limited data and computational resources, the available data must also be used to access the quality of the information given by the fitted surrogate. Our research focus on (i) cross-validation and bootstrap for design of conservative surrogates (metamodels that safely predicts the actual response); (ii) cross-validation for ranking the quality of prediction and correlation of uncertainty estimates.