Dimension Reduction for Aerodynamic Design Optimization

The search for an optimal design in a high-dimensional design space of a multivariate problem requires a sample size proportional or even exponential to the number of variables of the problem. This ‘curse of dimensionality’ places a computational burden on the cost of optimization, especially when the problem uses expensive high fidelity simulations and may force one to try to reduce the dimensions of a problem. Traditional variable screening techniques reduce the dimensionality of the problem by removing variables that seem irrelevant to the design problem. This practice fails when all the variables are equally relevant in the problem or when some variables are relevant only in some parts of the design space. The present work describes a dimension reduction method called generative topographic mapping based on non-linear latent models which transform a high-dimensional data set into a low-dimensional latent space, without removing any variables. It is first illustrated on a two dimensional Branin function and then applied to a thirty-dimensional airfoil problem. The method is then compared with a global optimizer (a genetic algorithm), other dimension reduction methods (principle component analysis and Gaussian process latent variable models) and with Kriging surrogate models. The method improves when the initial sample used for dimension reduction is filtered to contain only good designs.