From Uncertainty Quantification to Shape Optimization: Cross-Fertilization of Methods for Dimensionality Reduction

In shape optimization of complex industrial products (such as aerial vehicles or ship hulls), there exists an inherent similarity between global optimization (GO) and uncertainty quantification (UQ): they rely on an extensive exploration of the design and operational spaces, respectively; often, they need local refinements to ensure accurate identification of optimal solutions or probability density regions (such as distribution tails), respectively; they both are dramatically affected by the curse of dimensionality as GO and UQ algorithms' complexity and especially computational cost rapidly increase with the problem dimension. Therefore, there exists a natural ground for cross-fertilization of UQ and GO methods specifically aimed at (or using) dimensionality reduction. These enable the efficient exploration of large design spaces in shape optimization, which, in turn, enable global multidisciplinary optimization under uncertainty. The paper reviews and discusses recent techniques for design-space dimensionality reduction in shape optimization, based on the Karhunen-Loeve expansion (equivalent to proper orthogonal decomposition and, at the discrete level, principal component analysis), spanning from geometry-based approaches to physics-informed formulations. An example is shown and discussed for the hydrodynamic optimization of a ship hull.